chapter 2 resource masters - math class · cases, you have used inductive reasoning. however, you...

Chapter 2Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

22

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

biconditional

conjecture

kuhn·JEK·chur

conjunction

contrapositive

converse

counterexample

deductive reasoning

disjunction

if-then statement

(continued on the next page)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

inductive reasoning

inverse

negation

paragraph proof

postulate

statement

theorem

truth table

truth value

two-column proof

Reading to Learn MathematicsVocabulary Builder (continued)

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Learning to Read MathematicsProof Builder

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© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 2. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 2.1Midpoint Theorem

Theorem 2.2

Theorem 2.3Supplement Theorem

Theorem 2.4Complement Theorem

Theorem 2.5

Theorem 2.6

Theorem 2.7

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 2.8Vertical Angle Theorem

Theorem 2.9

Theorem 2.10

Theorem 2.11

Theorem 2.12

Theorem 2.13

Postulate 2.9Segment Addition Postulate

Learning to Read MathematicsProof Builder (continued)

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Study Guide and InterventionInductive Reasoning and Conjecture

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© Glencoe/McGraw-Hill 57 Glencoe Geometry

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Make Conjectures A conjecture is a guess based on analyzing information orobserving a pattern. Making a conjecture after looking at several situations is calledinductive reasoning.

Make a conjecture aboutthe next number in the sequence 1, 3, 9,27, 81.Analyze the numbers:Notice that each number is a power of 3.

1 3 9 27 8130 31 32 33 34

Conjecture: The next number will be 35 or 243.

Make a conjecture about the number of smallsquares in the next figure.Observe a pattern: The sides of the squareshave measures 1, 2, and 3 units.Conjecture: For the next figure, the side ofthe square will be 4 units, so the figurewill have 16 small squares.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Describe the pattern. Then make a conjecture about the next number in thesequence.

1. !5, 10, !20, 40

2. 1, 10, 100, 1000

3. 1, "65", "

75", "

85"

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture.

4. A(!1, !1), B(2, 2), C(4, 4) 5. !1 and !2 form a right angle.

6. !ABC and !DBE are vertical angles. 7. !E and !F are right angles.

FERT

QP

DC

EBA

T W

12

P

R

A(–1, –1)

B(2, 2)

C(4, 4)


Find Counterexamples A conjecture is false if there is even one situation in which theconjecture is not true. The false example is called a counterexample.

Determine whether the conjecture is true or false.If it is false, give a counterexample.Given: A!B! " B!C!Conjecture: B is the midpoint of A!C!.Is it possible to draw a diagram with A!B! " B!C! such that B is not the midpoint? This diagram is a counterexample because point B is not on A!C!. The conjecture is false.

Determine whether each conjecture is true or false. Give a counterexample forany false conjecture.

1. Given: Points A, B, and C are collinear. 2. Given: !R and !S are supplementary.Conjecture: AB # BC $ AC !R and !T are supplementary.

Conjecture: !T and !S are congruent.

3. Given: !ABC and !DEF are 4. Given: D!E! ⊥ E!F!supplementary. Conjecture: !DEF is a right angle.Conjecture: !ABC and !DEF form a linear pair.

EB FA

DC

AC

B

A

C

3 cm3 cm

B

Study Guide and Intervention (continued)

Inductive Reasoning and Conjecture

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ExampleExample

ExercisesExercises

Skills PracticeInductive Reasoning and Conjecture

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Make a conjecture about the next item in each sequence.

1.

2. !4, !1, 2, 5, 8 3. 6, "121", 5, "

92", 4 4. !2, 4, !8, 16, !32


5. Points A, B, and C are collinear, 6. Point P is the midpoint of N!Q!.and D is between B and C.

7. !1, !2, !3, and !4 form four 8. !3 " !4linear pairs.


9. Given: !ABC and !CBD form a linear pair.Conjecture: !ABC " !CBD

10. Given: A!B!, B!C!, and A!C! are congruent.Conjecture: A, B, and C are collinear.

11. Given: AB # BC $ ACConjecture: AB $ BC

12. Given: !1 is complementary to !2, and !1 is complementary to !3.Conjecture: !2 " !3

A CB

431 243

N QPA C D B


Make a conjecture about the next item in each sequence.

1.

2. 5, !10, 15, !20 3. !2, 1, !"12", "

14", !"

18" 4. 12, 6, 3, 1.5, 0.75


5. !ABC is a right angle. 6. Point S is between R and T.

7. P, Q, R, and S are noncollinear 8. ABCD is a parallelogram.and P!Q! " Q!R! " R!S! " S!P!.


9. Given: S, T, and U are collinear and ST $ TU.Conjecture: T is the midpoint of S!U!.

10. Given: !1 and !2 are adjacent angles.Conjecture: !1 and !2 form a linear pair.

11. Given: G!H! and J!K! form a right angle and intersect at P.Conjecture: G!H! ⊥ J!K!

12. ALLERGIES Each spring, Rachel starts sneezing when the pear trees on her street blossom.She reasons that she is allergic to pear trees. Find a counterexample to Rachel’s conjecture.

D

A

C

B

S

P

R

Q

TSRA

CB

Practice Inductive Reasoning and Conjecture

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Reading to Learn MathematicsInductive Reasoning and Conjecture

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Pre-Activity How can inductive reasoning help predict weather conditions?

Read the introduction to Lesson 2-1 at the top of page 62 in your textbook.

• What kind of weather patterns do you think meteorologists look at tohelp predict the weather?

• What is a factor that might contribute to long-term changes in theweather?

Reading the Lesson1. Explain in your own words the relationship between a conjecture, a counterexample, and

inductive reasoning.

2. Make a conjecture about the next item in each sequence.

a. 5, 9, 13, 17 b. 1, "13", "

19", "2

17"

c. 0, 1, 3, 6, 10 d. 8, 3, !2, !7e. 1, 8, 27, 64 f. 1, !2, 4, !8g. h.

3. State whether each conjecture is true or false. If the conjecture is false, give acounterexample.a. The sum of two odd integers is even.

b. The product of an odd integer and an even integer is odd.

c. The opposite of an integer is a negative integer.

d. The perfect squares (squares of whole numbers) alternate between odd and even.

Helping You Remember4. Write a short sentence that can help you remember why it only takes one counterexample

to prove that a conjecture is false.


CounterexamplesWhen you make a conclusion after examining several specificcases, you have used inductive reasoning. However, you must becautious when using this form of reasoning. By finding only onecounterexample, you disprove the conclusion.

Is the statement !1x! " 1 true when you replace x with

1, 2, and 3? Is the statement true for all reals? If possible, find a counterexample.

"11

" $ 1, "12

" % 1, and "13

" % 1. But when x $ "12

", then "1x

" $ 2. This counterexample

shows that the statement is not always true.

Answer each question.

1. The coldest day of the year in Chicago 2. Suppose John misses the school busoccurred in January for five straight four Tuesdays in a row. Can youyears. Is it safe to conclude that the safely conclude that John misses the coldest day in Chicago is always in school bus every Tuesday?January?

3. Is the equation #k2! $ k true when you 4. Is the statement 2x $ x # x true whenreplace k with 1, 2, and 3? Is the you replace x with "

12", 4, and 0.7? Is the

equation true for all integers? If statement true for all real numbers? possible, find a counterexample. If possible, find a counterexample.

5. Suppose you draw four points A, B, C, 6. Suppose you draw a circle, mark three and D and then draw A!B!, B!C!, C!D!, and points on it, and connect them. Will theD!A!. Does this procedure give a angles of the triangle be acute? Explainquadrilateral always or only sometimes? your answers with figures.Explain your answers with figures.

Enrichment

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ExampleExample

Study Guide and InterventionLogic

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Determine Truth Values A statement is any sentence that is either true or false. Thetruth or falsity of a statement is its truth value. A statement can be represented by using aletter. For example,

Statement p: Chicago is a city in Illinois. The truth value of statement p is true.

Several statements can be joined in a compound statement.

Statement p and statement q joined Statement p and statement q joined Negation: not p is the negation ofby the word and is a conjunction. by the word or is a disjunction. the statement p.

Symbols: p $ q (Read: p and q) Symbols: p % q (Read: p or q) Symbols: &p (Read: not p)

The conjunction p $ q is true only The disjunction p % q is true if p is The statements p and &p have when both p and q are true. true, if q is true, or if both are true. opposite truth values.

Write a compoundstatement for each conjunction. Thenfind its truth value.p: An elephant is a mammal.q: A square has four right angles.

a. p ! qJoin the statements with and: An elephantis a mammal and a square has four rightangles. Both parts of the statement aretrue so the compound statement is true.

b. "p ! q&p is the statement “An elephant is not amammal.” Join &p and q with the wordand: An elephant is not a mammal and asquare has four right angles. The firstpart of the compound statement, &p, isfalse. Therefore the compound statementis false.

Write a compoundstatement for each disjunction. Thenfind its truth value.p: A diameter of a circle is twice the radius.q: A rectangle has four equal sides.

a. p # qJoin the statements p and q with theword or: A diameter of a circle is twicethe radius or a rectangle has four equalsides. The first part of the compoundstatement, p, is true, so the compoundstatement is true.

b. "p # qJoin &p and q with the word or: Adiameter of a circle is not twice theradius or a rectangle has four equalsides. Neither part of the disjunction istrue, so the compound statement is false.


ExercisesExercises

Write a compound statement for each conjunction and disjunction.Then find its truth value.p: 10 # 8 $ 18 q: September has 30 days. r: A rectangle has four sides.

1. p and q .

2. p or r

3. q or r

4. q and &r


Truth Tables One way to organize the truth values of statements is in a truth table. The truth tables for negation, conjunction, and disjunction are shown at the right.

Disjunction

p q p # q

T T TT F TF T TF F F

Conjunction

p q p ! q

T T TT F FF T FF F F

Negation

p "p

T FF T


Logic

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Construct a truthtable for the compoundstatement q or r. Use thedisjunction table.

q r q or r

T T TT F TF T TF F F

Construct a truth table for thecompound statement p and (q or r).Use the disjunction table for (q or r). Then use theconjunction table for p and (q or r).

p q r q or r p and (q or r )

T T T T TT T F T TT F T T TT F F F FF T T T FF T F T FF F T T FF F F F F


ExercisesExercises

Contruct a truth table for each compound statement.

1. p or r 2. &p % q 3. q $ &r

4. &p $ & r 5. ( p and r) or q

Skills PracticeLogic

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Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value.p: #3 # 2 $ #5q: Vertical angles are congruent.r: 2 % 8 & 10s: The sum of the measures of complementary angles is 90°.

1. p and q

2. p $ r

3. p or s

4. r % s

5. p $ &q

6. q % &r

Copy and complete each truth table.

7. 8.

Construct a truth table for each compound statement.

9. &q $ r 10. &p % &r

p q "q p # "q

T T F

T F T

F T F

F F T

p q "p "p ! q "("p ! q)

T T

T F

F T

F F


Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value.p: 60 seconds $ 1 minuteq: Congruent supplementary angles each have a measure of 90.r : #12 % 11 ' #1

1. p $ q

2. q % r

3. &p % q

4. &p $ &r

Copy and complete each truth table.

5. 6.

Construct a truth table for each compound statement.

7. q % (p $ &q) 8. &q $ (&p % q)

SCHOOL For Exercises 9 and 10, use the following information.The Venn diagram shows the number of students in the band who work after school or on the weekends.

9. How many students work after school and on weekends?

10. How many students work after school or on weekends?

WorkWeekends

17

WorkAfter

School5

3

p q "p "p # q p ! ("p # q)

T T

T F

F T

F F

p q "p "q "p # "q

T T

T F

F T

F F

Practice Logic

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Reading to Learn MathematicsLogic

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Pre-Activity How does logic apply to school?


How can you use logic to help you answer a multiple-choice question on astandardized test if you are not sure of the correct answer?

Reading the Lesson1. Supply one or two words to complete each sentence.

a. Two or more statements can be joined to form a statement.b. A statement that is formed by joining two statements with the word or is called a

.c. The truth or falsity of a statement is called its .d. A statement that is formed by joining two statements with the word and is called a

.e. A statement that has the opposite truth value and the opposite meaning from a given

statement is called the of the statement.

2. Use true or false to complete each sentence.a. If a statement is true, then its negation is .b. If a statement is false, then its negation is .c. If two statements are both true, then their conjunction is and

their disjunction is .d. If two statements are both false, then their conjunction is and

their disjunction is .e. If one statement is true and another is false, then their conjunction is

and their disjunction is .

3. Consider the following statements:p: Chicago is the capital of Illinois. q: Sacramento is the capital of California.Write each statement symbolically and then find its truth value.a. Sacramento is not the capital of California.b. Sacramento is the capital of California and Chicago is not the capital of Illinois.

Helping You Remember4. Prefixes can often help you to remember the meaning of words or to distinguish between

similar words. Use your dictionary to find the meanings of the prefixes con and dis andexplain how these meanings can help you remember the difference between aconjunction and a disjunction.


Letter PuzzlesAn alphametic is a computation puzzle using letters instead ofdigits. Each letter represents one of the digits 0–9, and twodifferent letters cannot represent the same digit. Some alphameticpuzzles have more than one answer.

Solve the alphametic puzzle at the right.

Since R # E $ E, the value of R must be 0. Notice that thethousands digit must be the same in the first addend and thesum. Since the value of I is 9 or less, O must be 4 or less. Use trial and error to find values that work.

F $ 8, O $ 3, U $ 1, R $ 0

N $ 4, E $ 7, I $ 6, and V $ 5.

Can you find other solutions to this puzzle?

Find a value for each letter in each alphametic.

1. 2.

H $ A $ L $ T $ W $ O $

F $ W $ O $ F $ U $ R $

E $

3. 4.

T $ H $ R $ S $ E $ N $

E $ O $ N $ D $ M $ O $

S $ V $ R $ Y $

5. Do research to find more alphametic puzzles, or create your own puzzles. Challengeanother student to solve them.

SEND# MORE"""""

MONEY

THREETHREE

# ONE""""

SEVEN

TWO# TWO""""

FOUR

HALF# HALF"""""

WHOLE

8310# 347"""

8657

FOUR# ONE"""

F I VE

Enrichment

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ExampleExample

Study Guide and InterventionConditional Statements

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If-then Statements An if-then statement is a statement such as “If you are readingthis page, then you are studying math.” A statement that can be written in if-then form iscalled a conditional statement. The phrase immediately following the word if is thehypothesis. The phrase immediately following the word then is the conclusion.

A conditional statement can be represented in symbols as p → q, which is read “p implies q”or “if p, then q.”

Identify the hypothesis and conclusion of the statement.

If !X " !R and !R " !S, then !X " !S.hypothesis conclusion

Identify the hypothesis and conclusion.Write the statement in if-then form.

You receive a free pizza with 12 coupons.

If you have 12 coupons, then you receive a free pizza.hypothesis conclusion

Example 1Example 1

Example 2Example 2

ExercisesExercises

Identify the hypothesis and conclusion of each statement.

1. If it is Saturday, then there is no school.

2. If x ! 8 $ 32, then x $ 40.

3. If a polygon has four right angles, then the polygon is a rectangle.

Write each statement in if-then form.

4. All apes love bananas.

5. The sum of the measures of complementary angles is 90.

6. Collinear points lie on the same line.

Determine the truth value of the following statement for each set of conditions.If it does not rain this Saturday, we will have a picnic.

7. It rains this Saturday, and we have a picnic.

8. It rains this Saturday, and we don’t have a picnic.

9. It doesn’t rain this Saturday, and we have a picnic.

10. It doesn’t rain this Saturday, and we don’t have a picnic.


Converse, Inverse, and Contrapositive If you change the hypothesis or conclusionof a conditional statement, you form a related conditional. This chart shows the threerelated conditionals, converse, inverse, and contrapositive, and how they are related to aconditional statement.

Symbols Formed by Example

Conditional p → q using the given hypothesis and conclusion If two angles are vertical angles, then they are congruent.

Converse q → p exchanging the hypothesis and conclusion If two angles are congruent, then they are vertical angles.

Inverse &p → &q replacing the hypothesis with its negation If two angles are not vertical angles,and replacing the conclusion with its negation then they are not congruent.

Contrapositive &q → &p negating the hypothesis, negating the If two angles are not congruent,conclusion, and switching them then they are not vertical angles.

Just as a conditional statement can be true or false, the related conditionals also can be trueor false. A conditional statement always has the same truth value as its contrapositive, andthe converse and inverse always have the same truth value.

Write the converse, inverse, and contrapositive of each conditional statement. Tellwhich statements are true and which statements are false.

1. If you live in San Diego, then you live in California.

2. If a polygon is a rectangle, then it is a square.

3. If two angles are complementary, then the sum of their measures is 90.


Conditional Statements

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ExercisesExercises

Skills PracticeConditional Statements

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1. If you purchase a computer and do not like it, then you can return it within 30 days.

2. If x # 8 $ 4, then x $ !4.

3. If the drama class raises $2000, then they will go on tour.


4. A polygon with four sides is a quadrilateral.

5. “Those who stand for nothing fall for anything.” (Alexander Hamilton)

6. An acute angle has a measure less than 90.

Determine the truth value of the following statement for each set of conditions.If you finish your homework by 5 P.M., then you go out to dinner.

7. You finish your homework by 5 P.M. and you go out to dinner.

8. You finish your homework by 4 P.M. and you go out to dinner.

9. You finish your homework by 5 P.M. and you do not go out to dinner.

10. Write the converse, inverse, and contrapositive of the conditional statement. Determinewhether each statement is true or false. If a statement is false, find a counterexample.If 89 is divisible by 2, then 89 is an even number.



1. If 3x # 4 $ !5, then x $ !3.

2. If you take a class in television broadcasting, then you will film a sporting event.


3. “Those who do not remember the past are condemned to repeat it.” (George Santayana)

4. Adjacent angles share a common vertex and a common side.

Determine the truth value of the following statement for each set of conditions.If DVD players are on sale for less than $100, then you buy one.

5. DVD players are on sale for $95 and you buy one.

6. DVD players are on sale for $100 and you do not buy one.

7. DVD players are not on sale for under $100 and you do not buy one.

8. Write the converse, inverse, and contrapositive of the conditional statement. Determinewhether each statement is true or false. If a statement is false, find a counterexample.If (!8)2 " 0, then !8 " 0.

SUMMER CAMP For Exercises 9 and 10, use the following information.Older campers who attend Woodland Falls Camp are expected to work. Campers who arejuniors wait on tables.

9. Write a conditional statement in if-then form.

10. Write the converse of your conditional statement.

Practice Conditional Statements

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Reading to Learn MathematicsConditional Statements

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Pre-Activity How are conditional statements used in advertisements?


Does the second advertising statement in the introduction mean that youwill not get a free phone if you sign a contract for only six months ofservice? Explain your answer.

Reading the Lesson1. Identify the hypothesis and conclusion of each statement.

a. If you are a registered voter, then you are at least 18 years old.

b. If two integers are even, their product is even.

2. Complete each sentence.a. The statement that is formed by replacing both the hypothesis and the conclusion of a

conditional with their negations is the .b. The statement that is formed by exchanging the hypothesis and conclusion of a

conditional is the .

3. Consider the following statement:You live in North America if you live in the United States.a. Write this conditional statement in if-then form and give its truth value. If the

statement is false, give a counterexample.

b. Write the inverse of the given conditional statement in if-then form and give its truthvalue. If the statement is false, give a counterexample.

c. Write the contrapositive of the given conditional statement in if-then form and giveits truth value. If the statement is false, give a counterexample.

d. Write the converse of the given conditional statement in if-then form and give itstruth value. If the statement is false, give a counterexample.

Helping You Remember4. When working with a conditional statement and its three related conditionals, what is

an easy way to remember which statements are logically equivalent to each other?


Venn DiagramsA type of drawing called a Venn diagram can be useful in explaining conditionalstatements. A Venn diagram uses circles to represent sets of objects.

Consider the statement “All rabbits have long ears.” To make a Venn diagram for thisstatement, a large circle is drawn to represent all animals with long ears. Then asmaller circle is drawn inside the first to represent all rabbits. The Venn diagramshows that every rabbit is included in the group of long-eared animals.

The set of rabbits is called a subset of the set of long-eared animals.

The Venn diagram can also explain how to write thestatement, “All rabbits have long ears,” in if-then form. Everyrabbit is in the group of long-eared animals, so if an animal isa rabbit, then it has long ears.

For each statement, draw a Venn diagram. Then write the sentence in if-then form.

1. Every dog has long hair. 2. All rational numbers are real.

3. People who live in Iowa like corn. 4. Staff members are allowed in the faculty lounge.

people in thecafeteria

staffmembers

people wholike corn

people wholive in Iowa

real numbers

rationalnumbers

animals withlong hair

dogs

animals withlong ears

rabbits

Enrichment

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Study Guide and InterventionDeductive Reasoning

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Law of Detachment Deductive reasoning is the process of using facts, rules,definitions, or properties to reach conclusions. One form of deductive reasoning that drawsconclusions from a true conditional p → q and a true statement p is called the Law ofDetachment.

Law of Detachment If p → q is true and p is true, then q is true.

Symbols [(p → q)] $ p] → q

The statement If two angles are supplementary to the same angle,then they are congruent is a true conditional. Determine whether each conclusionis valid based on the given information. Explain your reasoning.

a. Given: !A and !C are supplementary to !B.Conclusion: !A is congruent to !C.

