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Unit 1

Proof, Parallel and Perpendicular Lines SpringBoard Geometry Pages 1-100 (add in comma after the course and write the unit and dash

before pages)

Overview

In this unit, students study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. They are then introduced to mathematical proof by applying formal definitions and logical reasoning to develop proofs about basic figures. Finally, students learn how to write equations of parallel and perpendicular lines.

Standards:

Standards in this Unit:

MAFS.912.G-CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MAFS.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

MAFS.912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

MAFS.912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Embedded Assessment 1:

Geometric Figures and Basic Reasoning-The Art

and Math of Folding Paper (page 37-38)

Make use of geometric figures (Lesson 1-1, 1-2)(describe

step by step process to create geometric figure)

Complete two column Algebraic proofs (Lesson 2-1, 2-

2)(write a prove statement when given a statement)

Axiomatic system of geometry(write an if then statement,

given a conditional statement write the converse, inverse,

and contrapositive, write a biconditional statement)(Ex:

That means that from a small, basic set of agreed-upon

assumptions and premises, an entire structure of logic is

devised.) (Lesson 3-1, 3-2, 3-3)

2

Vocabulary (Academic/Math):

compare and contrast, justify, argument, interchange, negate, format, confirm, inductive reasoning,

conjecture, deductive reasoning, proof, theorem, axiomatic system, undefined terms, two-column

proof, conditional statement, hypothesis, conclusion, counterexample, converse, inverse,

contrapositive, truth value, bi-conditional statement, postulates, midpoint, congruent, bisect,

bisector of an angle, parallel, transversal, same-side interior angles, alternate interior angles,

corresponding angles, perpendicular, perpendicular bisector(no bold)


Distance, Midpoint, and Angle Measurement- A

Walk in the Park (61-62) (include “page”)

(no bold)Use Segment, Midpoint, and Angle Bisectors

(include more detail about…used for finding

distance)(Lesson 4-1, 4-2)

Use Distance and Midpoint formulas on a coordinate

plane (Lesson 5-1, 5-2)


Angles, Parallel Lines, and Perpendicular Lines- Graph of Steel (99-100) (include “page”)

(no bold)Create Geometric proofs about line segments and angles (Lesson 6-1, 6-2)

Using and Proving Parallel and perpendicular lines with proofs (Lesson 7-1, 7-2, 7-3)

Writing Equations with the slops of parallel and perpendicular lines (Lesson 8-1, 8-2)

(Refer to finding measure of angles, justify reasoning of equations)

1

Support for Lesson 1-1

Learning Targets for lesson are found on page 3.

Main Ideas for success in 1-1

Identify, describe, and name points, lines, line segments, rays, and planes using correct notation.

Identify and name angles.

Vocabulary used in this lesson includes: point, line, plane, line segment, ray, and angle

Standards for Mathematical Practice to be demonstrated are: Critique the reasoning of others, and Reason abstractly

Practice Support for Lesson 1-1

This example is a model to help solve Practice problem 16.

EXAMPLE:

Identify all possible names of this geometric figure.

Since the figure has points of J, K, M, and L the figure should be called plane JKLM or you can use the figure letter R and

call it plane R.


EXAMPLE:

Identify all rays in the figure.

To identify a ray you have to look for the endpoint that the ray starts at and then extends through a point. You then use

the endpoint as the first letter and the point it extends through as the second point. So there are three rays , , and

.


EXAMPLE:

Explain why in the figure above is not a correct name for .

B is not correct because it describes the entire angle ABD, not the smaller portion of ABC.

2

Practice Support for Lesson 1-1 Continued:

3

Additional Practice:

Answers to additional practice:

1. a. angle; T, STQ, QTS

b. line segment; AC,CA

2. C

4



Main Ideas for success in 1-2.

Describe angles and angle pairs.

Identify and name parts of circles.

Vocabulary used in this lesson includes: “compare and contrast” complementary angles, supplementary angles, acute

angles, right angles, obtuse angles, straight angles, chord, diameter, radius, and justify

Standards for Mathematical Practice to be demonstrated are: Make sense of problems, Model with mathematics,

Construct viable arguments, Reason quantitatively



EXAMPLE:

a. List all possible correct names for segment .

a. Since has the endpoint on the sides of the circle we can call it a chord,

but it also goes through the center of the circle so we can call it a diameter.

b. all possible correct names for segment .

b. Since has its endpoints at the center and on the edge of the circle we call it a radius.

c. List all possible correct names for segment .

c. Since has endpoints on the circle but does not go through the center

we call it a chord.

5


This example is a model to help solve Practice problem 12

EXAMPLE:

Identify the pairs of Supplementary angles in the figure above.

Since supplementary angles have to be 180 degrees we look for linear pairs which equals 180.

PKO, PKM; PKL, LKN; OKL, LKM; PKM, NKM are all linear pairs.


EXAMPLE:

Name the angles that appear to be right in the figure above.

PKL, and NKL because they both equal 90 degrees.


EXAMPLE:

Can two acute angles be Supplementary?

No, for an angle to be acute it must be less than 90 degrees, so therefore two angles less than 90 cannot add to more

than 178 degrees.


EXAMPLE:

Is it possible to draw two non-adjacent angles that share vertex A, and are complementary?

Yes it is possible. As shown below two angles can be non-adjacent (meaning not directly next to each other and touching)

and still have a sum of 90 degrees. If angle 1 was 45 degrees, and angle 2 was 45 degrees then the two would add to 90

and still be non-adjacent.

6

Additional Practice 1-2:

1.

2.

3.

additional practice answers:

1.

2.

3.

7



Main Ideas for success in 2-1:

Make conjectures by applying inductive reasoning.

Recognize the limits of inductive reasoning.

Vocabulary used in this lesson includes: Inductive Reasoning, conjecture,

Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Critique the reasoning of others,

and Make use of structure,


This example is a model to help solve Practice problems 11.

EXAMPLE:

Use inductive reasoning to find the next 3 items in the sequence.

a. A, C, E, G,I, …

. Since we see A then C we skipped B, also we see C then E so we skipped D. therefore J, L, N would be next.

b. 2, 4, 6, 10, 16…

b. Since we see 2 then 4 followed by 6 we might think the sequence is even numbers, but we skip 8 and go to 10 followed

by 16. If we look at it as a sum we see 2+4=6, and 4+6=10, and 6+10=16. Therefore 26, 36, 62 would be the next three

items in the sequence.


EXAMPLE:

Write the first five terms of two different sequences that have 2 as the second term.

1, 2, 3, 4, 5 (you would just need to count up from ones.)

0, 2, 2, 4, 6, 10 (you would start at 0 + 2, and add the two adjacent terms together to get the next.)


EXAMPLE

Generate a sequence with 4 terms using this description: The first term in the sequence is 2, and each other term is three

more than twice the previous term.

(ie. 2 x + 3 if x is the previous term)

2, 7, 17, 37

8



EXAMPLE:

The diagram shows the first three figures in a pattern. Each figure is made of small triangles. How many small triangles will

be in the 4th

figure of the pattern? Support your answer.

We notice that the first is 1 and the second is 4 then 9. From there we try to demine the pattern is the pattern the x is

the number in the sequence ie: and , so from this patern we take =16. So 16 triangles will exist in

the 4th

figure of this sequence.


EXAMPLE:

The first two terms of a number pattern are 3 and 5. Joe conjectures that the next term will be 7. Mike conjectures that

the next term will be 8. Whose conjecture is reasonable? Explain.

Both students can be correct 3+5=8 and with a series of odd numbers we get 3,5,7.

9


1.

2.

3.

additional practice answers:

1.

2.

3.

10




Use deductive reasoning to prove that a conjecture is true.

Develop geometric and algebraic arguments based on deductive reasoning.

Vocabulary used in this lesson includes: argument, deductive reasoning, proof, and theorem

Standards for Mathematical Practice to be demonstrated are: Express regularity in repeated reasoning, Model with mathematics,

Reason abstractly, Construct viable arguments, Critique the reasoning of others, and Reason quantitatively



EXAMPLE:

Use expressions for even and odd integers to confirm the conjecture that the sum of an odd integer and an odd integer is an even

integer.

m + m+2 = 3m+2 and if m is odd all products of (3m+2) will equal an even integer.


EXAMPLE:

Prove this conjecture geometrically: Any even integer can be expressed as the sum of an even integer and an even integer.