The statement !A and !C are supplementary to !B is the hypothesis of the conditional. Therefore, by the Lawof Detachment, the conclusion is true.

b. Given: !A is congruent to !C.Conclusion: !A and !C are supplementary to !B.The statement !A is congruent to !C is not the hypothesis of the conditional, so the Law of Detachment cannot be used.The conclusion is not valid.

Determine whether each conclusion is valid based on the true conditional given.If not, write invalid. Explain your reasoning.If two angles are complementary to the same angle, then the angles are congruent.

1. Given: !A and !C are complementary to !B.Conclusion: !A is congruent to !C.

2. Given: !A " !CConclusion: !A and !C are complements of !B.

3. Given: !E and !F are complementary to !G.Conclusion: !E and !F are vertical angles.

A D

EF

H

J

C

GB

ExampleExample

ExercisesExercises


Law of Syllogism Another way to make a valid conclusion is to use the Law ofSyllogism. It is similar to the Transitive Property.

Law of Syllogism If p → q is true and q → r is true, then p → r is also true.

Symbols [(p → q)] $ (q → r )] → (p → r )

The two conditional statements below are true. Use the Law ofSyllogism to find a valid conclusion. State the conclusion.(1) If a number is a whole number, then the number is an integer.(2) If a number is an integer, then it is a rational number.

p: A number is a whole number.q: A number is an integer.r: A number is a rational number.

The two conditional statements are p → q and q → r. Using the Law of Syllogism, a validconclusion is p → r. A statement of p → r is “if a number is a whole number, then it is arational number.”

Determine whether you can use the Law of Syllogism to reach a valid conclusionfrom each set of statements.

1. If a dog eats Superdog Dog Food, he will be happy.Rover is happy.

2. If an angle is supplementary to an obtuse angle, then it is acute.If an angle is acute, then its measure is less than 90.

3. If the measure of !A is less than 90, then !A is acute.If !A is acute, then !A " !B.

4. If an angle is a right angle, then the measure of the angle is 90.If two lines are perpendicular, then they form a right angle.

5. If you study for the test, then you will receive a high grade.Your grade on the test is high.


Deductive Reasoning

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Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning.If the sum of the measures of two angles is 180, then the angles are supplementary.

1. Given: m!A # m!B is 180.Conclusion: !A and !B are supplementary.

2. Given: m!ABC is 95 and m!DEF is 90.Conclusion: !ABC and !DEF are supplementary.

3. Given: !1 and !2 are a linear pair.Conclusion: !1 and !2 are supplementary.

Use the Law of Syllogism to determine whether a valid conclusion can be reachedfrom each set of statements. If a valid conclusion is possible, write it.

4. If two angles are complementary, then the sum of their measures is 90.If the sum of the measures of two angles is 90, then both of the angles are acute.

5. If the heat wave continues, then air conditioning will be used more frequently.If air conditioning is used more frequently, then energy costs will be higher.

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid.

6. (1) If it is Tuesday, then Marla tutors chemistry.(2) If Marla tutors chemistry, then she arrives home at 4 P.M.(3) If Marla arrives at home at 4 P.M., then it is Tuesday.

7. (1) If a marine animal is a starfish, then it lives in the intertidal zone of the ocean.(2) The intertidal zone is the least stable of the ocean zones.(3) If a marine animal is a starfish, then it lives in the least stable of the ocean zones.


Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning.If a point is the midpoint of a segment, then it divides the segment into twocongruent segments.

1. Given: R is the midpoint of Q!S!.Conclusion: Q!R! " R!S!

2. Given: A!B! " B!C!Conclusion: B divides A!C! into two congruent segments.

Use the Law of Syllogism to determine whether a valid conclusion can be reachedfrom each set of statements. If a valid conclusion is possible, write it.

3. If two angles form a linear pair, then the two angles are supplementary.If two angles are supplementary, then the sum of their measures is 180.

4. If a hurricane is Category 5, then winds are greater than 155 miles per hour.If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down.

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid.

5. (1) If a whole number is even, then its square is divisible by 4.(2) The number I am thinking of is an even whole number.(3) The square of the number I am thinking of is divisible by 4.

6. (1) If the football team wins its homecoming game, then Conrad will attend the schooldance the following Friday.(2) Conrad attends the school dance on Friday.(3) The football team won the homecoming game.

7. BIOLOGY If an organism is a parasite, then it survives by living on or in a hostorganism. If a parasite lives in or on a host organism, then it harms its host. Whatconclusion can you draw if a virus is a parasite?

Practice Deductive Reasoning

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Reading to Learn MathematicsDeductive Reasoning

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Pre-Activity How does deductive reasoning apply to health?


Suppose a doctor wants to use the dose chart in your textbook to prescribean antibiotic, but the only scale in her office gives weights in pounds. Howcan she use the fact that 1 kilogram is about 2.2 pounds to determine thecorrect dose for a patient?

.

Reading the LessonIf s, t, and u are three statements, match each description from the list on the leftwith a symbolic statement from the list on the right.1. negation of t a. s % u2. conjunction of s and u b. [(s → t) $ s] → t3. converse of s → t c. &s → &u4. disjunction of s and u d. &u → &s5. Law of Detachment e. &t6. contrapositive of s → t f. [(u → t) $ (t → s)] → (u → s)7. inverse of s → u g. s $ u8. contrapositive of s → u h. t → s9. Law of Syllogism i. t

10. negation of &t j. &t → &s

11. Determine whether statement (3) follows from statements (1) and (2) by the Law ofDetachment or the Law of Syllogism. If it does, state which law was used. If it does not,write invalid.a. (1) Every square is a parallelogram.

(2) Every parallelogram is a polygon.(3) Every square is a polygon.

b. (1)If two lines that lie in the same plane do not intersect, they are parallel.(2) Lines ! and m lie in plane U and do not intersect.(3) Lines ! and m are parallel.

c. (1) Perpendicular lines intersect to form four right angles.(2) !A, !B, !C, and !D are four right angles.(3) !A, !B, !C, and !D are formed by intersecting perpendicular lines.

Helping You Remember12. A good way to remember something is to explain it to someone else. Suppose that a

classmate is having trouble remembering what the Law of Detachment means?


Valid and Faulty ArgumentsConsider the statements at the right.What conclusions can you make?

From statements 1 and 3, it is correct to conclude that Bootspurrs if it is happy. However, it is faulty to conclude from only statements 2 and 3 that Boots is happy. The if-then form of statement 3 is If a cat is happy, then it purrs.

Advertisers often use faulty logic in subtle ways to help selltheir products. By studying the arguments, you can decide whether the argument is valid or faulty.

Decide if each argument is valid or faulty.

1. (1) If you buy Tuff Cote luggage, it 2. (1) If you buy Tuff Cote luggage, itwill survive airline travel. will survive airline travel.

(2) Justin buys Tuff Cote luggage. (2) Justin’s luggage survived airline travel.Conclusion: Justin’s luggage will Conclusion: Justin has Tuff Cotesurvive airline travel. luggage.

3. (1) If you use Clear Line long 4. (1) If you read the book Beautiful distance service, you will have Braids, you will be able to makeclear reception. beautiful braids easily.

(2) Anna has clear long distance (2) Nancy read the book Beautifulreception. Braids.

Conclusion: Anna uses Clear Line Conclusion: Nancy can make long distance service. beautiful braids easily.

5. (1) If you buy a word processor, you 6. (1) Great swimmers wear AquaLinewill be able to write letters faster. swimwear.

(2) Tania bought a word processor. (2) Gina wears AquaLine swimwear.Conclusion: Tania will be able to Conclusion: Gina is a great swimmer.write letters faster.

7. Write an example of faulty logic thatyou have seen in an advertisement.

(1) Boots is a cat.(2) Boots is purring.(3) A cat purrs if it is happy.

Enrichment

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Study Guide and InterventionPostulates and Paragraph Proofs

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Points, Lines, and Planes In geometry, a postulate is a statement that is accepted astrue. Postulates describe fundamental relationships in geometry.

Postulate: Through any two points, there is exactly one line.Postulate: Through any three points not on the same line, there is exactly one plane.Postulate: A line contains at least two points.Postulate: A plane contains at least three points not on the same line.Postulate: If two points lie in a plane, then the line containing those points lies in the plane.Postulate: If two lines intersect, then their intersection is exactly one point.Postulate: If two planes intersect, then their intersection is a line.

Determine whether each statement is always,sometimes, or never true.a. There is exactly one plane that contains points A, B, and C.

Sometimes; if A, B, and C are collinear, they are contained in many planes. If they arenoncollinear, then they are contained in exactly one plane.

b. Points E and F are contained in exactly one line.Always; the first postulate states that there is exactly one line through any two points.

c. Two lines intersect in two distinct points M and N.Never; the intersection of two lines is one point.

Use postulates to determine whether each statement is always, sometimes, ornever true.

1. A line contains exactly one point.

2. Noncollinear points R, S, and T are contained in exactly one plane.

3. Any two lines ! and m intersect.

4. If points G and H are contained in plane M, then G!H! is perpendicular to plane M.

5. Planes R and S intersect in point T.

6. If points A, B, and C are noncollinear, then segments A!B!, B!C!, and C!A! are contained inexactly one plane.

In the figure, A$C$ and D$E$ are in plane Q and A$C$ || D$E$.State the postulate that can be used to show each statement is true.

7. Exactly one plane contains points F, B, and E.

8. BE!"# lies in plane Q.

QB

C

A

DE

F

G

ExampleExample

ExercisesExercises


Paragraph Proofs A statement that can be proved true is called a theorem. You canuse undefined terms, definitions, postulates, and already-proved theorems to prove otherstatements true.

A logical argument that uses deductive reasoning to reach a valid conclusion is called aproof. In one type of proof, a paragraph proof, you write a paragraph to explain why astatement is true.

In "ABC, B$D$ is an angle bisector. Write a paragraph proof to show that !ABD % !CBD.By definition, an angle bisector divides an angle into two congruentangles. Since B!D! is an angle bisector, !ABC is divided into twocongruent angles. Thus, !ABD " !CBD.

1. Given that !A " !D and !D " !E, write a paragraph proof to show that !A " !E.

2. It is given that B!C! " E!F!, M is the midpoint of B!C!, and N is the midpoint of E!F!. Write a paragraph proof to show that BM $ EN.

3. Given that S is the midpoint of Q!P!, T is the midpoint of P!R!,and P is the midpoint of S!T!, write a paragraph proof to show that QS $ TR. Q

RTPS

B

CM

EN

F

B

AD C


Postulates and Paragraph Proofs

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Determine the number of line segments that can be drawn connecting each pairof points.

1. 2.

Determine whether the following statements are always, sometimes, or never true.Explain.

3. Three collinear points determine a plane.

4. Two points A and B determine a line.

5. A plane contains at least three lines.

In the figure, DG!"# and DP""# lie in plane J and H lies on DG!"#. State the postulate that can be used to show each statement is true.

6. G and P are collinear.

7. Points D, H, and P are coplanar.

8. PROOF In the figure at the right, point B is the midpoint of A!C! and point C is the midpoint of B!D!. Write a paragraph proof to prove thatAB $ CD.

DCBA

J

PD

H

G


Determine the number of line segments that can be drawn connecting each pairof points.

1. 2.

Determine whether the following statements are always, sometimes, or never true.Explain.

3. The intersection of two planes contains at least two points.

4. If three planes have a point in common, then they have a whole line in common.

In the figure, line m and TQ""# lie in plane A . State the postulate that can be used to show that each statement is true.

5. L, T, and line m lie in the same plane.

6. Line m and S!T! intersect at T.

7. In the figure, E is the midpoint of A!B! and C!D!, and AB $ CD. Write a paragraph proof to prove that A!E! " E!D!.

8. LOGIC Points A, B, and C are not collinear. Points B, C, and D are not collinear. PointsA, B, C, and D are not coplanar. Describe two planes that intersect in line BC.

BE

C

D

A

AmT

QL

S

Practice Postulates and Paragraph Proofs

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Reading to Learn MathematicsPostulates and Paragraph Proofs

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Pre-Activity How are postulates used by the founding fathers of the United States?


Postulates are often described as statements that are so basic and so clearlycorrect that people will be willing to accept them as true without asking forevidence or proof. Give a statement about numbers that you think mostpeople would accept as true without evidence.

Reading the Lesson1. Determine whether each of the following is a correct or incorrect statement of a

geometric postulate. If the statement is incorrect, replace the underlined words to makethe statement correct.a. A plane contains at least that do not lie on the same line.b. If intersect, then the intersection is a line.c. Through any not on the same line, there is exactly one plane.d. A line contains at least .e. If two lines , then their intersection is exactly one point.f. Through any two points, there is one line.

2. Determine whether each statement is always, sometimes, or never true. If the statementis not always true, explain why.a. If two planes intersect, their intersection is a line.b. The midpoint of a segment divides the segment into two congruent segments.c. There is exactly one plane that contains three collinear points.

d. If two lines intersect, their intersection is one point.

3. Use the walls, floor, and ceiling of your classroom to describe a model for each of thefollowing geometric situations.a. two planes that intersect in a line

b. two planes that do not intersect

c. three planes that intersect in a point

Helping You Remember4. A good way to remember a new mathematical term is to relate it to a word you already

know. Explain how the idea of a mathematical theorem is related to the idea of a scientifictheory.

at mostare parallel

one pointfour points

two planestwo points


Logic ProblemsThe following problems can be solved by eliminating possibilities.It may be helpful to use charts such as the one shown in the firstproblem. Mark an X in the chart to eliminate a possible answer.

Solve each problem.

Enrichment

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Nancy Olivia Mario Kenji

Peach

Orange

Banana

Apple

1. Nancy, Olivia, Mario, and Kenji each haveone piece of fruit in their school lunch.They have a peach, an orange, a banana,and an apple. Mario does not have apeach or a banana. Olivia and Mario justcame from class with the student who hasan apple. Kenji and Nancy are sittingnext to the student who has a banana.Nancy does not have a peach. Whichstudent has each piece of fruit?

3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza, andMrs. Riva have jobs of doctor, accountant,teacher, and office manager. Mr. Mirzalives near the doctor and the teacher.Mrs. Riva is not the doctor or the officemanager. Mrs. Hakoi is not theaccountant or the office manager. Mr.Guthrie went to lunch with the doctor.Mrs. Riva’s son is a high school studentand is only seven years younger thanhis algebra teacher. Which person haseach occupation?

2. Victor, Leon, Kasha, and Sheri each playone instrument. They play the viola,clarinet, trumpet, and flute. Sheri doesnot play the flute. Kasha lives near thestudent who plays flute and the onewho plays trumpet. Leon does not playa brass or wind instrument. Whichstudent plays each instrument?

4. Yvette, Lana, Boris, and Scott each havea dog. The breeds are collie, beagle,poodle, and terrier. Yvette and Boriswalked to the library with the studentwho has a collie. Boris does not have apoodle or terrier. Scott does not have acollie. Yvette is in math class with thestudent who has a terrier. Whichstudent has each breed of dog?

Study Guide and InterventionAlgebraic Proof

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Algebraic Proof The following properties of algebra can be used to justify the stepswhen solving an algebraic equation.

Property Statement

Reflexive For every number a, a $ a.

Symmetric For all numbers a and b, if a $ b then b $ a.

Transitive For all numbers a, b, and c, if a $ b and b $ c then a $ c.

Addition and Subtraction For all numbers a, b, and c, if a $ b then a # c $ b # c and a ! c $ b ! c.

Multiplication and Division For all numbers a, b, and c, if a $ b then a & c $ b & c, and if c ' 0 then "ac" $ "

bc".

Substitution For all numbers a and b, if a $ b then a may be replaced by b in any equation or expression.

Distributive For all numbers a, b, and c, a(b # c) $ ab # ac.

Solve 6x % 2(x # 1) $ 30.

Algebraic Steps Properties6x # 2(x ! 1) $ 30 Given

6x # 2x ! 2 $ 30 Distributive Property

8x ! 2 $ 30 Substitution

8x ! 2 # 2 $ 30 # 2 Addition Property

8x $ 32 Substitution

"88x" $ "

382" Division Property

x $ 4 Substitution

Complete each proof.

ExampleExample

ExercisesExercises

1. Given: "4x

2# 6" $ 9

Prove: x $ 3

Statements Reasons

a. "4x

2# 6" $ 9 a.

b. '"4x2# 6"( $ 2(9) b. Mult. Prop.

c. 4x # 6 $ 18 c.

d. 4x # 6 ! 6 $ 18 ! 6 d.

e. 4x $ e. Substitution

f. "44x" $ f. Div. Prop.

g. g. Substitution

2. Given: 4x # 8 $ x # 2Prove: x $ !2

Statements Reasons

a. 4x # 8 $ x # 2 a.

b. 4x # 8 ! x $x # 2 ! x b.

c. 3x # 8 $ 2 c. Substitution

d.d. Subtr. Prop.

e. e. Substitution

f. "33x" $ "

!36" f.

g. g. Substitution


Geometric Proof Geometry deals with numbers as measures, so geometric proofs useproperties of numbers. Here are some of the algebraic properties used in proofs.

Property Segments Angles

Reflexive AB $ AB m!A $ m!A

Symmetric If AB $ CD, then CD $ AB. If m!A $ m!B, then m!B $ m!A.

Transitive If AB $ CD and CD $ EF, then AB $ EF. If m!1$ m!2 and m!2 $ m!3, then m!1$ m!3.

Write a two-column proof.Given: m!1 $ m!2, m!2 $ m!3Prove: m!1 $ m!3Proof:

Statements Reasons

1. m!1 $ m!2 1. Given2. m!2 $ m!3 2. Given3. m!1 $ m!3 3. Transitive Property

State the property that justifies each statement.

1. If m!1 $ m!2, then m!2 $ m!1.

2. If m!1$ 90 and m!2 $ m!1, then m!2$ 90.

3. If AB $ RS and RS $ WY, then AB $ WY.

4. If AB $ CD, then "12"AB $ "

12"CD.

5. If m!1 # m!2 $ 110 and m!2 $ m!3, then m!1 # m!3 $ 110.

6. RS $ RS

7. If AB $ RS and TU $ WY, then AB # TU $ RS # WY.

8. If m!1 $ m!2 and m!2 $ m!3, then m!1 $ m!3.

9. A formula for the area of a triangle is A $ "

12"bh. Prove that bh is equal

to 2 times the area of the triangle.

DA

R

TS

CB

1 23


Algebraic Proof

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State the property that justifies each statement.

1. If 80 $ m!A, then m!A $ 80.

2. If RS $ TU and TU $ YP, then RS $ YP.

3. If 7x $ 28, then x $ 4.

4. If VR # TY $ EN # TY, then VR $ EN.

5. If m!1 $ 30 and m!1 $ m!2, then m!2 $ 30.

Complete the following proof.

6. Given: 8x ! 5 $ 2x # 1Prove: x $ 1Proof:

Statements Reasonsa. 8x ! 5 $ 2x # 1 a.

b. 8x ! 5 ! 2x $ 2x # 1 ! 2x b.

c. c. Substitution Property

d. d. Addition Property

e. 6x $ 6 e.

f. "66x" $ "

66" f.

g. g.

Write a two-column proof for the following.

7. If P!Q! " Q!S! and Q!S! " S!T!, then PQ $ ST. QP

TS


PROOF Write a two-column proof.

1. If m!ABC # m!CBD $ 90, m!ABC $ 3x ! 5,and m!CBD $ "

x #2

1", then x $ 27.

2. FINANCE The formula for simple interest is I $ prt, where I is interest, p is principal,r is rate, and t is time. Solve the formula for r and justify each step.

A

D C

B

Practice Algebraic Proof

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Reading to Learn MathematicsAlgebraic Proof

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Pre-Activity How is mathematical evidence similar to evidence in law?


What are some of the things that lawyers might use in presenting theirclosing arguments to a trial jury in addition to evidence gathered prior tothe trial and testimony heard during the trial?

Reading the Lesson1. Name the property illustrated by each statement.

a. If a $ 4.75 and 4.75 $ b, then a $ b.b. If x $ y, then x # 8 $ y # 8.c. 5(12 # 19) $ 5 & 12 # 5 & 19d. If x $ 5, then x may be replaced with 5 in any equation or expression.e. If x $ y, then 8x $ 8y.f. If x $ 23.45, then 23.45 $ x.g. If 5x $ 7, then x $ "

75".

h. If x $ 12, then x ! 3 $ 9.

2. Give the reason for each statement in the following two-column proof.Given: 5(n ! 3) $ 4(2n ! 7) ! 14Prove: n $ 9Statements Reasons

1. 5(n ! 3) $ 4(2n ! 7) ! 14 1.

2. 5n ! 15 $ 8n ! 28 ! 14 2.

3. 5n ! 15 $ 8n ! 42 3.

4. 5n ! 15 # 15 $ 8n ! 42 # 15 4.

5. 5n $ 8n ! 27 5.

6. 5n ! 8n $ 8n ! 27 ! 8n 6.

7. !3n $ !27 7.

8. "!!33n

" $ "!!237

" 8.

9. n $ 9 9.

Helping You Remember3. A good way to remember mathematical terms is to relate them to words you already know.

Give an everyday word that is related in meaning to the mathematical term reflexive andexplain how this word can help you to remember the Reflexive Property and to distinguishit from the Symmetric and Transitive Properties.


Symmetric, Reflexive, and Transitive PropertiesEquality has three important properties.

ReflexiveSymmetricTransitive

Other relations have some of the same properties. Consider therelation “is next to” for objects labeled X, Y, and Z. Which of theproperties listed above are true for this relation?

X is next to X. FalseIf X is next to Y, then Y is next to X. TrueIf X is next to Y and Y is next to Z, then X is next to Z. False

Only the symmetric property is true for the relation “is next to.”

For each relation, state which properties (symmetric, reflexive,transitive) are true.

1. is the same size as 2. is a family descendant of

3. is in the same room as 4. is the identical twin of

5. is warmer than 6. is on the same line as

7. is a sister of 8. is the same weight as

9. Find two other examples of relations,and tell which properties are true foreach relation.

a $ aIf a $ b, then b $ a.If a $ b and b $ c, then a $ c.

Enrichment

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Study Guide and InterventionProving Segment Relationships

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Segment Addition Two basic postulates for working with segments and lengths arethe Ruler Postulate, which establishes number lines, and the Segment Addition Postulate,which describes what it means for one point to be between two other points.

Ruler PostulateThe points on any line or line segment can be paired with real numbers so that, given any twopoints A and B on a line, A corresponds to zero and B corresponds to a positive real number.

Segment Addition Postulate

B is between A and C if and only if AB # BC $ AC.

Write a two-column proof.Given: Q is the midpoint of P!R!.

R is the midpoint of Q!S!.Prove: PR $ QS

Statements Reasons

1. Q is the midpoint of P!R!. 1. Given2. PQ $ QR 2. Definition of midpoint3. R is the midpoint of Q!S!. 3. Given4. QR $ RS 4. Definition of midpoint5. PQ # QR $ QR # RS 5. Addition Property6. PQ # QR $ PR, QR # RS $ QS 6. Segment Addition Postulate7. PR $ QS 7. Substitution


PQ

RS

ExampleExample

ExercisesExercises

1. Given: BC $ DEProve: AB # DE $ AC

Statements Reasonsa. BC $ DE a.

b. b. Seg. Add. Post.

c. AB # DE $ AC c.

A BC

DE

2. Given: Q is betweenP and R, R is between Q and S, PR $ QS.Prove: PQ $ RS

Statements Reasons

a.Q is between a. GivenP and R.

b.PQ # QR $ PR b.c. R is between c.

Q and S.d. d. Seg. Add. Post.e. PR $ QS e.f. PQ # QR $ f.

QR # RSg.PQ # QR ! QR $ g.

QR # RS ! QRh. h. Substitution

P QR S


Segment Congruence Three properties of algebra—the Reflexive, Symmetric, andTransitive Properties of Equality—have counterparts as properties of geometry. Theseproperties can be proved as a theorem. As with other theorems, the properties can then beused to prove relationships among segments.

Segment Congruence Theorem Congruence of segments is reflexive, symmetric, and transitive.

Reflexive Property A!B! " A!B!Symmetric Property If A!B! " C!D!, then C!D! " A!B!.

Transitive Property If A!B! " C!D! and C!D! " E!F!, then A!B! " E!F!.

Write a two-column proof.Given: A!B! " D!E!; B!C! " E!F!Prove: A!C! " D!F!Statements Reasons1. A!B! " D!E! 1. Given2. AB $ DE 2. Definition of congruence of segments3. B!C! " E!F! 3. Given4. BC $ EF 4. Definition of congruence of segments5. AB # BC $ DE # EF 5. Addition Property6. AB # BC $ AC, DE # EF $ DF 6. Segment Addition Postulate7. AC $ DF 7. Substitution8. A!C! " D!F! 8. Definition of congruence of segments

Justify each statement with a property of congruence.

1. If D!E! " G!H!, then G!H! " D!E!.

2. If A!B! " R!S! and R!S! " W!Y!, then A!B! " W!Y!.

3. R!S! " R!S! .

4. Complete the proof.Given: P!R! " Q!S!Prove: P!Q! " R!S!

Statements Reasons

a. P!R! " Q!S! a.b. PR $ QS b.c. PQ # QR $ PR c.d. d. Segment Addition Postulatee. PQ # QR $ QR # RS e.f. f. Subtraction Propertyg. g. Definition of congruence of segments

P QR S

DE FA

B C


Proving Segment Relationships

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Justify each statement with a property of equality, a property of congruence, or apostulate.

1. QA $ QA

2. If A!B! " B!C! and B!C! " C!E!, then A!B! " C!E!.

3. If Q is between P and R, then PR $ PQ # QR.

4. If AB # BC $ EF # FG and AB # BC $ AC, then EF # FG $ AC.


5. Given: S!U! " L!R!T!U! " L!N!

Prove: S!T! " N!R!Proof:

Statements Reasons

a. S!U! " L!R!, T!U! " L!N! a.b. b. Definition of " segmentsc. SU $ ST # TU c.

LR $ LN # NRd. ST # TU $ LN # NR d.e. ST # LN $ LN # NR e.f. ST # LN ! LN $ LN # NR ! LN f.g. g. Substitution Propertyh. S!T! " N!R! h.

6. Given: A!B! " C!D!Prove: C!D! " A!B!Proof:

Statements Reasons

a. a. Givenb. AB $ CD b.c. CD $ AB c.d. d. Definition of " segments

US T

RL N


Complete the following proof.

1. Given: A!B! " D!E!B is the midpoint of A!C!.E is the midpoint of D!F!.

Prove: B!C! " E!F!Proof:

Statements Reasons

a. a. Given

b. AB $ DE b.

c. c. Definition of Midpoint

d. AC $ AB # BC d.

DF $ DE # EF

e. AB # BC $ DE # EF e.

f. AB # BC $ AB # EF f.

g. AB # BC ! AB $ AB # EF ! AB g. Subtraction Property

h. BC $ EF h.

i. i.

2. TRAVEL Refer to the figure. DeAnne knows that the distance from Grayson to Apex is the same as the distancefrom Redding to Pine Bluff. Prove that the distance fromGrayson to Redding is equal to the distance from Apex to Pine Bluff.

Grayson Apex Redding Pine Bluff

G A R P

CA B

FD

E

Practice Proving Segment Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Reading to Learn MathematicsProving Segment Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Pre-Activity How can segment relationships be used for travel?


• What is the total distance that the plane will fly to get from San Diego toDallas?

• Before leaving home, a passenger used a road atlas to determine that thedistance between San Diego and Dallas is about 1350 miles. Why is theflying distance greater than that?

Reading the Lesson1. If E is between Y and S, which of the following statements are always true?

A. YS # ES $ YE B. YS ! ES $ YEC. YE ( ES D. YE & ES $ YSE. SE # EY $ SY F. E is the midpoint of Y!S!.

2. Give the reason for each statement in the following two-column proof.Given: C is the midpoint of B!D!.

D is the midpoint of C!E!.Prove: B!D! " C!E!Statements Reasons

1. C is the midpoint of B!D!. 1.

2. BC $ CD 2.

3. D is the midpoint of C!E!. 3.

4. CD $ DE 4.

5. BC $ DE 5.

6. BC # CD $ CD # DE 6.7. BC # CD $ BD 7.

CD # DE $ CE

8. BD $ CE 8.

9. B!D! " C!E! 9.

Helping You Remember3. One way to keep the names of related postulates straight in your mind is to associate

something in the name of the postulate with the content of the postulate. How can you usethis idea to distinguish between the Ruler Postulate and the Segment Addition Postulate?

A

B EDC


Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Geometry Crossword Puzzle

ACROSS

3. Points on the same line are!!!!!!!!.

4. A point on a line and allpoints of the line to one side of it.

9. An angle whose measure isgreater than 90.

10. Two endpoints and all pointsbetween them.

16. A flat figure with no thicknessthat extends indefinitely in alldirections.

17. Segments of equal length are!!!!!!!! segments.

18. Two noncollinear rays with acommon endpoint.

19. If m!A # m!B $ 180, then!A and !B are !!!!!!!! angles.

DOWN

1. The set of all points collinear to two pointsis a !!!!!!!!.

2. The point where the x- and y-axis meet.5. An angle whose measure is less than 90.6. If m!A # m!D $ 90, then !A and !D are

!!!!!!!! angles.7. Lines that meet at a 90° angle are !!!!!!!!.8. Two angles with a common side but no common

interior points are !!!!!!!!.10. An “angle” formed by opposite rays is a

!!!!!!!! angle.11. The middle point of a line segment.12. Points that lie in the same plane are !!!!!!!!.13. The four parts of a coordinate plane.14. Two nonadjacent angles formed by two

intersecting lines are !!!!!!!! angles.15. In angle ABC, point B is the !!!!!!!!.

1 2

3 4 5

6 7

9

14

18

15

17

1110

8

1312

16

19

Study Guide and InterventionProving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8


Less

on

2-8

Supplementary and Complementary Angles There are two basic postulates forworking with angles. The Protractor Postulate assigns numbers to angle measures, and theAngle Addition Postulate relates parts of an angle to the whole angle.

Protractor Given AB""# and a number r between 0 and 180, there is exactly one ray Postulate with endpoint A, extending on either side of AB""#, such that the measure

of the angle formed is r.

Angle Addition R is in the interior of !PQS if and only if Postulate m!PQR # m!RQS $ m!PQS.

The two postulates can be used to prove the following two theorems.

Supplement If two angles form a linear pair, then they are supplementary angles.Theorem If !1 and !2 form a linear pair, then m!1 # m!2 $ 180.

Complement If the noncommon sides of two adjacent angles form a right angle, Theorem then the angles are complementary angles.

If GF!"# ⊥ GH!"#, then m!3 # m!4 $ 90. HG 43

FJ

CB1 2

A

D

S

R

P

Q

If !1 and !2 form alinear pair and m!2 $ 115, find m!1.

m!1 # m!2 $ 180 Suppl. Theorem

m!1 # 115 $ 180 Substitution

m!1 $ 65 Subtraction Prop.

P

Q

NM12

If !1 and !2 form aright angle and m!2 $ 20, find m!1.

m!1 # m!2 $ 90 Compl. Theorem

m!1 # 20 $ 90 Substitution

m!1 $ 70 Subtraction Prop.

T

WR

S 12


ExercisesExercises

Find the measure of each numbered angle.

1. 2. 3.

m!7 $ 5x # 5, m!5 $ 5x, m!6 $ 4x # 6, m!11 $ 11x,m!8 $ x ! 5 m!7 $ 10x, m!12 $ 10x # 10

m!8 $ 12x ! 12

F CJ

HA 11

1213

WU

X YZ

V58

67R

TP

QS

87


Congruent and Right Angles Three properties of angles can be proved as theorems.

Congruence of angles is reflexive, symmetric, and transitive.

Angles supplementary to the same angle Angles complementary to the same angle or to or to congruent angles are congruent. congruent angles are congruent.

If !1 and !2 are supplementary to !3, If !4 and !5 are complementary to !6, then !1 " !2. then !4 " !5.

Write a two-column proof.Given: !ABC and !CBD are complementary.

!DBE and !CBD form a right angle.Prove: !ABC " !DBE

Statements Reasons1. !ABC and !CBD are complementary. 1. Given

!DBE and !CBD form a right angle.2. !DBE and !CBD are complementary. 2. Complement Theorem3. !ABC " !DBE 3. Angles complementary to the same ! are ".


B E

D

CA

ML

NK

S

TR

PQ

45

6A B

C

F

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1

2 3


Proving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8

ExampleExample

ExercisesExercises

1. Given: A!B! ⊥ B!C!;!1 and !3 are complementary.Prove: !2 " !3

Statements Reasons

a. A!B! ⊥ B!C! a.b. b. Definition of ⊥

c. m!1 # m!2 $ c.m!ABC

d. !1 and !2 form d.a rt !.

e. !1 and !2 are e.compl.

f. f. Given

g. !2 " !3 g.

B C

EA1 2

3D

2. Given: !1 and !2 form a linear pair.m!1 # m!3 $ 180Prove: !2 " !3

Statements Reasons

a. !1 and !2 form a. Givena linear pair.m!1 # m!3 $ 180

b. b. Suppl.Theorem

c. !1 is suppl. c.to !3.

d. d." suppl. to thesame ! are ".

BR

T

L

1 23

P

Skills PracticeProving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8


Less

on

2-8


1. m!2 $ 57 2. m!5 $ 22 3. m!1 $ 38

4. m!13 $ 4x # 11, 5. !9 and !10 are 6. m!2 $ 4x ! 26,m!14 $ 3x # 1 complementary. m!3 $ 3x # 4

!7 " !9, m!8 $ 41

Determine whether the following statements are always, sometimes, or never true.

7. Two angles that are supplementary form a linear pair.

8. Two angles that are vertical are adjacent.

9. Copy and complete the following proof.Given: !QPS " !TPRProve: !QPR " !TPSProof:

Statements Reasons

a. a.b. m!QPS $ m!TPR b.c. m!QPS $ m!QPR # m!RPS c.

m!TPR $ m!TPS # m!RPSd. d. Substitution

e. e.f. f.

Q P

RS

T

237

8 910

13 14

12

651 2



1. m!1 $ x # 10 2. m!4 $ 2x ! 5 3. m!6 $ 7x ! 24m!2 $ 3x # 18 m!5 $ 4x ! 13 m!7 $ 5x # 14

Determine whether the following statements are always, sometimes, or never true.

4. Two angles that are supplementary are complementary.

5. Complementary angles are congruent.

6. Write a two-column proof.Given: !1 and !2 form a linear pair.

!2 and !3 are supplementary.Prove: !1 " !3

7. STREETS Refer to the figure. Barton Road and Olive Tree Lane form a right angle at their intersection. Tryon Street forms a 57°angle with Olive Tree Lane. What is the measure of the acute angleTryon Street forms with Barton Road? Olive Tree Lane

BartonRd

TryonSt

1 23

67

453

1 2

Practice Proving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8

Reading to Learn MathematicsProving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8


Less

on

2-8

Pre-Activity How do scissors illustrate supplementary angles?


Is it possible to open a pair of scissors so that the angles formed by the twoblades, a blade and a handle, and the two handles, are all congruent? If so,explain how this could happen.

Reading the Lesson1. Complete each sentence to form a statement that is always true.

a. If two angles form a linear pair, then they are adjacent and .b. If two angles are complementary to the same angle, then they are .c. If D is a point in the interior of !ABC, then m!ABC $ m!ABD # .d. Given RS""# and a number x between and , there is exactly one ray

with endpoint R, extended on either side of RS, such that the measure of the angleformed is x.

e. If two angles are congruent and supplementary, then each angle is a(n) angle.

f. lines form congruent adjacent angles.g. “Every angle is congruent to itself” is a statement of the Property

of angle congruence.h. If two congruent angles form a linear pair, then the measure of each angle is .i. If the noncommon sides of two adjacent angles form a right angle, then the angles are

.

2. Determine whether each statement is always, sometimes, or never true.a. Supplementary angles are congruent.b. If two angles form a linear pair, they are complementary.c. Two vertical angles are supplementary.d. Two adjacent angles form a linear pair.e. Two vertical angles form a linear pair.f. Complementary angles are congruent.g. Two angles that are congruent to the same angle are congruent to each other.h. Complementary angles are adjacent angles.

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that a

classmate thinks that two angles can only be vertical angles if one angle lies above theother. How can you explain to him the meaning of vertical angles, using the word vertexin your explanation?


Bisecting a Hidden AngleThe vertex of !BAD at the right is hidden in a region.Within the region, you are not allowed to use a compass.Can you bisect the angle?

Follow these instructions to bisect !BAD.

1. Use a straightedge to draw lines CE and BD.

2. Construct the bisectors of !DEC and !BCE.

3. Label the intersection of the two bisectors as point P.

4. Construct the bisectors of !BDE and !DBC.

5. Label the intersection of the two previous bisectors as point Q.

6. Use a straightedge to draw line PQ, which bisects the hidden angle.

7. Another hidden angle is shown at right. Constructthe bisector using themethod above, or deviseyour own method.

AD

E

C

PQ

B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8

© Glencoe/McGraw-Hill A2 Glencoe Geometry

Stu

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give

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w a

fig

ure

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illu

stra

teyo

ur

con

ject

ure

.5–

8.S

ampl

e an

swer

s ar

e gi

ven.

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AB

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a r

ight

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le.

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int

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wee

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and

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BA

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e no

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r 8.

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ents

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ine

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eth

er e

ach

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ject

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e or

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ive

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un

tere

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ple

for

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e co

nje

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re.

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iven

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,and

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e co

lline

ar a

nd S

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.C

onje

ctur

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is t

he m

idpo

int

of S !

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true

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iven

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and

!2

are

adja

cent

ang

les.

Con

ject

ure:

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and

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form

a li

near

pai

r.Fa

lse;

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and

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coul

d ea

ch m

easu

re 6

0°.

11.G

iven

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and

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rm a

rig

ht a

ngle

and

inte

rsec

t at

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ject

ure:

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⊥J!K!

true

12.A

LLER

GIE

SE

ach

spri

ng,R

ache

l sta

rts

snee

zing

whe

n th

e pe

ar t

rees

on

her

stre

et b

loss

om.

She

reas

ons

that

she

is a

llerg

ic t

o pe

ar t

rees

.Fin

d a

coun

tere

xam

ple

to R

ache

l’s c

onje

ctur

e.S

ampl

e an

swer

:Rac

hel c

ould

be

alle

rgic

to

othe

r ty

pes

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lant

s th

atbl

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hen

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pear

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es b

loss

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A

C

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ctiv

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easo

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ject

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____

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2-1

2-1



Rea

din

g t

o L

earn

Math

emati

csIn

duct

ive

Rea

soni

ng a

nd C

onje

ctur

e

NA

ME

____

____

____

____

____

____

____

____

____

____

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ATE

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RIO

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___

2-1

2-1

©G

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Gle

ncoe

Geo

met

ry

Lesson 2-1

Pre-

Act

ivit

yH

ow c

an i

nd

uct

ive

reas

onin

g h

elp

pre

dic

t w

eath

er c

ond

itio

ns?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

1 at

the

top

of

page

62

in y

our

text

book

.

•W

hat

kind

of

wea

ther

pat

tern

s do

you

thi

nk m

eteo

rolo

gist

s lo

ok a

t to

help

pre

dict

the

wea

ther

? S

ampl

e an

swer

:pat

tern

s of

hig

h an

dlo

w t

empe

ratu

res,

incl

udin

g he

at s

pells

and

col

d sp

ells

;pa

tter

ns o

f pr

ecip

itatio

n,in

clud

ing

wet

spe

lls a

nd d

ry s

pells

•W

hat

is a

fac

tor

that

mig

ht c

ontr

ibut

e to

long

-ter

m c

hang

es in

the

wea

ther

? S

ampl

e an

swer

:glo

bal w

arm

ing

due

to h

igh

usag

eof

foss

il fu

els

Rea

din

g t

he

Less

on

1.E

xpla

in in

you

r ow

n w

ords

the

rel

atio

nshi

p be

twee

n a

conj

ectu

re,a

cou

nter

exam

ple,

and

indu

ctiv

e re

ason

ing.

Sam

ple

answ

er:A

con

ject

ure

is a

n ed

ucat

ed g

uess

bas

ed o

n sp

ecifi

cex

ampl

es o

r in

form

atio

n.A

cou

nter

exam

ple

is a

n ex

ampl

e th

at s

how

sth

at a

con

ject

ure

is f

alse

.Ind

uctiv

e re

ason

ing

is t

he p

roce

ss o

f m

akin

g a

conj

ectu

re b

ased

on

spec

ific

exam

ples

or

info

rmat

ion.

2.M

ake

a co

njec

ture

abo

ut t

he n

ext

item

in e

ach

sequ

ence

.

a.5,

9,13

,17

21b.

1,"1 3" ,

"1 9" ," 21 7"

" 81 1"

c.0,

1,3,

6,10

15d.

8,3,

!2,

!7

!12

e.1,

8,27

,64

125

f.1,

!2,

4,!

816

g.h

.

3.St

ate

whe

ther

eac

h co

njec

ture

is t

rue

or f

alse

.If

the

conj

ectu

re is

fal

se,g

ive

aco

unte

rexa

mpl

e.a.

The

sum

of

two

odd

inte

gers

is e

ven.

true

b.T

he p

rodu

ct o

f an

odd

inte

ger

and

an e

ven

inte

ger

is o

dd.

Fals

e;sa

mpl

e an

swer

:5 %

8 #

40,w

hich

is e

ven.

c.T

he o

ppos

ite

of a

n in

tege

r is

a n

egat

ive

inte

ger.

Fals

e;sa

mpl

e an

swer

:The

oppo

site

of

the

inte

ger

!5

is 5

,whi

ch is

a p

ositi

ve in

tege

r.d.

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per

fect

squ

ares

(sq

uare

s of

who

le n

umbe

rs)

alte

rnat

e be

twee

n od

d an

d ev

en.

true

Hel

pin

g Y

ou

Rem

emb

er4.

Wri

te a

sho

rt s

ente

nce

that

can

hel

p yo

u re

mem

ber

why

it o

nly

take

s on

e co

unte

rexa

mpl

eto

pro

ve t

hat

a co

njec

ture

is f

alse

.S

ampl

e an

swer

:Tru

e m

eans

alw

ays

true

.

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Geo

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Cou

nter

exam

ples

Whe

n yo

u m

ake

a co

nclu

sion

aft

er e

xam

inin

g se

vera

l spe

cifi

cca

ses,

you

have

use

d in

du

ctiv

e re

ason

ing.

How

ever

,you

mus

t be

caut

ious

whe

n us

ing

this

for

m o

f re

ason

ing.

By

find

ing

only

one

cou

nte

rexa

mp

le,y

ou d

ispr

ove

the

conc

lusi

on.

Is t

he

stat

emen

t "1 x"

&1

tru

e w

hen

you

rep

lace

xw

ith

1,

2,an

d 3

? Is

th

e st

atem

ent

tru

e fo

r al

l re

als?

If

pos

sibl

e,fi

nd

a

cou

nte

rexa

mp

le.

"1 1"$

1,"1 2"

%1,

and

"1 3"%

1.B

ut w

hen

x$

"1 2" ,th

en "1 x"

$2.

Thi

s co

unte

rexa

mpl

e

show

s th

at t

he s

tate

men

t is

not

alw

ays

true

.

An

swer

eac

h q

ues

tion

.

1.T

he c

olde

st d

ay o

f th

e ye

ar in

Chi

cago

2.Su

ppos

e Jo

hn m

isse

s th

e sc

hool

bus

occu

rred

in J

anua

ry f

or f

ive

stra

ight

four

Tue

sday

s in

a r

ow.C

an y

ouye

ars.

Is it

saf

e to

con

clud

e th

at t

he

safe

ly c

oncl

ude

that

Joh

n m

isse

s th

e co

ldes

t da

y in

Chi

cago

is a

lway

s in

sc

hool

bus

eve

ry T

uesd

ay?

noJa

nuar

y?no

3.Is

the

equ

atio

n #

k2 !$

ktr

ue w

hen

you

4.Is

the

sta

tem

ent

2x$

x#

xtr

ue w

hen

repl

ace

kw

ith

1,2,

and

3? I

s th

eyo

u re

plac

e x

wit

h "1 2" ,

4,an

d 0.

7? I

s th

eeq

uati

on t

rue

for

all i

nteg

ers?

If

stat

emen

t tr

ue f

or a

ll re

al n

umbe

rs?

poss

ible

,fin

d a

coun

tere

xam

ple.

If p

ossi

ble,

find

a c

ount

erex

ampl

e.

It is

tru

e fo

r 1,

2,an

d 3.

It is

not

It

is t

rue

for

all r

eal n

umbe

rs.

true

for

nega

tive

inte

gers

.S

ampl

e an

swer

:!2

5.Su

ppos

e yo

u dr

aw f

our

poin

ts A

,B,C

,6.

Supp

ose

you

draw

a c

ircl

e,m

ark

thre

e an

d D

and

then

dra

w A !

B!,B!

C!,C!

D!,a

nd

poin

ts o

n it

,and

con

nect

the

m.W

ill t

heD!

A!.D

oes

this

pro

cedu

re g

ive

a an

gles

of

the

tria

ngle

be

acut

e? E

xpla

inqu

adri

late

ral a

lway

s or

onl

y so

met

imes

? yo

ur a

nsw

ers

wit

h fi

gure

s.E

xpla

in y

our

answ

ers

wit

h fi

gure

s.no

,onl

y so

met

imes

only

som

etim

esE

xam

ple:

Cou

nter

exam

ple:

Exa

mpl

e:C

ount

erex

ampl

e:A

A

B

BC

C

D

D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-1

2-1

Exam

ple

Exam

ple



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Logi

c

NA

ME

____

____

____

____

____

____

____

____

____

____

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__D

ATE

____

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RIO

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___

2-2

2-2

©G

lenc

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w-H

ill63

Gle

ncoe

Geo

met

ry

Lesson 2-2

Det

erm

ine

Tru

th V

alu

esA

sta

tem

ent

is a

ny s

ente

nce

that

is e

ithe

r tr

ue o

r fa

lse.

The

trut

h or

fal

sity

of

a st

atem

ent

is it

s tr

uth

val

ue.

A s

tate

men

t ca

n be

rep

rese

nted

by

usin

g a

lett

er.F

or e

xam

ple,

Sta

tem

entp

:Chi

cago

is a

cit

y in

Illi

nois

.The

tru

th v

alue

of

stat

emen

t p

is t

rue.

Seve

ral s

tate

men

ts c

an b

e jo

ined

in a

com

pou

nd

sta

tem

ent.

Sta

tem

ent p

and

stat

emen

t qjo

ined

S

tate

men

t pan

d st

atem

ent q

join

ed

Neg

atio

n:no

t pis

the

nega

tion

ofby

the

wor

d an

dis

a c

onju

nctio

n.by

the

wor

d or

is a

dis

junc

tion.

the

stat

emen

t p.

Sym

bols

: p$

q(R

ead:

p a

nd q

)S

ymbo

ls: p

%q

(Rea

d: p

or

q)S

ymbo

ls: &

p(R

ead:

not

p)

The

con

junc

tion

p$

qis

true

onl

yT

he d

isju

nctio

n p

%q

is tr

ue if

pis

T

he s

tate

men

ts p

and

&p

have

w

hen

both

pan

d q

are

true

.tr

ue, i

f qis

true

, or

if bo

th a

re tr

ue.

oppo

site

trut

h va

lues

.

Wri

te a

com

pou

nd

stat

emen

t fo

r ea

ch c

onju

nct

ion

.Th

enfi

nd

its

tru

th v

alu

e.p:

An

elep

hant

is a

mam

mal

.q:

A s

quar

e ha

s fo

ur r

ight

ang

les.

a.p

"q

Join

the

sta

tem

ents

wit

h an

d:A

n el

epha

ntis

a m

amm

al a

nd a

squ

are

has

four

rig

htan

gles

.Bot

h pa

rts

of t

he s

tate

men

t ar

etr

ue s

o th

e co

mpo

und

stat

emen

t is

tru

e.

b.#

p"

q&

pis

the

sta

tem

ent

“An

elep

hant

is n

ot a

mam

mal

.”Jo

in &

pan

d q

wit

h th

e w

ord

and:

An

elep

hant

is n

ot a

mam

mal

and

asq

uare

has

fou

r ri

ght

angl

es.T

he f

irst

part

of

the

com

poun

d st

atem

ent,

&p,

isfa

lse.

The

refo

re t

he c

ompo

und

stat

emen

tis

fal

se.

Wri

te a

com

pou

nd

stat

emen

t fo

r ea

ch d

isju

nct

ion

.Th

enfi

nd

its

tru

th v

alu

e.p:

A d

iam

eter

of

a ci

rcle

is t

wic

e th

e ra

dius

.q:

A r

ecta

ngle

has

fou

r eq

ual s

ides

.

a.p

$q

Join

the

sta

tem

ents

pan

d q

wit

h th

ew

ord

or:A

dia

met

er o

f a

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le is

tw

ice

the

radi

us o

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e ha

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ur e

qual

side

s.T

he f

irst

par

t of

the

com

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e,so

the

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pan

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diam

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ot t

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e th

era

dius

or

a re

ctan

gle

has

four

equ

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des.

Nei

ther

par

t of

the

dis

junc

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istr

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o th

e co

mpo

und

stat

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fal

se.

Exam

ple1

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ple2

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Exercis

esExercis

es

Wri

te a

com

pou

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sta

tem

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for

each

con

jun

ctio

n a

nd

dis

jun

ctio

n.

Th

en f

ind

its

tru

th v

alu

e.p:

10 #

8 $

18q:

Sept

embe

r ha

s 30

day

s.r:

A r

ecta

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has

fou

r si

des.

1.p

and

q10

$8

#18

and

Sep

tem

ber

has

30 d

ays;

true

.

2.p

or r

10 $

8 #

18 o

r a

rect

angl

e ha

s fo

ur s

ides

;tru

e.

3.q

or r

Sep

tem

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has

30 d

ays

or a

rec

tang

le h

as fo

ur s

ides

;tru

e.

4.q

and

&r

Sep

tem

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has

30 d

ays

and

a re

ctan

gle

does

not

hav

e fo

ursi

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fals

e.

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Geo

met

ry

Tru

th T

able

sO

ne w

ay t

o or

gani

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the

trut

h va

lues

of

stat

emen

ts is

in a

tr

uth

tab

le.T

he t

ruth

tab

les

for

nega

tion

,con

junc

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,and

dis

junc

tion

ar

e sh

own

at t

he r

ight

.

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junc

tion

pq

p$

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TT

FT

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TF

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junc

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TT

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atio

n

p#

p

TF

FT

Stu

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Gu

ide

and I

nte

rven

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(con

tinu

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Logi

c

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

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PE

RIO

D__

___

2-2

2-2

Con

stru

ct a

tru

thta

ble

for

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com

pou

nd

stat

emen

t q

or r

.Use

th

ed

isju

nct

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tab

le.

qr

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r

TT

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FT

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stru

ct a

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able

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mp

oun

d s

tate

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t p

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(q

or r

).U

se t

he d

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on t

able

for

(qor

r).

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n us

e th

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ctio

n ta

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).

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)

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FT

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ple1

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ple1

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Con

tru

ct a

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th t

able

for

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h c

omp

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p%

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4.&

p$

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5.(p

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.T

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ram

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ws

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tude

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and

who

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k af

ter

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wee

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ow m

any

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ents

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k af

ter

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2-2

2-2



An

swer

s

Rea

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2-2

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Lesson 2-2

Pre-

Act

ivit

yH

ow d

oes

logi

c ap

ply

to

sch

ool?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

2 at

the

top

of

page

67

in y

our

text

book

.

How

can

you

use

logi

c to

hel

p yo

u an

swer

a m

ulti

ple-

choi

ce q

uest

ion

on a

stan

dard

ized

tes

t if

you

are

not

sure

of t

he c

orre

ct a

nsw

er?

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ple

answ

er:

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inat

e th

e ch

oice

s th

at y

ou k

now

are

wro

ng.T

hen

choo

se th

eon

e yo

u th

ink

is m

ost l

ikel

y co

rrec

t fro

m th

e on

es th

at a

re le

ft.

Rea

din

g t

he

Less

on

1.Su

pply

one

or

two

wor

ds t

o co

mpl

ete

each

sen

tenc

e.a.

Tw

o or

mor

e st

atem

ents

can

be

join

ed t

o fo

rm a

st

atem

ent.

b.A

sta

tem

ent

that

is f

orm

ed b

y jo

inin

g tw

o st

atem

ents

wit

h th

e w

ord

oris

cal

led

a .

c.T

he t

ruth

or

fals

ity

of a

sta

tem

ent

is c

alle

d it

s

.d.

A s

tate

men

t th

at is

for

med

by

join

ing

two

stat

emen

ts w

ith

the

wor

d an

dis

cal

led

a

.e.

A s

tate

men

t th

at h

as t

he o

ppos

ite

trut

h va

lue

and

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oppo

site

mea

ning

fro

m a

giv

en

stat

emen

t is

cal

led

the

of

the

sta

tem

ent.

2.U

se t

rue

or f

alse

to c

ompl

ete

each

sen

tenc

e.a.

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tem

ent

is t

rue,

then

its

nega

tion

is

.b.

If a

sta

tem

ent

is f

alse

,the

n it

s ne

gati

on is

.

c.If

tw

o st

atem

ents

are

bot

h tr

ue,t

hen

thei

r co

njun

ctio

n is

an

d th

eir

disj

unct

ion

is

.d.

If t

wo

stat

emen

ts a

re b

oth

fals

e,th

en t

heir

con

junc

tion

is

and

thei

r di

sjun

ctio

n is

.

e.If

one

sta

tem

ent

is t

rue

and

anot

her

is f

alse

,the

n th

eir

conj

unct

ion

is

and

thei

r di

sjun

ctio

n is

.

3.C

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der

the

follo

win

g st

atem

ents

:p:

Chi

cago

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he c

apit

al o

f Il

linoi

s.q:

Sacr

amen

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the

cap

ital

of

Cal

ifor

nia.

Wri

te e

ach

stat

emen

t sy

mbo

lical

ly a

nd t

hen

find

its

trut

h va

lue.

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ento

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ot t

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a.#

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alse

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is t

he c

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a an

d C

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e

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D__

___

2-2

2-2

Exam

ple

Exam

ple



Stu

dy

Gu

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nte

rven

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Con

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tate

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ME

____

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2-3

2-3

©G

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Gle

ncoe

Geo

met

ry

Lesson 2-3

If-t

hen

Sta

tem

ents

An

if-t

hen

stat

emen

t is

a s

tate

men

t su

ch a

s “I

f yo

u ar

e re

adin

gth

is p

age,

then

you

are

stu

dyin

g m

ath.

”A s

tate

men

t th

at c

an b

e w

ritt

en in

if-t

hen

form

isca

lled

a co

nd

itio

nal

sta

tem

ent.

The

phr

ase

imm

edia

tely

fol

low

ing

the

wor

d if

is t

heh

ypot

hes

is.T

he p

hras

e im

med

iate

ly f

ollo

win

g th

e w

ord

then

is t

he c

oncl

usi

on.

A c

ondi

tion

al s

tate

men

t ca

n be

rep

rese

nted

in s

ymbo

ls a

s p

→q,

whi

ch is

rea

d “p

impl

ies

q”or

“if

p,t

hen

q.”

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tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

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stat

emen

t.

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X"

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and

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S,t

hen

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S.

hypo

thes

isco

nclu

sion

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tify

th

e h

ypot

hes

is a

nd

con

clu

sion

.W

rite

th

e st

atem

ent

in i

f-th

en f

orm

.

You

rece

ive

a fr

ee p

izza

wit

h 12

cou

pons

.

If y

ou h

ave

12 c

oupo

ns,t

hen

you

rece

ive

a fr

ee p

izza

.hy

poth

esis

conc

lusi

on

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ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

it is

Sat

urda

y,th

en t

here

is n

o sc

hool

.H

:it

is S

atur

day;

C:t

here

is n

o sc

hool

2.If

x!

8 $

32,t

hen

x$

40.

H:x

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#32

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3.If

a p

olyg

on h

as f

our

righ

t an

gles

,the

n th

e po

lygo

n is

a r

ecta

ngle

.H

:a p

olyg

on h

as fo

ur r

ight

ang

les;

C:t

he p

olyg

on is

a r

ecta

ngle

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

4.A

ll ap

es lo

ve b

anan

as.

If an

ani

mal

is a

n ap

e,th

en it

love

s ba

nana

s.

5.T

he s

um o

f th

e m

easu

res

of c

ompl

emen

tary

ang

les

is 9

0.If

two

angl

esar

e co

mpl

emen

tary

,the

n th

e su

m o

f th

eir

mea

sure

s is

90.

6.C

ollin

ear

poin

ts li

e on

the

sam

e lin

e.If

poin

ts a

re c

ollin

ear,

then

the

y lie

on

the

sam

e lin

e.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If i

t do

es n

ot r

ain

this

Sat

urda

y,w

e w

ill

have

a p

icni

c.

7.It

rai

ns t

his

Satu

rday

,and

we

have

a p

icni

c.tr

ue8.

It r

ains

thi

s Sa

turd

ay,a

nd w

e do

n’t

have

a p

icni

c.tr

ue9.

It d

oesn

’t ra

in t

his

Satu

rday

,and

we

have

a p

icni

c.tr

ue10

.It

does

n’t

rain

thi

s Sa

turd

ay,a

nd w

e do

n’t

have

a p

icni

c.fa

lse

©G

lenc

oe/M

cGra

w-H

ill70

Gle

ncoe

Geo

met

ry

Co

nve

rse,

Inve

rse,

an

d C

on

trap

osi

tive

If y

ou c

hang

e th

e hy

poth

esis

or

conc

lusi

onof

a c

ondi

tion

al s

tate

men

t,yo

u fo

rm a

rel

ated

con

dit

ion

al.T

his

char

t sh

ows

the

thre

ere

late

d co

ndit

iona

ls,c

onve

rse,

inve

rse,

and

cont

rapo

siti

ve,a

nd h

ow t

hey

are

rela

ted

to a

cond

itio

nal s

tate

men

t.

Sym

bols

Form

ed b

yE

xam

ple

Con

ditio

nal

p→

qus

ing

the

give

n hy

poth

esis

and

con

clus

ion

If tw

o an

gles

are

ver

tical

ang

les,

th

en th

ey a

re c

ongr

uent

.

Con

vers

eq

→p

exch

angi

ng th

e hy

poth

esis

and

con

clus

ion

If tw

o an

gles

are

con

grue

nt, t

hen

th

ey a

re v

ertic

al a

ngle

s.

Inve

rse

&p

→&

qre

plac

ing

the

hypo

thes

is w

ith it

s ne

gatio

n If

two

angl

es a

re n

ot v

ertic

al a

ngle

s,an

d re

plac

ing

the

conc

lusi

on w

ith it

s ne

gatio

nth

en th

ey a

re n

ot c

ongr

uent

.

Con

trap

ositi

ve&

q→

&p

nega

ting

the

hypo

thes

is, n

egat

ing

the

If tw

o an

gles

are

not

con

grue

nt,

conc

lusi

on, a

nd s

witc

hing

them

then

they

are

not

ver

tical

ang

les.

Just

as

a co

ndit

iona

l sta

tem

ent

can

be t

rue

or f

alse

,the

rel

ated

con

diti

onal

s al

so c

an b

e tr

ueor

fal

se.A

con

diti

onal

sta

tem

ent

alw

ays

has

the

sam

e tr

uth

valu

e as

its

cont

rapo

siti

ve,a

ndth

e co

nver

se a

nd in

vers

e al

way

s ha

ve t

he s

ame

trut

h va

lue.

Wri

te t

he

con

vers

e,in

vers

e,an

d c

ontr

apos

itiv

e of

eac

h c

ond

itio

nal

sta

tem

ent.

Tell

wh

ich

sta

tem

ents

are

tru

ean

d w

hic

h s

tate

men

ts a

re f

als

e.

1.If

you

live

in S

an D

iego

,the

n yo

u liv

e in

Cal

ifor

nia.

Con

vers

e:If

you

live

in C

alifo

rnia

,the

n yo

u liv

e in

San

Die

go.I

nver

se:

If yo

u do

not

live

in S

an D

iego

,the

n yo

u do

not

live

in C

alifo

rnia

.C

ontr

apos

itive

:If

you

do n

ot li

ve in

Cal

iforn

ia,t

hen

you

do n

ot li

ve

in S

an D

iego

.Tru

e:st

atem

ent

and

cont

rapo

sitiv

e;fa

lse:

conv

erse

an

d in

vers

e

2.If

a p

olyg

on is

a r

ecta

ngle

,the

n it

is a

squ

are.

Con

vers

e:If

a po

lygo

n is

a s

quar

e,th

en it

is a

rec

tang

le.I

nver

se:

If a

poly

gon

is n

ot a

rec

tang

le,t

hen

it is

not

a s

quar

e.C

ontr

apos

itive

:If

a po

lygo

n is

not

a s

quar

e,th

en it

is n

ot a

rec

tang

le.T

rue:

conv

erse

an

d in

vers

e;fa

lse:

stat

emen

t an

d co

ntra

posi

tive

3.If

tw

o an

gles

are

com

plem

enta

ry,t

hen

the

sum

of

thei

r m

easu

res

is 9

0.C

onve

rse:

If th

e su

m o

f th

e m

easu

res

of t

wo

angl

es is

90,

then

the

angl

es a

re c

ompl

emen

tary

.Inv

erse

:If

two

angl

es a

re n

ot c

ompl

emen

tary

,th

en t

he s

um o

f th

eir

mea

sure

s is

not

90.

Con

trap

ositi

ve:i

f th

e su

m o

fth

e m

easu

res

of t

wo

angl

e is

not

90,

then

the

ang

les

are

not

com

plem

enta

ry.T

rue:

all;

fals

e:no

ne

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Con

ditio

nal S

tate

men

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-3

2-3

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Con

ditio

nal S

tate

men

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill71

Gle

ncoe

Geo

met

ry

Lesson 2-3

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

you

pur

chas

e a

com

pute

r an

d do

not

like

it,t

hen

you

can

retu

rn it

wit

hin

30 d

ays.

H:y

ou p

urch

ase

a co

mpu

ter

and

do n

ot li

ke it

;C

:you

can

ret

urn

it w

ithin

30

days

2.If

x#

8 $

4,th

en x

$!

4.H

:x$

8 #

4;C

:x#

!4

3.If

the

dra

ma

clas

s ra

ises

$20

00,t

hen

they

will

go

on t

our.

H:t

he d

ram

a cl

ass

rais

es $

2000

;C:t

hey

will

go

on t

our

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

4.A

pol

ygon

wit

h fo

ur s

ides

is a

qua

drila

tera

l.If

a po

lygo

n ha

s fo

ur s

ides

,the

n it

is a

qua

drila

tera

l.

5.“T

hose

who

sta

nd f

or n

othi

ng f

all f

or a

nyth

ing.

”(A

lexa

nder

Ham

ilto

n)If

you

stan

d fo

r no

thin

g,th

en y

ou w

ill f

all f

or a

nyth

ing.

6.A

n ac

ute

angl

e ha

s a

mea

sure

less

tha

n 90

.If

an a

ngle

is a

cute

,the

n its

mea

sure

is le

ss t

han

90.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If y

ou f

inis

h y

our

hom

ewor

k by

5 P

.M.,

then

you

go

out

to d

inn

er.

7.Yo

u fi

nish

you

r ho

mew

ork

by 5

P.M

.and

you

go

out

to d

inne

r.tr

ue

8.Yo

u fi

nish

you

r ho

mew

ork

by 4

P.M

.and

you

go

out

to d

inne

r.tr

ue

9.Yo

u fi

nish

you

r ho

mew

ork

by 5

P.M

.and

you

do

not

go o

ut t

o di

nner

.fa

lse

10.W

rite

the

con

vers

e,in

vers

e,an

d co

ntra

posi

tive

of

the

cond

itio

nal s

tate

men

t.D

eter

min

ew

heth

er e

ach

stat

emen

t is

tru

e or

fal

se.I

f a

stat

emen

t is

fal

se,f

ind

a co

unte

rexa

mpl

e.If

89

is d

ivis

ible

by

2,th

en 8

9 is

an

even

num

ber.

Con

vers

e:If

89 is

an

even

num

ber,

then

89

is d

ivis

ible

by

2;tr

ue.

Inve

rse:

If 89

is n

ot d

ivis

ible

by

2,th

en 8

9 is

not

an

even

num

ber;

true

.C

ontr

apos

itive

:If

89 is

not

an

even

num

ber,

then

89

is n

ot d

ivis

ible

by

2;tr

ue.

©G

lenc

oe/M

cGra

w-H

ill72

Gle

ncoe

Geo

met

ry

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

3x

#4

$!

5,th

en x

$!

3.H

:3x

$4

#!

5;C

:x#

!3

2.If

you

tak

e a

clas

s in

tel

evis

ion

broa

dcas

ting

,the

n yo

u w

ill f

ilm a

spo

rtin

g ev

ent.

H:y

ou t

ake

a cl

ass

in t

elev

isio

n br

oadc

astin

g;C

:you

will

film

a s

port

ing

even

t

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

3.“T

hose

who

do

not

rem

embe

r th

e pa

st a

re c

onde

mne

d to

rep

eat

it.”

(Geo

rge

San

taya

na)

If yo

u do

not

rem

embe

r th

e pa

st,t

hen

you

are

cond

emne

d to

rep

eat

it.

4.A

djac

ent

angl

es s

hare

a c

omm

on v

erte

x an

d a

com

mon

sid

e.If

two

angl

es a

re a

djac

ent,

then

the

y sh

are

a co

mm

on v

erte

x an

d a

com

mon

sid

e.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If D

VD

pla

yers

are

on

sa

le f

or l

ess

tha

n $

100,

then

you

bu

y on

e.

5.D

VD

pla

yers

are

on

sale

for

$95

and

you

buy

one

.tr

ue

6.D

VD

pla

yers

are

on

sale

for

$10

0 an

d yo

u do

not

buy

one

.tr

ue

7.D

VD

pla

yers

are

not

on

sale

for

und

er $

100

and

you

do n

ot b

uy o

ne.

true

8.W

rite

the

con

vers

e,in

vers

e,an

d co

ntra

posi

tive

of

the

cond

itio

nal s

tate

men

t.D

eter

min

ew

heth

er e

ach

stat

emen

t is

tru

e or

fal

se.I

f a

stat

emen

t is

fal

se,f

ind

a co

unte

rexa

mpl

e.If

(!

8)2

"0,

then

!8

"0.

Con

vers

e:If

!8

'0,

then

(!

8)2

'0;

true

.In

vers

e:If

(!8)

2&

0,th

en !

8 &

0;tr

ue.

Con

trap

ositi

ve:I

f !

8 &

0,th

en (

!8)

2&

0;fa

lse.

SUM

MER

CA

MP

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

Old

er c

ampe

rs w

ho a

tten

d W

oodl

and

Falls

Cam

p ar

e ex

pect

ed t

o w

ork.

Cam

pers

who

are

juni

ors

wai

t on

tab

les.

9.W

rite

a c

ondi

tion

al s

tate

men

t in

if-t

hen

form

.S

ampl

e an

swer

:If

you

are

a ju

nior

,the

n yo

u w

ait

on t

able

s.

10.W

rite

the

con

vers

e of

you

r co

ndit

iona

l sta

tem

ent.

If yo

u w

ait

on t

able

s,th

en y

ou a

re a

juni

or.

Pra

ctic

e (A

vera

ge)

Con

ditio

nal S

tate

men

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-3

2-3



Rea

din

g t

o L

earn

Math

emati

csC

ondi

tiona

l Sta

tem

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill73

Gle

ncoe

Geo

met

ry

Lesson 2-3

Pre-

Act

ivit

yH

ow a

re c

ond

itio

nal

sta

tem

ents

use

d i

n a

dve

rtis

emen

ts?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

3 at

the

top

of

page

75

in y

our

text

book

.

Doe

s th

e se

cond

adv

erti

sing

sta

tem

ent

in t

he in

trod

ucti

on m

ean

that

you

will

not

get

a f

ree

phon

e if

you

sig

n a

cont

ract

for

onl

y si

x m

onth

s of

serv

ice?

Exp

lain

you

r an

swer

.N

o;it

only

tel

ls y

ou w

hat

happ

ens

ifyo

u si

gn u

p fo

r on

e ye

ar.

Rea

din

g t

he

Less

on

1.Id

enti

fy t

he h

ypot

hesi

s an

d co

nclu

sion

of

each

sta

tem

ent.

a.If

you

are

a r

egis

tere

d vo

ter,

then

you

are

at

leas

t 18

yea

rs o

ld. H

ypot

hesi

s:yo

uar

e a

regi

ster

ed v

oter

;Con

clus

ion:

you

are

at le

ast

18 y

ears

old

b.If

tw

o in

tege

rs a

re e

ven,

thei

r pr

oduc

t is

eve

n.H

ypot

hesi

s:tw

o in

tege

rs a

reev

en;C

oncl

usio

n:th

eir

prod

uct

is e

ven

2.C

ompl

ete

each

sen

tenc

e.a.

The

sta

tem

ent

that

is f

orm

ed b

y re

plac

ing

both

the

hyp

othe

sis

and

the

conc

lusi

on o

f a

cond

itio

nal w

ith

thei

r ne

gati

ons

is t

he

.b.

The

sta

tem

ent

that

is f

orm

ed b

y ex

chan

ging

the

hyp

othe

sis

and

conc

lusi

on o

f a

cond

itio

nal i

s th

e .

3.C

onsi

der

the

follo

win

g st

atem

ent:

You

live

in N

orth

Am

eric

a if

you

live

in t

he U

nite

d St

ates

.a.

Wri

te t

his

cond

itio

nal s

tate

men

t in

if-t

hen

form

and

giv

e it

s tr

uth

valu

e.If

the

stat

emen

t is

fal

se,g

ive

a co

unte

rexa

mpl

e.If

you

live

in t

he U

nite

d S

tate

s,th

enyo

u liv

e in

Nor

th A

mer

ica;

fals

e:Yo

u liv

e in

Haw

aii.

b.W

rite

the

inve

rse

of t

he g

iven

con

diti

onal

sta

tem

ent

in if

-the

n fo

rm a

nd g

ive

its

trut

hva

lue.

If t

he s

tate

men

t is

fal

se,g

ive

a co

unte

rexa

mpl

e.If

you

do n

ot li

ve in

the

Uni

ted

Sta

tes,

then

you

do

not

live

in N

orth

Am

eric

a;fa

lse;

sam

ple

answ

er:Y

ou li

ve in

Mex

ico.

c.W

rite

the

con

trap

osit

ive

of t

he g

iven

con

diti

onal

sta

tem

ent

in if

-the

n fo

rm a

nd g

ive

its

trut

h va

lue.

If t

he s

tate

men

t is

fal

se,g

ive

a co

unte

rexa

mpl

e.If

you

do n

ot li

vein

Nor

th A

mer

ica,

then

you

do

not

live

in t

he U

nite

d S

tate

s;fa

lse:

You

live

in H

awai

i.d.

Wri

te t

he c

onve

rse

of t

he g

iven

con

diti

onal

sta

tem

ent

in if

-the

n fo

rm a

nd g

ive

its

trut

h va

lue.

If t

he s

tate

men

t is

fal

se,g

ive

a co

unte

rexa

mpl

e.If

you

live

in N

orth

Am

eric

a,th

en y

ou li

ve in

the

Uni

ted

Sta

tes;

fals

e;sa

mpl

e an

swer

:You

live

in C

anad

a.

Hel

pin

g Y

ou

Rem

emb

er4.

Whe

n w

orki

ng w

ith

a co

ndit

iona

l sta

tem

ent

and

its

thre

e re

late

d co

ndit

iona

ls,w

hat

isan

eas

y w

ay t

o re

mem

ber

whi

ch s

tate

men

ts a

re lo

gica

lly e

quiv

alen

t to

eac

h ot

her?

Sam

ple

answ

er:T

he t

wo

stat

emen

ts w

hose

nam

es c

onta

in v

erse

(the

conv

erse

and

the

inve

rse)

are

a lo

gica

lly e

quiv

alen

t pa

ir.Th

e ot

her

two

(the

ori

gina

l con

ditio

nal a

nd t

he c

ontr

apos

itive

) ar

e th

e ot

her

logi

cally

equi

vale

nt p

air.

conv

erse

inve

rse

©G

lenc

oe/M

cGra

w-H

ill74

Gle

ncoe

Geo

met

ry

Venn

Dia

gram

sA

typ

e of

dra

win

g ca

lled

a V

enn

dia

gram

can

be u

sefu

l in

expl

aini

ng c

ondi

tion

alst

atem

ents

.A V

enn

diag

ram

use

s ci

rcle

s to

rep

rese

nt s

ets

of o

bjec

ts.

Con

side

r th

e st

atem

ent

“All

rabb

its

have

long

ear

s.”T

o m

ake

a V

enn

diag

ram

for

thi

sst

atem

ent,

a la

rge

circ

le is

dra

wn

to r

epre

sent

all

anim

als

wit

h lo

ng e

ars.

The

n a

smal

ler

circ

le is

dra

wn

insi

de t

he f

irst

to

repr

esen

t al

l rab

bits

.The

Ven

n di

agra

msh

ows

that

eve

ry r

abbi

t is

incl

uded

in t

he g

roup

of

long

-ear

ed a

nim

als.

The

set

of

rabb

its

is c

alle

d a

subs

et o

f th

e se

t of

long

-ear

ed a

nim

als.

The

Ven

n di

agra

m c

an a

lso

expl

ain

how

to

wri

te t

hest

atem

ent,

“All

rabb

its

have

long

ear

s,”

in if

-the

n fo

rm.E

very

rabb

it is

in t

he g

roup

of

long

-ear

ed a

nim

als,

so if

an

anim

al is

a ra

bbit

,the

n it

has

long

ear

s.

For

eac

h s

tate

men

t,d

raw

a V

enn

dia

gram

.Th

en w

rite

th

e se

nte

nce

in

if-

then

for

m.

1.E

very

dog

has

long

hai

r.2.

All

rati

onal

num

bers

are

rea

l.If

an a

nim

al is

a d

og,t

hen

itIf

a nu

mbe

r is

rat

iona

l,th

enha

s lo

ng h

air.

it is

rea

l.

3.Pe

ople

who

live

in I

owa

like

corn

.4.

Staf

f m

embe

rs a

re a

llow

ed in

the

fa

cult

y lo

unge

.

If a

pers

on li

ves

in Io

wa,

then

the

per

son

likes

cor

n.If

a pe

rson

is a

sta

ff m

embe

r,th

en t

he p

erso

n is

allo

wed

inth

e fa

culty

caf

eter

ia.

peop

le in

the

cafe

teria staf

fm

embe

rs

peop

le w

holik

e co

rn

peop

le w

holiv

e in

Iow

a

real

num

bers

ratio

nal

num

bers

anim

als

with

long

hai

r

dogs

anim

als

with

long

ear

s

rabb

itsEn

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-3

2-3



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Ded

uctiv

e R

easo

ning

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill75

Gle

ncoe

Geo

met

ry

Lesson 2-4

Law

of

Det

ach

men

tD

edu

ctiv

e re

ason

ing

is t

he p

roce

ss o

f us

ing

fact

s,ru

les,

defi

niti

ons,

or p

rope

rtie

s to

rea

ch c

oncl

usio

ns.O

ne f

orm

of

dedu

ctiv

e re

ason

ing

that

dra

ws

conc

lusi

ons

from

a t

rue

cond

itio

nal p

→q

and

a tr

ue s

tate

men

t p

is c

alle

d th

e L

aw o

fD

etac

hm

ent.

Law

of

Det

achm

ent

If p

→q

is tr

ue a

nd p

is tr

ue, t

hen

qis

true

.

Sym

bols

[(p

→q)

] $p]

→q

Th

e st

atem

ent

If t

wo

an

gles

are

su

pple

men

tary

to

the

sam

e a

ngl

e,th

en t

hey

are

con

gru

ent

is a

tru

e co

nd

itio

nal

.Det

erm

ine

wh

eth

er e

ach

con

clu

sion

is v

alid

bas

ed o

n t

he

give

n i

nfo

rmat

ion

.Exp

lain

you

r re

ason

ing.

a.G

iven

:!A

and

!C

are

sup

ple

men

tary

to

!B

.C

oncl

usi

on:!

Ais

con

gru

ent

to !

C.

The

sta

tem

ent

!A

and

!C

are

sup

plem

enta

ry t

o !

Bis

th

e hy

poth

esis

of

the

cond

itio

nal.

The

refo

re,b

y th

e L

awof

Det

achm

ent,

the

conc

lusi

on is

tru

e.

b.G

iven

:!A

is c

ongr

uen

t to

!C

.C

oncl

usi

on:!

Aan

d !

Car

e su

pp

lem

enta

ry t

o !

B.

The

sta

tem

ent

!A

is

cong

ruen

t to

!C

is n

ot t

he h

ypot

hesi

s of

the

con

diti

onal

,so

the

Law

of

Det

achm

ent

cann

ot b

e us

ed.

The

con

clus

ion

is n

ot v

alid

.

Det

erm

ine

wh

eth

er e

ach

con

clu

sion

is

vali

d b

ased

on

th

e tr

ue

con

dit

ion

al g

iven

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If t

wo

angl

es a

re c

ompl

emen

tary

to

the

sam

e an

gle,

then

the

ang

les

are

cong

ruen

t.

1.G

iven

:!A

and

!C

are

com

plem

enta

ry t

o !

B.

Con

clu

sion

:!A

is c

ongr

uent

to

!C

.

The

give

n st

atem

ent

is t

he h

ypot

hesi

s of

the

con

ditio

nal s

tate

men

t.S

ince

the

con

ditio

nal i

s tr

ue,t

he c

oncl

usio

n!

A%

!C

is t

rue.

2.G

iven

:!A

"!

CC

oncl

usi

on:!

Aan

d !

Car

e co

mpl

emen

ts o

f !B

.

The

give

n st

atem

ent

is n

ot t

he h

ypot

hesi

s of

the

con

ditio

nal.

Ther

efor

e,th

e co

nclu

sion

is in

valid

.

3.G

iven

:!E

and

!F

are

com

plem

enta

ry t

o !

G.

Con

clu

sion

:!E

and

!F

are

vert

ical

ang

les.

Whi

le t

he g

iven

sta

tem

ent

is t

he h

ypot

hesi

s of

the

con

ditio

nal s

tate

men

t,th

e st

atem

ent

that

!E

and

!F

are

vert

ical

ang

les

is n

ot t

he c

oncl

usio

n of

the

cond

ition

al.T

he c

oncl

usio

n is

inva

lid.

ADE

F

H J

C

GB

Exam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

oe/M

cGra

w-H

ill76

Gle

ncoe

Geo

met

ry

Law

of

Syllo

gis

mA

noth

er w

ay t

o m

ake

a va

lid c

oncl

usio

n is

to

use

the

Law

of

Syl

logi

sm.I

t is

sim

ilar

to t

he T

rans

itiv

e P

rope

rty.

Law

of

Syl

logi

smIf

p→

qis

true

and

q→

ris

true

, the

n p

→r

is a

lso

true

.

Sym

bols

[(p

→q)

] $(q

→r)

] →(p

→r)

Th

e tw

o co

nd

itio

nal

sta

tem

ents

bel

ow a

re t

rue.

Use

th

e L

aw o

fS

yllo

gism

to

fin

d a

val

id c

oncl

usi

on.S

tate

th

e co

ncl

usi

on.

(1)

If a

num

ber

is a

who

le n

umbe

r,th

en t

he n

umbe

r is

an

inte

ger.

(2)

If a

num

ber

is a

n in

tege

r,th

en it

is a

rat

iona

l num

ber.

p:A

num

ber

is a

who

le n

umbe

r.q:

A n

umbe

r is

an

inte

ger.

r:A

num

ber

is a

rat

iona

l num

ber.

The

tw

o co

ndit

iona

l sta

tem

ents

are

p→

qan

d q

→r.

Usi

ng t

he L

aw o

f Sy

llogi

sm,a

val

idco

nclu

sion

is p

→r.

A s

tate

men

t of

p→

ris

“if

a n

umbe

r is

a w

hole

num

ber,

then

it is

ara

tion

al n

umbe

r.”

Det

erm

ine

wh

eth

er y

ou c

an u

se t

he

Law

of

Syl

logi

sm t

o re

ach

a v

alid

con

clu

sion

from

eac

h s

et o

f st

atem

ents

.

1.If

a d

og e

ats

Supe

rdog

Dog

Foo

d,he

will

be

happ

y.R

over

is h

appy

.N

o co

nclu

sion

is p

ossi

ble.

2.If

an

angl

e is

sup

plem

enta

ry t

o an

obt

use

angl

e,th

en it

is a

cute

.If

an

angl

e is

acu

te,t

hen

its

mea

sure

is le

ss t

han

90.

A c

oncl

usio

n is

pos

sibl

e:If

an a

ngle

is s

uppl

emen

tary

to

an o

btus

ean

gle,

then

its

mea

sure

is le

ss t

han

90.

3.If

the

mea

sure

of !

Ais

less

tha

n 90

,the

n !

Ais

acu

te.

If !

Ais

acu

te,t

hen

!A

"!

B.

A c

oncl

usio

n is

pos

sibl

e:If

the

mea

sure

of

!A

is le

ss 9

0,th

en !

A%

!B

.

4.If

an

angl

e is

a r

ight

ang

le,t

hen

the

mea

sure

of

the

angl

e is

90.

If t

wo

lines

are

per

pend

icul

ar,t

hen

they

for

m a

rig

ht a

ngle

.A

con

clus

ion

is p

ossi

ble:

If tw

o lin

es a

re p

erpe

ndic

ular

,the

n th

e an

gle

they

form

has

mea

sure

90.

5.If

you

stu

dy f

or t

he t

est,

then

you

will

rec

eive

a h

igh

grad

e.Yo

ur g

rade

on

the

test

is h

igh.

No

conc

lusi

on is

pos

sibl

e.

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Ded

uctiv

e R

easo

ning

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

Ded

uctiv

e R

easo

ning

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill77

Gle

ncoe

Geo

met

ry

Lesson 2-4

Det

erm

ine

wh

eth

er t

he

stat

ed c

oncl

usi

on i

s va

lid

bas

ed o

n t

he

give

n i

nfo

rmat

ion

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If t

he

sum

of

the

mea

sure

s of

tw

o a

ngl

es i

s 18

0,th

en t

he

an

gles

are

su

pple

men

tary

.

1.G

iven

:m!

A#

m!

Bis

180

.C

oncl

usi

on:!

Aan

d !

Bar

e su

pple

men

tary

.Va

lid;t

he c

oncl

usio

n fo

llow

s by

the

Law

of

Det

achm

ent.

2.G

iven

:m!

AB

Cis

95

and

m!

DE

Fis

90.

Con

clu

sion

:!A

BC

and

!D

EF

are

supp

lem

enta

ry.

Inva

lid;t

he g

iven

info

rmat

ion

show

s th

at t

he s

um o

f th

e an

gle

mea

sure

sis

185

,not

180

.You

can

not

use

#p

and

p→

qto

con

clud

e q.

3.G

iven

:!1

and

!2

are

a lin

ear

pair

.C

oncl

usi

on:!

1 an

d !

2 ar

e su

pple

men

tary

.Va

lid;s

ince

the

sum

of

the

mea

sure

s of

the

ang

les

of a

line

ar p

air

is 1

80,

the

angl

es a

re s

uppl

emen

tary

.

Use

th

e L

aw o

f S

yllo

gism

to

det

erm

ine

wh

eth

er a

val

id c

oncl

usi

on c

an b

e re

ach

edfr

om e

ach

set

of

stat

emen

ts.I

f a

vali

d c

oncl

usi

on i

s p

ossi

ble,

wri

te i

t.

4.If

tw

o an

gles

are

com

plem

enta

ry,t

hen

the

sum

of

thei

r m

easu

res

is 9

0.If

the

sum

of

the

mea

sure

s of

tw

o an

gles

is 9

0,th

en b

oth

of t

he a

ngle

s ar

e ac

ute.

If tw

o an

gles

are

com

plem

enta

ry,t

hen

both

of

the

angl

es a

re a

cute

.

5.If

the

hea

t w

ave

cont

inue

s,th

en a

ir c

ondi

tion

ing

will

be

used

mor

e fr

eque

ntly

.If

air

con

diti

onin

g is

use

d m

ore

freq

uent

ly,t

hen

ener

gy c

osts

will

be

high

er.

If th

e he

at w

ave

cont

inue

s,th

en e

nerg

y co

sts

will

be

high

er.

Det

erm

ine

wh

eth

er s

tate

men

t (3

) fo

llow

s fr

om s

tate

men

ts (

1) a

nd

(2)

by

the

Law

of D

etac

hm

ent

or t

he

Law

of

Syl

logi

sm.I

f it

doe

s,st

ate

wh

ich

law

was

use

d.I

f it

doe

s n

ot,w

rite

in

vali

d.

6.(1

) If

it is

Tue

sday

,the

n M

arla

tut

ors

chem

istr

y.(2

) If

Mar

la t

utor

s ch

emis

try,

then

she

arr

ives

hom

e at

4 P

.M.

(3)

If M

arla

arr

ives

at

hom

e at

4 P

.M.,

then

it is

Tue

sday

.in

valid

7.(1

) If

a m

arin

e an

imal

is a

sta

rfis

h,th

en it

live

s in

the

inte

rtid

al z

one

of t

he o

cean

.(2

) The

inte

rtid

al z

one

is t

he le

ast

stab

le o

f th

e oc

ean

zone

s.(3

) If

a m

arin

e an

imal

is a

sta

rfis

h,th

en it

live

s in

the

leas

t st

able

of

the

ocea

n zo

nes.

yes;

Law

of

Syl

logi

sm

©G

lenc

oe/M

cGra

w-H

ill78

Gle

ncoe

Geo

met

ry

Det

erm

ine

wh

eth

er t

he

stat

ed c

oncl

usi

on i

s va

lid

bas

ed o

n t

he

give

n i

nfo

rmat

ion

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If a

poi

nt

is t

he

mid

poin

t of

a s

egm

ent,

then

it

div

ides

th

e se

gmen

t in

to t

wo

con

gru

ent

segm

ents

.

1.G

iven

:Ris

the

mid

poin

t of

Q !S!.

Con

clu

sion

:Q !R!

"R!

S!Va

lid;s

ince

Ris

the

mid

poin

t of

Q!S!,

the

Law

of

Det

achm

ent

indi

cate

sth

at it

div

ides

Q!S!

into

tw

o co

ngru

ent

segm

ents

.

2.G

iven

:A!B!

"B!

C!C

oncl

usi

on:B

divi

des

A!C!

into

tw

o co

ngru

ent

segm

ents

.In

valid

;the

poi

nts

A,B

,and

Cm

ay n

ot b

e co

lline

ar,a

nd if

the

y ar

e no

t,th

en B

will

not

be

the

mid

poin

t of

A!C!

.

Use

th

e L

aw o

f S

yllo

gism

to

det

erm

ine

wh

eth

er a

val

id c

oncl

usi

on c

an b

e re

ach

edfr

om e

ach

set

of

stat

emen

ts.I

f a

vali

d c

oncl

usi

on i

s p

ossi

ble,

wri

te i

t.

3.If

tw

o an

gles

for

m a

line

ar p

air,

then

the

tw

o an

gles

are

sup

plem

enta

ry.

If t

wo

angl

es a

re s

uppl

emen

tary

,the

n th

e su

m o

f th

eir

mea

sure

s is

180

.If

two

angl

es fo

rm a

line

ar p

air,

then

the

sum

of

thei

r m

easu

res

is 1

80.

4.If

a h

urri

cane

is C

ateg

ory

5,th

en w

inds

are

gre

ater

tha

n 15

5 m

iles

per

hour

.If

win

ds a

re g

reat

er t

han

155

mile

s pe

r ho

ur,t

hen

tree

s,sh

rubs

,and

sig

ns a

re b

low

n do

wn.

If a

hurr

ican

e is

Cat

egor

y 5,

then

tree

s,sh

rubs

,and

sig

ns a

re b

low

n do

wn.

Det

erm

ine

wh

eth

er s

tate

men

t (3

) fo

llow

s fr

om s

tate

men

ts (

1) a

nd

(2)

by

the

Law

of D

etac

hm

ent

or t

he

Law

of

Syl

logi

sm.I

f it

doe

s,st

ate

wh

ich

law

was

use

d.I

f it

doe

s n

ot,w

rite

in

vali

d.

5.(1

) If

a w

hole

num

ber

is e

ven,

then

its

squa

re is

div

isib

le b

y 4.

(2) T

he n

umbe

r I

am t

hink

ing

of is

an

even

who

le n

umbe

r.(3

) The

squ

are

of t

he n

umbe

r I

am t

hink

ing

of is

div

isib

le b

y 4.

yes;

Law

of

Det

achm

ent

6.(1

) If

the

foo

tbal

l tea

m w

ins

its

hom

ecom

ing

gam

e,th

en C

onra

d w

ill a

tten

d th

e sc

hool

danc

e th

e fo

llow

ing

Fri

day.

(2)

Con

rad

atte

nds

the

scho

ol d

ance

on

Fri

day.

(3) T

he f

ootb

all t

eam

won

the

hom

ecom

ing

gam

e.in

valid

7.B

IOLO

GY

If a

n or

gani

sm is

a p

aras

ite,

then

it s

urvi

ves

by li

ving

on

or in

a h

ost

orga

nism

.If

a pa

rasi

te li

ves

in o

r on

a h

ost

orga

nism

,the

n it

har

ms

its

host

.Wha

tco

nclu

sion

can

you

dra

w if

a v

irus

is a

par

asit

e?If

a vi

rus

is a

par

asite

,the

n it

harm

s its

hos

t.

Pra

ctic

e (A

vera

ge)

Ded

uctiv

e R

easo

ning

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csD

educ

tive

Rea

soni

ng

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill79

Gle

ncoe

Geo

met

ry

Lesson 2-4

Pre-

Act

ivit

yH

ow d

oes

ded

uct

ive

reas

onin

g ap

ply

to

hea

lth

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

4 at

the

top

of

page

82

in y

our

text

book

.

Supp

ose

a do

ctor

wan

ts t

o us

e th

e do

se c

hart

in y

our

text

book

to

pres

crib

ean

ant

ibio

tic,

but

the

only

sca

le in

her

off

ice

give

s w

eigh

ts in

pou

nds.

How

can

she

use

the

fact

tha

t 1

kilo

gram

is a

bout

2.2

pou

nds

to d

eter

min

e th

eco

rrec

t do

se f

or a

pat

ient

?S

ampl

e an

swer

:The

doc

tor

can

divi

deth

e pa

tient

’s w

eigh

t in

pou

nds

by 2

.2 t

o fin

d th

e eq

uiva

lent

mas

s in

kilo

gram

s.S

he c

an t

hen

use

the

dose

cha

rt.

Rea

din

g t

he

Less

on

If s

,t,a

nd

uar

e th

ree

stat

emen

ts,m

atch

eac

h d

escr

ipti

on f

rom

th

e li

st o

n t

he

left

wit

h a

sym

boli

c st

atem

ent

from

th

e li

st o

n t

he

righ

t.1.

nega

tion

of t

ea.

s%

u2.

conj

unct

ion

of s

and

ug

b.[(

s→

t) $

s] →

t3.

conv

erse

of s

→t

hc.

&s

→&

u4.

disj

unct

ion

of s

and

ua

d.&

u→

&s

5.L

aw o

f D

etac

hmen

tb

e.&

t6.

cont

rapo

siti

ve o

f s→

tj

f.[(

u→

t) $

(t→

s)] →

(u→

s)7.

inve

rse

of s

→u

cg.

s$

u8.

cont

rapo

siti

ve o

f s→

ud

h.

t→

s9.

Law

of

Syllo

gism

fi.

t10

.neg

atio

n of

&t

ij.

&t

→&

s

11.D

eter

min

e w

heth

er s

tate

men

t (3

) fo

llow

s fr

om s

tate

men

ts (

1) a

nd (

2) b

y th

e L

aw o

fD

etac

hmen

t or

the

Law

of

Syllo

gism

.If

it d

oes,

stat

e w

hich

law

was

use

d.If

it d

oes

not,

wri

te in

vali

d.a.

(1)

Eve

ry s

quar

e is

a p

aral

lelo

gram

.(2

) E

very

par

alle

logr

am is

a p

olyg

on.

(3)

Eve

ry s

quar

e is

a p

olyg

on.

yes;

Law

of

Syl

logi

smb.

(1)I

f tw

o lin

es t

hat

lie in

the

sam

e pl

ane

do n

ot in

ters

ect,

they

are

par

alle

l.(2

) L

ines

!an

d m

lie in

pla

neU

and

do n

ot in

ters

ect.

(3)

Lin

es !

and

mar

e pa

ralle

l.ye

s;La

w o

f D

etac

hmen

tc.

(1)

Perp

endi

cula

r lin

es in

ters

ect

to f

orm

fou

r ri

ght

angl

es.

(2)

!A

,!B

,!C

,and

!D

are

four

rig

ht a

ngle

s.(3

) !

A,!

B,!

C,a

nd !

Dar

e fo

rmed

by

inte

rsec

ting

per

pend

icul

ar li

nes.

inva

lid

Hel

pin

g Y

ou

Rem

emb

er12

.A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Su

ppos

e th

at a

clas

smat

e is

hav

ing

trou

ble

rem

embe

ring

wha

t th

e L

aw o

f D

etac

hmen

t m

eans

?S

ampl

e an

swer

:The

wor

d de

tach

mea

ns t

o ta

ke s

omet

hing

off

of

anot

her

thin

g.Th

e La

w o

f D

etac

hmen

t sa

ys t

hat

whe

n a

cond

ition

al a

nd it

shy

poth

esis

are

bot

h tr

ue,y

ou c

an d

etac

h th

e co

nclu

sion

and

fee

lco

nfid

ent

that

it t

oo is

a t

rue

stat

emen

t.

©G

lenc

oe/M

cGra

w-H

ill80

Gle

ncoe

Geo

met

ry

Valid

and

Fau

lty A

rgum

ents

Con

side

r th

e st

atem

ents

at

the

righ

t.W

hat

conc

lusi

ons

can

you

mak

e?

Fro

m s

tate

men

ts 1

and

3,i

t is

cor

rect

to

conc

lude

tha

t B

oots

purr

s if

it is

hap

py.H

owev

er,i

t is

fau

lty

to c

oncl

ude

from

onl

y st

atem

ents

2 a

nd 3

tha

t B

oots

is h

appy

.The

if-t

hen

form

of

stat

emen

t 3

is I

f a

cat

is h

appy

,the

n it

pur

rs.

Adv

erti

sers

oft

en u

se f

ault

y lo

gic

in s

ubtl

e w

ays

to h

elp

sell

thei

r pr

oduc

ts.B

y st

udyi

ng t

he a

rgum

ents

,you

can

dec

ide

whe

ther

the

arg

umen

t is

val

id o

r fa

ulty

.

Dec

ide

if e

ach

arg

um

ent

is v

alid

or

fau

lty.

1.(1

)If

you

buy

Tuf

f C

ote

lugg

age,

it2.

(1)

If y

ou b

uy T

uff

Cot

e lu

ggag

e,it

will

sur

vive

air

line

trav

el.

will

sur

vive

air

line

trav

el.

(2)

Just

in b

uys

Tuf

f C

ote

lugg

age.

(2)

Just

in’s

lugg

age

surv

ived

air

line

trav

el.

Con

clus

ion:

Just

in’s

lugg

age

will

Con

clus

ion:

Just

in h

as T

uff

Cot

esu

rviv

e ai

rlin

e tr

avel

.va

lidlu

ggag

e.fa

ulty

3.(1

)If

you

use

Cle

ar L

ine

long

4.(1

)If

you

rea

d th

e bo

ok B

eaut

iful

di

stan

ce s

ervi

ce,y

ou w

ill h

ave

B

raid

s,yo

u w

ill b

e ab

le t

o m

ake

clea

r re

cept

ion.

beau

tifu

l bra

ids

easi

ly.

(2)

Ann

a ha

s cl

ear

long

dis

tanc

e (2

)N

ancy

rea

d th

e bo

ok B

eaut

iful

rece

ptio

n.B

raid

s.C

oncl

usio

n:A

nna

uses

Cle

ar L

ine

Con

clus

ion:

Nan

cy c

an m

ake

long

dis

tanc

e se

rvic

e.fa

ulty

beau

tifu

l bra

ids

easi

ly.

valid

5.(1

)If

you

buy

a w

ord

proc

esso

r,yo

u6.

(1)

Gre

at s

wim

mer

s w

ear

Aqu

aLin

ew

ill b

e ab

le t

o w

rite

lett

ers

fast

er.

swim

wea

r.(2

)Ta

nia

boug

ht a

wor

d pr

oces

sor.

(2)

Gin

a w

ears

Aqu

aLin

e sw

imw

ear.

Con

clus

ion:

Tani

a w

ill b

e ab

le t

o C

oncl

usio

n:G

ina

is a

gre

at s

wim

mer

.w

rite

lett

ers

fast

er.

valid

faul

ty

7.W

rite

an

exam

ple

of f

ault

y lo

gic

that

you

have

see

n in

an

adve

rtis

emen

t.S

ee s

tude

nts’

wor

k.

(1)

Boo

ts is

a c

at.

(2)

Boo

ts is

pur

ring

.(3

)A

cat

pur

rs if

it is

hap

py.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-4

2-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Post

ulat

es a

nd P

arag

raph

Pro

ofs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5

©G

lenc

oe/M

cGra

w-H

ill81

Gle

ncoe

Geo

met

ry

Lesson 2-5

Poin

ts, L

ines

, an

d P

lan

esIn

geo

met

ry,a

pos

tula

teis

a s

tate

men

t th

at is

acc

epte

d as

true

.Pos

tula

tes

desc

ribe

fun

dam

enta

l rel

atio

nshi

ps in

geo

met

ry.

Pos

tula

te:

Thr

ough

any

two

poin

ts, t

here

is e

xact

ly o

ne li

ne.

Pos

tula

te:

Thr

ough

any

thre

e po

ints

not

on

the

sam

e lin

e, th

ere

is e

xact

ly o

ne p

lane

.P

ostu

late

:A

line

cont

ains

at l

east

two

poin

ts.

Pos

tula

te:

Apl

ane

cont

ains

at l

east

thre

e po

ints

not

on

the

sam

e lin

e.P

ostu

late

:If

two

poin

ts li

e in

a p

lane

, the

n th

e lin

e co

ntai

ning

thos

e po

ints

lies

in th

e pl

ane.

Pos

tula

te:

If tw

o lin

es in

ters

ect,

then

thei

r in

ters

ectio

n is

exa

ctly

one

poi

nt.

Pos

tula

te:

If tw

o pl

anes

inte

rsec

t, th

en th

eir

inte

rsec

tion

is a

line

.

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is a

lwa

ys,

som

etim

es,o

r n

ever

tru

e.a.

Th

ere

is e

xact

ly o

ne

pla

ne

that

con

tain

s p

oin

ts A

,B,a

nd

C.

Som

etim

es;i

f A

,B,a

nd C

are

colli

near

,the

y ar

e co

ntai

ned

in m

any

plan

es.I

f th

ey a

reno

ncol

linea

r,th

en t

hey

are

cont

aine

d in

exa

ctly

one

pla

ne.

b.P

oin

ts E

and

Far

e co

nta

ined

in

exa

ctly

on

e li

ne.

Alw

ays;

the

firs

t po

stul

ate

stat

es t

hat

ther

e is

exa

ctly

one

line

thr

ough

any

tw

o po

ints

.

c.T

wo

lin

es i

nte

rsec

t in

tw

o d

isti

nct

poi

nts

Man

d N

.N

ever

;the

inte

rsec

tion

of

two

lines

is o

ne p

oint

.

Use

pos

tula

tes

to d

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays

,som

etim

es,o

rn

ever

tru

e.

1.A

line

con

tain

s ex

actl

y on

e po

int.

neve

r

2.N

onco

lline

ar p

oint

s R

,S,a

nd T

are

cont

aine

d in

exa

ctly

one

pla

ne.a

lway

s

3.A

ny t

wo

lines

!an

d m

inte

rsec

t.so

met

imes

4.If

poi

nts

Gan

d H

are

cont

aine

d in

pla

ne M

,the

n G!

H!is

per

pend

icul

ar t

o pl

ane

M.n

ever

5.P

lane

s R

and

Sin

ters

ect

in p

oint

T.n

ever

6.If

poi

nts

A,B

,and

Car

e no

ncol

linea

r,th

en s

egm

ents

A!B!

,B!C!

,and

C!A!

are

cont

aine

d in

exac

tly

one

plan

e.al

way

s

In t

he

figu

re,A!

C!an

d D!

E!ar

e in

pla

ne

Qan

d A!

C!|| D!

E!.

Sta

te t

he

pos

tula

te t

hat

can

be

use

d t

o sh

ow e

ach

st

atem

ent

is t

rue.

7.E

xact

ly o

ne p

lane

con

tain

s po

ints

F,B

,and

E. T

hrou

gh

any

thre

e po

ints

not

on

the

sam

e lin

e,th

ere

isex

actly

one

pla

ne.

8.B

E!"

#lie

s in

pla

ne Q

.If

two

poin

ts li

e in

a p

lane

,the

n th

e lin

e co

ntai

ning

tho

se p

oint

s lie

s in

the

pla

ne.

QB

C

A

DE

F

G

Exam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

oe/M

cGra

w-H

ill82

Gle

ncoe

Geo

met

ry

Para

gra

ph

Pro

ofs

A s

tate

men

t th

at c

an b

e pr

oved

tru

e is

cal

led

a th

eore

m.Y

ou c

anus

e un

defi

ned

term

s,de

fini

tion

s,po

stul

ates

,and

alr

eady

-pro

ved

theo

rem

s to

pro

ve o

ther

stat

emen

ts t

rue.

A lo

gica

l arg

umen

t th

at u

ses

dedu

ctiv

e re

ason

ing

to r

each

a v

alid

con

clus

ion

is c

alle

d a

pro

of.I

n on

e ty

pe o

f pr

oof,

a p

arag

rap

h p

roof

,you

wri

te a

par

agra

ph t

o ex

plai

n w

hy a

stat

emen

t is

tru

e. In "

AB

C,B !

D!is

an

an

gle

bise

ctor

.Wri

te a

p

arag

rap

h p

roof

to

show

th

at !

AB

D%

!C

BD

.B

y de

fini

tion

,an

angl

e bi

sect

or d

ivid

es a

n an

gle

into

tw

o co

ngru

ent

angl

es.S

ince

B !D!

is a

n an

gle

bise

ctor

,!A

BC

is d

ivid

ed in

to t

wo

cong

ruen

t an

gles

.Thu

s,!

AB

D"

!C

BD

.

1.G

iven

tha

t !

A"

!D

and

!D

"!

E,w

rite

a p

arag

raph

pro

of t

o sh

ow t

hat

!A

"!

E.

Sin

ce !

A%

!D

and

!D

%!

E,t

hen

m!

A#

m!

Dan

d m

!D

#m

!E

byth

e de

f.of

con

grue

nce.

So

m!

A#

m!

Eby

the

Tra

nsiti

ve P

rope

rty,

and

!A

%!

Eby

the

def

.of

cong

ruen

ce.

2.It

is g

iven

tha

t B!

C!"

E!F!,

Mis

the

mid

poin

t of

B!C!

,and

Nis

the

m

idpo

int

of E !

F!.W

rite

a p

arag

raph

pro

of t

o sh

ow t

hat

BM

$E

N.

Mis

the

mid

poin

t of

B!C!

and

Nis

the

mid

poin

t of

E!F!,

so B

M#

"1 2" BC

and

EN

#"1 2" E

F.B!

C!%

E!F!so

BC

#E

Fby

the

def

.of

cong

ruen

ce,a

nd "1 2" B

C#

"1 2" EF

by t

he

mul

t.pr

op.T

hus

BM

#E

Nby

the

Tra

nsiti

ve P

rope

rty.

3.G

iven

tha

t S

is t

he m

idpo

int

of Q!

P!,T

is t

he m

idpo

int

of P!

R!,

and

Pis

the

mid

poin

t of

S !T!

,wri

te a

par

agra

ph p

roof

to

show

th

at Q

S$

TR

.P

is t

he m

idpo

int

of Q!

R!so

QP

#P

R."

1 2" QP

#"1 2" P

Rby

th

e M

ultip

licat

ion

Pro

pert

y.S

and

Tar

e th

e m

idpo

ints

of

Q!P!

and

P!R!,r

espe

ctiv

ely,

so Q

S#

"1 2" QP

and

TR#

"1 2" PR

.Th

en T

R#

"1 2" QP

by s

ubst

itutio

n,an

d Q

S#

TRby

the

Tr

ansi

tive

Pro

pert

y.

Q

RT

PS

B

CM E

NF

B

AD

C

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Post

ulat

es a

nd P

arag

raph

Pro

ofs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Post

ulat

es a

nd P

arag

raph

Pro

ofs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5

©G

lenc

oe/M

cGra

w-H

ill83

Gle

ncoe

Geo

met

ry

Lesson 2-5

Det

erm

ine

the

nu

mbe

r of

lin

e se

gmen

ts t

hat

can

be

dra

wn

con

nec

tin

g ea

ch p

air

of p

oin

ts.

1.6

2.15

Det

erm

ine

wh

eth

er t

he

foll

owin

g st

atem

ents

are

alw

ays

,som

etim

es,o

r n

ever

tru

e.E

xpla

in.

3.T

hree

col

linea

r po

ints

det

erm

ine

a pl

ane.

Nev

er;3

non

colli

near

poi

nts

dete

rmin

e a

plan

e.

4.T

wo

poin

ts A

and

Bde

term

ine

a lin

e.A

lway

s;th

roug

h an

y tw

o po

ints

the

re is

exa

ctly

one

line

.

5.A

pla

ne c

onta

ins

at le

ast

thre

e lin

es.

Alw

ays;

a pl

ane

cont

ains

at

leas

t th

ree

poin

ts n

ot o

n th

e sa

me

line,

and

each

pai

r of

the

se d

eter

min

es a

line

.

In t

he

figu

re,D

G#!

"an

d D

P!!

"li

e in

pla

ne

Jan

d H

lies

on

DG

#!" .

Sta

te

the

pos

tula

te t

hat

can

be

use

d t

o sh

ow e

ach

sta

tem

ent

is t

rue.

6.G

and

Par

e co

lline

ar.

Pos

tula

te 2

.1:t

hrou

gh a

ny t

wo

poin

ts,t

here

is e

xact

lyon

e lin

e.

7.Po

ints

D,H

,and

Par

e co

plan

ar.

Pos

tula

te 2

.2;T

hrou

gh a

ny t

hree

poi

nts

not

on t

he s

ame

line,

ther

e is

exac

tly o

ne p

lane

.

8.PR

OO

FIn

the

fig

ure

at t

he r

ight

,poi

nt B

is t

he m

idpo

int

of A!

C!an

d po

int

Cis

the

mid

poin

t of

B !D!

.Wri

te a

par

agra

ph p

roof

to

prov

e th

atA

B$

CD

.

Giv

en:B

is t

he m

idpo

int

of A!

C!.

Cis

the

mid

poin

t of

B!D!

.P

rove

: AB

#C

DP

roof

:Sin

ce B

is t

he m

idpo

int

of A!

C!an

d C

is t

he m

idpo

int

of B!

D!,w

ekn

ow b

y th

e M

idpo

int T

heor

em t

hat

A!B!

%B!

C!an

d B!

C!%

C!D!

.Sin

ceco

ngru

ent

segm

ents

hav

e eq

ual m

easu

res,

AB

#B

Can

d B

C#

CD

.Thu

s,by

the

Tra

nsiti

ve P

rope

rty

of E

qual

ity, A

B#

CD

.

DC

BAJ

PD

H

G

©G

lenc

oe/M

cGra

w-H

ill84

Gle

ncoe

Geo

met

ry

Det

erm

ine

the

nu

mbe

r of

lin

e se

gmen

ts t

hat

can

be

dra

wn

con

nec

tin

g ea

ch p

air

of p

oin

ts.

1.21

2.28

Det

erm

ine

wh

eth

er t

he

foll

owin

g st

atem

ents

are

alw

ays

,som

etim

es,o

r n

ever

tru

e.E

xpla

in.

3.T

he in

ters

ecti

on o

f tw

o pl

anes

con

tain

s at

leas

t tw

o po

ints

.A

lway

s;th

e in

ters

ectio

n of

tw

o pl

anes

is a

line

,and

a li

ne c

onta

ins

atle

ast

two

poin

ts.

4.If

thr

ee p

lane

s ha

ve a

poi

nt in

com

mon

,the

n th

ey h

ave

a w

hole

line

in c

omm

on.

Som

etim

es;t

hey

mig

ht h

ave

only

tha

t si

ngle

poi

nt in

com

mon

.

In t

he

figu

re,l

ine

man

d T

Q!!

"li

e in

pla

ne

A.S

tate

th

e p

ostu

late

th

at c

an b

e u

sed

to

show

th

at e

ach

sta

tem

ent

is t

rue.

5.L

,T,a

nd li

ne m

lie in

the

sam

e pl

ane.

Pos

tula

te 2

.5:I

f tw

o po

ints

lie

in a

pla

ne,t

hen

the

entir

e lin

e co

ntai

ning

tho

se p

oint

s lie

s in

tha

t pl

ane.

6.L

ine

man

d S!T!

inte

rsec

t at

T.

Pos

tula

te 2

.6:I

f tw

o lin

es in

ters

ect,

then

the

ir in

ters

ectio

n is

exa

ctly

one

poin

t.

7.In

the

fig

ure,

Eis

the

mid

poin

t of

A!B!

and

C!D!

,and

AB

$C

D.W

rite

a

para

grap

h pr

oof

to p

rove

tha

t A !

E!"

E !D!

.

Giv

en:E

is t

he m

idpo

int

of A!

B!an

d C!

D!A

B#

CD

Pro

ve:A!

E!%

E!D!P

roof

:Sin

ce E

is t

he m

idpo

int

of A!

B!an

d C!

D!,w

e kn

ow b

y th

e M

idpo

int

Theo

rem

,tha

t A!

E!%

E!B!an

d C!

E!%

E!D!.B

y th

e de

finiti

on o

f co

ngru

ent

segm

ents

,AE

#E

B#

"1 2" AB

and

CE

#E

D#

"1 2" CD

.Sin

ce A

B#

CD

,

"1 2" AB

#"1 2" C

Dby

the

Mul

tiplic

atio

n P

rope

rty.

So

AE

#E

D,a

nd b

y th

e

defin

ition

of

cong

ruen

t se

gmen

ts, A!

E!%

E!D!.

8.LO

GIC

Poin

ts A

,B,a

nd C

are

not

colli

near

.Poi

nts

B,C

,and

Dar

e no

t co

lline

ar.P

oint

sA

,B,C

,and

Dar

e no

t co

plan

ar.D

escr

ibe

two

plan

es t

hat

inte

rsec

t in

line

BC

.th

e pl

ane

that

con

tain

s A

,B,a

nd C

and

the

plan

e th

at c

onta

ins

B,C

,an

d D

BEC D

A

Am

TQ

L

S

Pra

ctic

e (A

vera

ge)

Post

ulat

es a

nd P

arag

raph

Pro

ofs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5



Rea

din

g t

o L

earn

Math

emati

csPo

stul

ates

and

Par

agra

ph P

roof

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5

©G

lenc

oe/M

cGra

w-H

ill85

Gle

ncoe

Geo

met

ry

Lesson 2-5

Pre-

Act

ivit

yH

ow a

re p

ostu

late

s u

sed

by t

he

fou

ndi

ng

fath

ers

of t

he

Un

ited

Sta

tes?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

5 at

the

top

of

page

89

in y

our

text

book

.

Post

ulat

es a

re o

ften

des

crib

ed a

s st

atem

ents

tha

t ar

e so

bas

ic a

nd s

o cl

earl

yco

rrec

t th

at p

eopl

e w

ill b

e w

illin

g to

acc

ept

them

as

true

wit

hout

ask

ing

for

evid

ence

or

proo

f.G

ive

a st

atem

ent

abou

t nu

mbe

rs t

hat

you

thin

k m

ost

peop

le w

ould

acc

ept

as t

rue

wit

hout

evi

denc

e.S

ampl

e an

swer

:Eve

rynu

mbe

r is

equ

al t

o its

elf.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

heth

er e

ach

of t

he f

ollo

win

g is

a c

orre

ctor

inc

orre

ctst

atem

ent

of a

geom

etri

c po

stul

ate.

If t

he s

tate

men

t is

inco

rrec

t,re

plac

e th

e un

derl

ined

wor

ds t

o m

ake

the

stat

emen

t co

rrec

t.in

corr

ect;

a.A

pla

ne c

onta

ins

at le

ast

that

do

not

lie o

n th

e sa

me

line.

thre

e po

ints

b.If

in

ters

ect,

then

the

inte

rsec

tion

is a

line

.co

rrec

tin

corr

ect;

c.T

hrou

gh a

ny

not

on t

he s

ame

line,

ther

e is

exa

ctly

one

pla

ne.

thre

e po

ints

d.A

line

con

tain

s at

leas

t .

inco

rrec

t;tw

o po

ints

inco

rrec

t;e.

If t

wo

lines

,t

hen

thei

r in

ters

ecti

on is

exa

ctly

one

poi

nt.

inte

rsec

tf.

Thr

ough

any

tw

o po

ints

,the

re is

on

e lin

e.in

corr

ect;

exac

tly

2.D

eter

min

e w

heth

er e

ach

stat

emen

t is

alw

ays,

som

etim

es,o

r ne

ver

true

.If

the

stat

emen

tis

not

alw

ays

true

,exp

lain

why

.a.

If t

wo

plan

es in

ters

ect,

thei

r in

ters

ecti

on is

a li

ne.

alw

ays

b.T

he m

idpo

int

of a

seg

men

t di

vide

s th

e se

gmen

t in

to t

wo

cong

ruen

t se

gmen

ts.a

lway

sc.

The

re is

exa

ctly

one

pla

ne t

hat

cont

ains

thr

ee c

ollin

ear

poin

ts.

neve

r;S

ampl

ean

swer

:The

re a

re in

finite

ly m

any

plan

es if

the

thr

ee p

oint

s ar

eco

lline

ar,b

ut o

nly

one

plan

e if

the

poin

ts a

re n

onco

lline

ar.

d.If

tw

o lin

es in

ters

ect,

thei

r in

ters

ecti

on is

one

poi

nt.

alw

ays

3.U

se t

he w

alls

,flo

or,a

nd c

eilin

g of

you

r cl

assr

oom

to

desc

ribe

a m

odel

for

eac

h of

the

follo

win

g ge

omet

ric

situ

atio

ns.

a.tw

o pl

anes

tha

t in

ters

ect

in a

line

Sam

ple

answ

er:t

wo

adja

cent

wal

ls t

hat

inte

rsec

t at

an

edge

of

both

wal

ls in

the

cor

ner

of t

he r

oom

b.tw

o pl

anes

tha

t do

not

inte

rsec

tS

ampl

e an

swer

:the

cei

ling

and

the

floor

(or

two

oppo

site

wal

ls)

c.th

ree

plan

es t

hat

inte

rsec

t in

a p

oint

Sam

ple

answ

er:t

he f

loor

(or

cei

ling)

and

two

adja

cent

wal

ls t

hat

inte

rsec

t at

a c

orne

r of

the

flo

or (

or c

eilin

g)

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is t

o re

late

it t

o a

wor

d yo

u al

read

ykn

ow.E

xpla

in h

ow t

he id

ea o

f a m

athe

mat

ical

theo

rem

is r

elat

ed t

o th

e id

ea o

f a s

cien

tific

theo

ry.

Sam

ple

answ

er:S

cien

tists

do

expe

rim

ents

to p

rove

theo

ries

;m

athe

mat

icia

ns u

se d

educ

tive

reas

onin

g to

pro

ve th

eore

ms.

Bot

hpr

oces

ses

invo

lve

usin

g ev

iden

ce to

sho

w th

at c

erta

in s

tate

men

ts a

re tr

ue.

at m

ost

are

para

llelon

e po

int

four

poi

nts

two

plan

estw

o po

ints

©G

lenc

oe/M

cGra

w-H

ill86

Gle

ncoe

Geo

met

ry

Logi

c P

robl

ems

The

fol

low

ing

prob

lem

s ca

n be

sol

ved

by e

limin

atin

g po

ssib

iliti

es.

It m

ay b

e he

lpfu

l to

use

char

ts s

uch

as t

he o

ne s

how

n in

the

fir

stpr

oble

m.M

ark

an X

in t

he c

hart

to

elim

inat

e a

poss

ible

ans

wer

.

Sol

ve e

ach

pro

blem

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-5

2-5

Nan

cyO

livia

Mar

ioK

enji

Pea

chX

XX

Ora

nge

XX

XB

anan

aX

XX

App

leX

XX

1.N

ancy

,Oliv

ia,M

ario

,and

Ken

ji ea

ch h

ave

one

piec

e of

frui

t in

the

ir s

choo

l lun

ch.

The

y ha

ve a

pea

ch,a

n or

ange

,a b

anan

a,an

d an

app

le.M

ario

doe

s no

t ha

ve a

peac

h or

a b

anan

a.O

livia

and

Mar

io ju

stca

me

from

cla

ss w

ith

the

stud

ent

who

has

an a

pple

.Ken

ji an

d N

ancy

are

sit

ting

next

to

the

stud

ent

who

has

a b

anan

a.N

ancy

doe

s no

t ha

ve a

pea

ch.W

hich

stud

ent

has

each

pie

ce o

f fru

it?

Nan

cy—

appl

e,O

livia

—ba

nana

,M

ario

—or

ange

,K

enji—

peac

h

3.M

r.G

uthr

ie,M

rs.H

akoi

,Mr.

Mir

za,a

ndM

rs.R

iva

have

jobs

of d

octo

r,ac

coun

tant

,te

ache

r,an

d of

fice

man

ager

.Mr.

Mir

zaliv

es n

ear

the

doct

or a

nd t

he t

each

er.

Mrs

.Riv

a is

not

the

doc

tor

or t

he o

ffic

em

anag

er.M

rs.H

akoi

is n

ot t

heac

coun

tant

or

the

offi

ce m

anag

er.M

r.G

uthr

ie w

ent

to lu

nch

wit

h th

e do

ctor

.M

rs.R

iva’

s so

n is

a h

igh

scho

ol s

tude

ntan

d is

onl

y se

ven

year

s yo

unge

r th

anhi

s al

gebr

a te

ache

r.W

hich

per

son

has

each

occ

upat

ion?

Mr.

Gut

hrie

—te

ache

r,M

rs.H

akoi

—do

ctor

,M

r.M

irza

—of

fice

man

ager

,M

rs.R

iva—

acco

unta

nt

2.V

icto

r,L

eon,

Kas

ha,a

nd S

heri

eac

h pl

ayon

e in

stru

men

t.T

hey

play

the

vio

la,

clar

inet

,tru

mpe

t,an

d fl

ute.

Sher

i doe

sno

t pl

ay t

he f

lute

.Kas

ha li

ves

near

the

stud

ent

who

pla

ys f

lute

and

the

one

who

pla

ys t

rum

pet.

Leo

n do

es n

ot p

lay

a br

ass

or w

ind

inst

rum

ent.

Whi

chst

uden

t pl

ays

each

inst

rum

ent?

Vic

tor—

flute

,Le

on—

viol

a,K

asha

—cl

arin

et,

She

ri—

trum

pet

4.Y

vett

e,L

ana,

Bor

is,a

nd S

cott

eac

h ha

vea

dog.

The

bre

eds

are

colli

e,be

agle

,po

odle

,and

ter

rier

.Yve

tte

and

Bor

isw

alke

d to

the

libr

ary

wit

h th

e st

uden

tw

ho h

as a

col

lie.B

oris

doe

s no

t ha

ve a

pood

le o

r te

rrie

r.Sc

ott

does

not

hav

e a

colli

e.Y

vett

e is

in m

ath

clas

s w

ith

the

stud

ent

who

has

a t

erri

er.W

hich

stud

ent

has

each

bre

ed o

f do

g?

Yve

tte—

pood

leLa

na—

colli

eB

oris

—be

agle

Sco

tt—

terr

ier



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Alg

ebra

ic P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-6

2-6

©G

lenc

oe/M

cGra

w-H

ill87

Gle

ncoe

Geo

met

ry

Lesson 2-6

Alg

ebra

ic P

roo

fT

he f

ollo

win

g pr

oper

ties

of

alge

bra

can

be u

sed

to ju

stif

y th

e st

eps

whe

n so

lvin

g an

alg

ebra

ic e

quat

ion.

Pro

pert

yS

tate

men

t

Ref

lexi

veF

or e

very

num

ber

a, a

$a.

Sym

met

ric

For

all

num

bers

aan

d b,

if a

$b

then

b$

a.

Tran

sitiv

eF

or a

ll nu

mbe

rs a

, b, a

nd c

, if a

$b

and

b$

cth

en a

$c.

Add

ition

and

Sub

trac

tion

For

all

num

bers

a, b

, and

c, i

f a$

bth

en a

#c

$b

#c

and

a!

c$

b!

c.

Mul

tiplic

atio

n an

d D

ivis

ion

For

all

num

bers

a, b

, and

c, i

f a$

bth

en a

&c

$b

&c,

and

if c

'0

then

"a c"$

"b c" .

Sub

stitu

tion

For

all

num

bers

aan

d b,

if a

$b

then

am

ay b

e re

plac

ed b

y b

in a

ny e

quat

ion

or e

xpre

ssio

n.

Dis

trib

utiv

eF

or a

ll nu

mbe

rs a

, b, a

nd c

, a(b

#c)

$ab

#ac

.

Sol

ve 6

x$

2(x

!1)

#30

.

Alg

ebra

ic S

teps

Pro

pert

ies

6x#

2(x

!1)

$30

Giv

en

6x#

2x!

2 $

30D

istr

ibut

ive

Pro

pert

y

8x!

2 $

30S

ubst

itutio

n

8x!

2 #

2 $

30 #

2A

dditi

on P

rope

rty

8x$

32S

ubst

itutio

n

"8 8x "$

"3 82 "D

ivis

ion

Pro

pert

y

x$

4S

ubst

itutio

n

Com

ple

te e

ach

pro

of.

Exam

ple

Exam

ple

Exercis

esExercis

es

1.G

iven

:"4x

2#6

"$

9P

rove

:x$

3

Stat

emen

tsR

easo

ns

a."4x

2#6

"$

9a.

Giv

en

b.2 '"

4x2#

6"

($2(

9)b.

Mul

t.P

rop.

c.4x

#6

$18

c.S

ubst

.d.

4x#

6 !

6 $

18 !

6d.

Sub

tr.P

rop.

e.4x

$12

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bsti

tuti

on

f."4 4x "

$"1 42 "

f.D

iv.P

rop.

g.x

#3

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bsti

tuti

on

2.G

iven

:4x

#8

$x

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ve:x

$!

2

Stat

emen

tsR

easo

ns

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$x

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iven

b.4x

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!x

$x

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!x

b.S

ubtr

.Pro

p.c.

3x#

8 $

2c.

Subs

titu

tion

d.3x

$8

!8

#2

!8

d.Su

btr.

Pro

p.

e.3x

#!

6e.

Subs

titu

tion

f."3 3x "

$"! 36 "

f.D

iv.P

rop.

g.x

#!

2g.

Subs

titu

tion

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lenc

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met

ry

Geo

met

ric

Pro

of

Geo

met

ry d

eals

wit

h nu

mbe

rs a

s m

easu

res,

so g

eom

etri

c pr

oofs

use

prop

erti

es o

f nu

mbe

rs.H

ere

are

som

e of

the

alg

ebra

ic p

rope

rtie

s us

ed in

pro

ofs.

Pro

pert

yS

egm

ents

Ang

les

Ref

lexi

veA

B$

AB

m!

A$

m!

A

Sym

met

ric

If A

B$

CD

, the

n C

D$

AB

.If

m!

A$

m!

B, t

hen

m!

B$

m!

A.

Tran

sitiv

eIf

AB

$C

Dan

d C

D$

EF,

then

AB

$E

F.If

m!

1$m

!2

and

m!

2 $

m!

3, th

en m

!1$

m!

3.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:m!

1 $

m!

2,m

!2

$m

!3

Pro

ve:m

!1

$m

!3

Pro

of:

Stat

emen

tsR

easo

ns

1.m

!1

$m

!2

1.G

iven

2.m

!2

$m

!3

2.G

iven

3.m

!1

$m

!3

3.T

rans

itiv

e P

rope

rty

Sta

te t

he

pro

per

ty t

hat

ju

stif

ies

each

sta

tem

ent.

1.If

m!

1 $

m!

2,th

en m

!2

$m

!1.

Sym

met

ric

Pro

pert

y

2.If

m!

1$90

and

m!

2 $

m!

1,th

en m

!2$

90.

Sub

stitu

tion

3.If

AB

$R

San

d R

S$

WY

,the

n A

B$

WY

.Tr

ansi

tive

Pro

pert

y

4.If

AB

$C

D,t

hen

"1 2" AB

$"1 2" C

D.

Mul

tiplic

atio

n P

rope

rty

5.If

m!

1 #

m!

2 $

110

and

m!

2 $

m!

3,th

en m

!1

#m

!3

$11

0.S

ubst

itutio

n

6.R

S$

RS

Ref

lexi

ve P

rope

rty

7.If

AB

$R

San

d T

U$

WY

,the

n A

B#

TU

$R

S#

WY

.A

dditi

on P

rope

rty

8.If

m!

1 $

m!

2 an

d m

!2

$m

!3,

then

m!

1 $

m!

3.Tr

ansi

tive

Pro

pert

y

9.A

for

mul

a fo

r th

e ar

ea o

f a

tria

ngle

is

A$

"1 2" bh.

Pro

ve t

hat

bhis

equ

al

to 2

tim

es t

he a

rea

of t

he t

rian

gle.

DA

R

TS

CB

12 3

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Alg

ebra

ic P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-6

2-6

Exam

ple

Exam

ple

Exercis

esExercis

es

Giv

en:A

#"1 2" b

hP

rove

:2A

#bh

Sta

tem

ents

Rea

sons

1.A

#"1 2" b

h1.

Giv

en

2.2A

#2 &"1 2" b

h '2.

Mul

t.P

rope

rty

3.2 A

#bh

3.S

ubst

itutio

n



Skil

ls P

ract

ice

Alg

ebra

ic P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-6

2-6

©G

lenc

oe/M

cGra

w-H

ill89

Gle

ncoe

Geo

met

ry

Lesson 2-6

Sta

te t

he

pro

per

ty t

hat

ju

stif

ies

each

sta

tem

ent.

1.If

80

$m

!A

,the

n m

!A

$80

.S

ymm

etri

c P

rope

rty

2.If

RS

$T

Uan

d T

U$

YP

,the

n R

S$

YP

.Tr

ansi

tive

Pro

pert

y

3.If

7x

$28

,the

n x

$4.

Div

isio

n P

rope

rty

4.If

VR

#T

Y$

EN

#T

Y,t

hen

VR

$E

N.

Sub

trac

tion

Pro

pert

y

5.If

m!

1 $

30 a

nd m

!1

$m

!2,

then

m!

2 $

30.

Sub

stitu

tion

Pro

pert

y

Com

ple

te t

he

foll

owin

g p

roof

.

6.G

iven

:8x

!5

$2x

#1

Pro

ve:x

$1

Pro

of:

Stat

emen

tsR

easo

ns

a.8x

!5

$2x

#1

a.G

iven

b.8x

!5

!2x

$2x

#1

!2x

b.S

ubtr

actio

n P

rope

rty

c.6x

!5

#1

c.Su

bsti

tuti

on P

rope

rty

d.6x

!5

$5

#1

$5

d.A

ddit

ion

Pro

pert

y

e.6x

$6

e.S

ubst

itutio

n P

rope

rty

f."6 6x "

$"6 6"

f.D

ivis

ion

Pro

pert

y

g.x

#1

g.S

ubst

itutio

n P

rope

rty

Wri

te a

tw

o-co

lum

n p

roof

for

th

e fo

llow

ing.

7.If

P!Q!

"Q!

S!an

d Q!

S!"

S!T!,t

hen

PQ

$S

T.

Giv

en:P!

Q!%

Q!S!

and

Q!S!

%S!T!

Pro

ve:PP

Q#

ST

Pro

of:

Sta

tem

ents

Rea

sons

1.P!Q!

%Q!

S!1.

Giv

enQ!

S!%

S!T!2.

PQ

#Q

S2.

Def

initi

on o

f co

ngru

ent

segm

ents

QS

#S

T3.

PQ

#S

T3.

Tran

sitiv

e P

rope

rty

QP

TS

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lenc

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Geo

met

ry

PRO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

1.If

m!

AB

C#

m!

CB

D$

90,m

!A

BC

$3x

!5,

and

m!

CB

D$

"x# 2

1"

,the

n x

$27

.

Giv

en:

m!

AB

C$

m!

CB

D#

90m

!A

BC

#3x

!5

m!

CB

D#

"x$ 2

1"

Pro

ve:x

#27

Pro

of:

Sta

tem

ents

Rea

sons

1.m

!A

BC

$m

!C

BD

#90

1.G

iven

m!

AB

C#

3x!

5m

!C

BD

#"x

$ 21

"

2.3x

!5

$"x

$ 21

"#

902.

Sub

stitu

tion

Pro

pert

y

3.(2

)(3x

!5)

$(2

) &"x$ 2

1"

'#(2

)90

3.M

ultip

licat

ion

Pro

pert

y4.

6x!

10 $

x$

1 #

180

4.S

ubst

itutio

n P

rope

rty

5.7x

!9

#18

05.

Sub

stitu

tion

Pro

pert

y6.

7x!

9 $

9 #

180

$9

6.A

dditi

on P

rope

rty

7.7x

#18

97.

Sub

stitu

tion

Pro

pert

y

8."7 7x "

#"18 79 "

8.D

ivis

ion

Pro

pert

y9.

x#

279.

Sub

stitu

tion

Pro

pert

y

2.FI

NA

NC

ET

he f

orm

ula

for

sim

ple

inte

rest

is I

$pr

t,w

here

Iis

inte

rest

,pis

pri

ncip

al,

ris

rat

e,an

d t

is t

ime.

Solv

e th

e fo

rmul

a fo

r r

and

just

ify

each

ste

p.G

iven

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prt

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ve:r

#" pI t"

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of:

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tem

ents

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sons

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prt

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#"p pr tt "

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ivis

ion

Pro

pert

y

3." pI t"

#r

3.S

ubst

itutio

n P

rope

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#" pI t"

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ymm

etri

c P

rope

rty

A

DC

B

Pra

ctic

e (A

vera

ge)

Alg

ebra

ic P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-6

2-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csA

lgeb

raic

Pro

of

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-6

2-6

©G

lenc

oe/M

cGra

w-H

ill91

Gle

ncoe

Geo

met

ry

Lesson 2-6

Pre-

Act

ivit

yH

ow i

s m

ath

emat

ical

evi

den

ce s

imil

ar t

o ev

iden

ce i

n l

aw?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

6 at

the

top

of

page

94

in y

our

text

book

.

Wha

t ar

e so

me

of t

he t

hing

s th

at la

wye

rs m

ight

use

in p

rese

ntin

g th

eir

clos

ing

argu

men

ts t

o a

tria

l jur

y in

add

itio

n to

evi

denc

e ga

ther

ed p

rior

to

the

tria

l and

tes

tim

ony

hear

d du

ring

the

tri

al?

Sam

ple

answ

er:T

hey

mig

ht t

ell t

he ju

ry a

bout

law

s re

late

d to

the

cas

e,co

urt

rulin

gs,a

nd p

rece

dent

s se

t by

ear

lier

tria

ls.

Rea

din

g t

he

Less

on

1.N

ame

the

prop

erty

illu

stra

ted

by e

ach

stat

emen

t.a.

If a

$4.

75 a

nd 4

.75

$b,

then

a$

b.Tr

ansi

tive

Pro

pert

y of

Equ

ality

b.If

x$

y,th

en x

#8

$y

#8.

Add

ition

Pro

pert

y of

Equ

ality

c.5(

12 #

19)

$5

&12

#5

&19

Dis

trib

utiv

e P

rope

rty

Sub

stitu

tion

Pro

pert

yd.

If x

$5,

then

xm

ay b

e re

plac

ed w

ith

5 in

any

equ

atio

n or

exp

ress

ion.

of E

qual

itye.

If x

$y,

then

8x

$8y

.M

ultip

licat

ion

Pro

pert

y of

Equ

ality

f.If

x$

23.4

5,th

en 2

3.45

$x.

Sym

met

ric

Pro

pert

y of

Equ

ality

g.If

5x

$7,

then

x$

"7 5" .D

ivis

ion

Pro

pert

y of

Equ

ality

h.

If x

$12

,the

n x

!3

$9.

Sub

trac

tion

Pro

pert

y of

Equ

ality

2.G

ive

the

reas

on f

or e

ach

stat

emen

t in

the

fol

low

ing

two-

colu

mn

proo

f.G

iven

:5(n

!3)

$4(

2n!

7) !

14P

rove

:n$

9St

atem

ents

Rea

son

s

1.5(

n!

3) $

4(2n

!7)

!14

1.G

iven

2.5n

!15

$8n

!28

!14

2.D

istr

ibut

ive

Pro

pert

y3.

5n!

15 $

8n!

423.

Sub

stitu

tion

Pro

pert

y4.

5n!

15 #

15 $

8n!

42 #

154.

Add

ition

Pro

pert

y5.

5n$

8n!

275.

Sub

stitu

tion

Pro

pert

y6.

5n!

8n$

8n!

27 !

8n6.

Sub

trac

tion

Pro

pert

y7.

!3n

$!

277.

Sub

stitu

tion

Pro

pert

y

8."! !

3 3n "$

"! !2 37 "

8.D

ivis

ion

Pro

pert

y

9.n

$9

9.S

ubst

itutio

n P

rope

rty

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r m

athe

mat

ical

ter

ms

is t

o re

late

the

m t

o w

ords

you

alr

eady

kno

w.

Giv

e an

eve

ryda

y w

ord

that

is r

elat

ed in

mea

ning

to

the

mat

hem

atic

al t

erm

ref

lexi

vean

dex

plai

n ho

w t

his

wor

d ca

n he

lp y

ou t

o re

mem

ber

the

Ref

lexi

ve P

rope

rty

and

to d

isti

ngui

shit

from

the

Sym

met

ric

and

Tra

nsit

ive

Pro

pert

ies.

Sam

ple

answ

er:R

efle

ctio

n:If

you

look

at

your

ref

lect

ion,

you

see

your

self.

The

Ref

lexi

ve P

rope

rty

says

tha

tev

ery

num

ber

is e

qual

to

itsel

f.Th

e R

efle

xive

Pro

pert

y in

volv

es o

nly

one

num

ber,

whi

le t

he S

ymm

etri

c an

d Tr

ansi

tive

Pro

pert

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each

invo

lve

two

or t

hree

num

bers

.

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Geo

met

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Sym

met

ric,

Ref

lexi

ve,a

nd T

rans

itive

Pro

pert

ies

Equ

alit

y ha

s th

ree

impo

rtan

t pr

oper

ties

.

Ref

lexi

veSy

mm

etri

cT

rans

itiv

e

Oth

er r

elat

ions

hav

e so

me

of t

he s

ame

prop

erti

es.C

onsi

der

the

rela

tion

“is

nex

t to

”fo

r ob

ject

s la

bele

d X

,Y,a

nd Z

.Whi

ch o

f th

epr

oper

ties

list

ed a

bove

are

tru

e fo

r th

is r

elat

ion?

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nex

t to

X.

Fals

eIf

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nex

t to

Y,t

hen

Yis

nex

t to

X.

True

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is n

ext

to Y

and

Yis

nex

t to

Z,t

hen

Xis

nex

t to

Z.

Fals

e

Onl

y th

e sy

mm

etri

c pr

oper

ty is

tru

e fo

r th

e re

lati

on “

is n

ext

to.”

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eac

h r

elat

ion

,sta

te w

hic

h p

rop

erti

es (

sym

met

ric,

refl

exiv

e,tr

an

siti

ve)

are

tru

e.

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the

sam

e si

ze a

s2.

is a

fam

ily d

esce

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t of

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met

ric,

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xive

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ansi

tive

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sitiv

e

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he s

ame

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as

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iden

tica

l tw

in o

f

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met

ric,

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xive

,sy

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c,tr

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tive

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sitiv

e

5.is

war

mer

tha

n6.

is o

n th

e sa

me

line

as

tran

sitiv

esy

mm

etri

c,re

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ve

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a s

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r of

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e w

eigh

t as

tran

sitiv

esy

mm

etri

c,re

flexi

vetr

ansi

tive

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ind

two

othe

r ex

ampl

es o

f re

lati

ons,

and

tell

whi

ch p

rope

rtie

s ar

e tr

ue f

orea

ch r

elat

ion.

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stu

dent

s’w

ork.

a$

aIf

a$

b,th

en b

$a.

If a

$b

and

b$

c,th

en a

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rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

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RIO

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___

2-6

2-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Pro

ving

Seg

men

t Rel

atio

nshi

ps

NA

ME

____

____

____

____

____

____

____

____

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____

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ATE

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RIO

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___

2-7

2-7

©G

lenc

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ncoe

Geo

met

ry

Lesson 2-7

Seg

men

t A

dd

itio

nT

wo

basi

c po

stul

ates

for

wor

king

wit

h se

gmen

ts a

nd le

ngth

s ar

eth

e R

uler

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tula

te,w

hich

est

ablis

hes

num

ber

lines

,and

the

Seg

men

t A

ddit

ion

Post

ulat

e,w

hich

des

crib

es w

hat

it m

eans

for

one

poi

nt t

o be

bet

wee

n tw

o ot

her

poin

ts.

Rul

er P

ostu

late

The

poin

ts o

n an

y lin

e or

line

seg

men

t can

be

paire

d w

ith r

eal n

umbe

rs s

o th

at, g

iven

any

two

poin

ts A

and

Bon

a li

ne, A

corr

espo

nds

to z

ero

and

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rres

pond

s to

a p

ositi

ve r

eal n

umbe

r.

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men

t A

dditi

on

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tula

teB

is b

etw

een

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d C

if an

d on

ly if

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.

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te a

tw

o-co

lum

n p

roof

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iven

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rove

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emen

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es

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emen

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ve:P

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emen

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d R

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lenc

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Seg

men

t C

on

gru

ence

Thr

ee p

rope

rtie

s of

alg

ebra

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e R

efle

xive

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met

ric,

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nsit

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pert

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qual

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pert

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n be

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ips

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ts.

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men

t C

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uenc

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eore

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ve, s

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ify

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2-7

2-7

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Pro

ving

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men

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atio

nshi

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ME

____

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ATE

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RIO

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2-7

2-7

©G

lenc

oe/M

cGra

w-H

ill95

Gle

ncoe

Geo

met

ry

Lesson 2-7

Just

ify

each

sta

tem

ent

wit

h a

pro

per

ty o

f eq

ual

ity,

a p

rop

erty

of

con

gru

ence

,or

ap

ostu

late

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lexi

ve P

rope

rty

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qual

ity

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d B!

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n A!

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ansi

tive

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pert

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grue

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wee

n P

and

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hen

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egm

ent

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ition

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tula

te

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d A

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hen

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ubst

itutio

n P

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ple

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ach

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ve:S!

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of:

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emen

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init

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init

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iven

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int

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ve:B !

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efer

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e kn

ows

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stan

ce f

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the

sam

e as

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dis

tanc

efr

om R

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rove

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e di

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qual

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pert

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initi

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son

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ving

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men

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nshi

ps

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ME

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ATE

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RIO

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___

2-7

2-7



Rea

din

g t

o L

earn

Math

emati

csP

rovi

ng S

egm

ent R

elat

ions

hips

NA

ME

____

____

____

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ATE

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RIO

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2-7

2-7

©G

lenc

oe/M

cGra

w-H

ill97

Gle

ncoe

Geo

met

ry

Lesson 2-7

Pre-

Act

ivit

yH

ow c

an s

egm

ent

rela

tion

ship

s be

use

d f

or t

rave

l?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

7 at

the

top

of

page

101

in y

our

text

book

.

•W

hat

is t

he t

otal

dis

tanc

e th

at t

he p

lane

will

fly

to

get

from

San

Die

go t

oD

alla

s?14

30 m

i•

Bef

ore

leav

ing

hom

e,a

pass

enge

r us

ed a

roa

d at

las

to d

eter

min

e th

at t

hedi

stan

ce b

etw

een

San

Die

go a

nd D

alla

s is

abo

ut 1

350

mile

s.W

hy is

the

flyi

ng d

ista

nce

grea

ter

than

tha

t?S

ampl

e an

swer

:Pho

enix

is n

oton

a s

trai

ght

line

betw

een

San

Die

go a

nd D

alla

s,so

the

sto

pad

ded

to t

he d

ista

nce

trav

eled

.A n

onst

op f

light

wou

ld h

ave

been

sho

rter

.

Rea

din

g t

he

Less

on

1.If

Eis

bet

wee

n Y

and

S,w

hich

of

the

follo

win

g st

atem

ents

are

alw

ays

true

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,EA

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S$

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B.Y

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ES

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EC

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SD

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the

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t of

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ive

the

reas

on f

or e

ach

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emen

t in

the

fol

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the

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emen

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easo

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late

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stul

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aigh

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r m

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o as

soci

ate

som

ethi

ng in

the

nam

e of

the

pos

tula

te w

ith

the

cont

ent

of t

he p

ostu

late

.How

can

you

use

this

idea

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dist

ingu

ish

betw

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the

Rul

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the

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men

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:The

re a

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wo

wor

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tula

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thre

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men

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atem

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of t

he R

uler

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tula

tem

entio

ns t

wo

poin

ts,a

nd t

he s

tate

men

t of

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men

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dditi

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2-7

Geo

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wor

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e lin

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e!!

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poi

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line

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one

sid

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n an

gle

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se m

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wo

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d al

l poi

nts

betw

een

them

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lat

figu

re w

ith

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ness

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ends

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fini

tely

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lldi

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ions

.17

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men

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engt

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e!!

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wo

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mon

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m!

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he s

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near

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a !

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he p

oint

whe

re t

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- an

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mee

t.5.

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hose

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ss t

han

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hen

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es.

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ines

tha

t m

eet

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e ar

e !!

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wo

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mm

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10.A

n “a

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a

!!!!

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angl

e.11

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mid

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ne s

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ent.

12.P

oint

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at li

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sam

e pl

ane

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fou

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e.14

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o no

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gles

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med

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s ar

e !!

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angl

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BC

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CO

LL

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2-8

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2-8

2-8



An

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2-8

2-8

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lenc

oe/M

cGra

w-H

ill10

3G

lenc

oe G

eom

etry

Lesson 2-8

Pre-

Act

ivit

yH

ow d

o sc

isso

rs i

llu

stra

te s

up

ple

men

tary

an

gles

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

8 at

the

top

of

page

107

in y

our

text

book

.

Is it

pos

sibl

e to

ope

n a

pair

of

scis

sors

so

that

the

ang

les

form

ed b

y th

e tw

obl

ades

,a b

lade

and

a h

andl

e,an

d th

e tw

o ha

ndle

s,ar

e al

l con

grue

nt?

If s

o,ex

plai

n ho

w t

his

coul

d ha

ppen

.S

ampl

e an

swer

:Yes

;ope

n th

esc

isso

rs s

o th

at t

he t

wo

blad

es a

re p

erpe

ndic

ular

.The

n al

l the

angl

es w

ill b

e ri

ght

angl

es a

nd w

ill b

e co

ngru

ent.

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sen

tenc

e to

for

m a

sta

tem

ent

that

is a

lway

s tr

ue.

a.If

tw

o an

gles

for

m a

line

ar p

air,

then

the

y ar

e ad

jace

nt a

nd

.b.

If t

wo

angl

es a

re c

ompl

emen

tary

to

the

sam

e an

gle,

then

the

y ar

e .

c.If

Dis

a p

oint

in t

he in

teri

or o

f !A

BC

,the

n m

!A

BC

$m

!A

BD

#.

d.G

iven

RS

""#

and

a nu

mbe

r x

betw

een

and

,the

re is

exa

ctly

one

ray

wit

h en

dpoi

nt R

,ext

ende

d on

eit

her

side

of R

S,s

uch

that

the

mea

sure

of

the

angl

efo

rmed

is x

.e.

If t

wo

angl

es a

re c

ongr

uent

and

sup

plem

enta

ry,t

hen

each

ang

le is

a(n

) an

gle.

f.lin

es f

orm

con

grue

nt a

djac

ent

angl

es.

g.“E

very

ang

le is

con

grue

nt t

o it

self

”is

a s

tate

men

t of

the

P

rope

rty

of a

ngle

con

grue

nce.

h.

If t

wo

cong

ruen

t an

gles

for

m a

line

ar p

air,

then

the

mea

sure

of

each

ang

le is

.

i.If

the

non

com

mon

sid

es o

f tw

o ad

jace

nt a

ngle

s fo

rm a

rig

ht a

ngle

,the

n th

e an

gles

are

.

2.D

eter

min

e w

heth

er e

ach

stat

emen

t is

alw

ays,

som

etim

es,o

r ne

ver

true

.a.

Supp

lem

enta

ry a

ngle

s ar

e co

ngru

ent.

som

etim

esb.

If t

wo

angl

es f

orm

a li

near

pai

r,th

ey a

re c

ompl

emen

tary

.ne

ver

c.T

wo

vert

ical

ang

les

are

supp

lem

enta

ry.

som

etim

esd.

Tw

o ad

jace

nt a

ngle

s fo

rm a

line

ar p

air.

som

etim

ese.

Tw

o ve

rtic

al a

ngle

s fo

rm a

line

ar p

air.

neve

rf.

Com

plem

enta

ry a

ngle

s ar

e co

ngru

ent.

som

etim

esg.

Tw

o an

gles

tha

t ar

e co

ngru

ent

to t

he s

ame

angl

e ar

e co

ngru

ent

to e

ach

othe

r.al

way

sh

.C

ompl

emen

tary

ang

les

are

adja

cent

ang

les.

som

etim

es

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Su

ppos

e th

at a

clas

smat

e th

inks

tha

t tw

o an

gles

can

onl

y be

ver

tica

lan

gles

if o

ne a

ngle

lies

abo

ve t

heot

her.

How

can

you

exp

lain

to

him

the

mea

ning

of

vert

ical

ang

les,

usin

g th

e w

ord

vert

exin

you

r ex

plan

atio

n?S

ampl

e an

swer

:Tw

o an

gles

are

ver

tical

angl

es if

the

ysh

are

the

sam

e ve

rtex

and

thei

r si

des

are

oppo

site

ray

s.It

does

n’t

mat

ter

how

the

ang

les

are

posi

tione

d.

com

plem

enta

ry

90

Ref

lexi

veP

erpe

ndic

ular

righ

t

180

0m

!D

BC

cong

ruen

tsu

pple

men

tary

©G

lenc

oe/M

cGra

w-H

ill10

4G

lenc

oe G

eom

etry

Bis

ectin

g a

Hid

den

Ang

leT

he v

erte

x of

!B

AD

at

the

ri

ght

is h

idde

n in

a r

egio

n.W

ithi

n th

e re

gion

,you

are

no

t al

low

ed t

o us

e a

com

pass

.C

an y

ou b

isec

t th

e an

gle?

Fol

low

th

ese

inst

ruct

ion

s to

bis

ect

!B

AD

.

1.U

se a

str

aigh

tedg

e to

dra

w li

nes

CE

and

BD

.

2.C

onst

ruct

the

bis

ecto

rs o

f !D

EC

and

!B

CE

.

3.L

abel

the

inte

rsec

tion

of

the

two

bise

ctor

s as

poi

nt P

.

4.C

onst

ruct

the

bis

ecto

rs o

f !B

DE

and

!D

BC

.

5.L

abel

the

inte

rsec

tion

of

the

two

prev

ious

bis

ecto

rs a

s po

int

Q.

6.U

se a

str

aigh

tedg

e to

dra

w li

ne P

Q,w

hich

bis

ects

the

hid

den

angl

e.

7.A

noth

er h

idde

n an

gle

is

show

n at

rig

ht.C

onst

ruct

the

bise

ctor

usi

ng t

hem

etho

d ab

ove,

or d

evis

eyo

ur o

wn

met

hod.

See

stu

dent

s’w

ork.

AD

E

C

PQ

B

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

2-8

2-8


chapter 2 resource masters - math class · cases, you have used inductive reasoning. however, you...

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