A figure representing an even integer can be broken apart into a rectangle representing an even integer and a square representing an

even integer. The even integer can be expressed as the sum of the even integer and the even integer 2.


EXAMPLE

Use deductive reasoning to prove that the solution of the equation x − 4 = − 2 is x = 2. Be sure to justify each step in your proof.

X – 4 = -2 Given

X – 4 + 4 = -2 + 4 Addition Property of Equality

X + 0 = 2 Addition

X = 2 Identity Property

11



EXAMPLE:

Based solely on the pattern in the table, Kevin states that the number of edges of a polyhedron is equal to its number of

Vertices plus the Faces minus 2. Is Andre’s statement a conjecture or a theorem? Explain.

Shape Vertices Faces Edges

Square Pyramid 5 5 8

Square Prism 8 6 12

Tetrahedron 4 4 6

A conjecture. Kevin makes the statement based on three types of polygons. He has not proved that the statement is true

for every polygon. A theorem is an item that first needs to proven to be used as a true statement.


EXAMPLE:

DNA found at a crime scene is consistent with those of an escaped prisoner. Based on this evidence, an investigator

concludes that the suspect was at the crime scene. Is this an example of inductive or deductive reasoning? Explain.

Inductive reasoning, the evidence shows that the hair could have come from the prisoner, but it does not prove that the

suspect was actually at the crime scene. It could have been planted there by a second person. So, the reasoning is

inductive rather than deductive. Deductive reasoning is when your case or idea is based on a proof that is built on rules of

logic

12


1. Use 2p and 2q to represent two even integers. Then (2p)(2q)=2(2pq)

We know that the expression 2pq

represents an integer because when you

find the product of two or more integers,

the result is also an integer. So the

expression “2(2pq)” is an even integer

because it is 2 times an integer.

2. Sample answer: 2x+1 ≤ 7

2x ≤ 6 Subtraction Property of Inequality

x ≤ 3 Division Property of Inequality

5 is not less than or equal to 3 because 5 is

to the right of 3 on the number line. So x =

5 is not in the solution set of x ≤ 3.

3. The student’s statement is a conjecture

because it is a generalization based on a

pattern of data. The statement is not a

theorem because it has not been proved

using deductive reasoning.

4. a. Deductive reasoning. The student’s

conclusion is based on a proof, so the

reasoning is deductive rather than

inductive.

1. Use expressions for even integers to show that the product of two even integers is an even integer.

2. Make use of structure. Use deductive reasoning to prove

that x = 5 is not in the solution set of the inequality 2x+1≤ 7.

Be sure to justify each step in your proof.

3. During the first month of school, students

recorded each day on which they had a quiz in

math class. A student stated that there is a math

quiz every Tuesday morning. Is the student’s

statement a conjecture or a theorem? Explain.

4.


1


Learning Targets for lesson are found on page 25

Main Ideas for success in:

Distinguish between undefined and defined terms

Use properties to complete and write algebraic two-column proofs

Use undefined terms to create definitions of defined terms

Vocabulary used in this lesson includes: axiomatic system, undefined terms, ray, collinear points, coplanar points,

angle, vertex, complementary angles, supplementary angles, two-column proofs.

Standards for Mathematical Practice to be demonstrated are: express regularity in repeated reasoning, construct

viable arguments, look for and make use of structure.



EXAMPLE:

Identify the property that justifies the statement: If , then .

The property that justifies the statement would be the Subtraction Property of Equality because 10 has been

subtracted to both sides of the equation.

-10 -10

This example is a model to help solve Practice problems 9 and 11.

EXAMPLE

Complete the prove statement and write a two-column proof for the equation:

Given: Prove:

Given: Prove:

Statements Reasons

1. 1. Given equation

2. 2. Distributive Property

3. 3. Subtraction Property of Equality

4. 4. Addition Property of Equality

5. 5. Division Property of Equality

Complete algebraic solutions:

2



EXAMPLE:

Explain why ray is considered a defined term in geometry.

Ray is a defined term because it can be defined in terms of the undefined terms line and point. A ray is a part of

a line bounded by one endpoint and extending infinitely in one direction.


EXAMPLE:

Suppose you are given that and . What can you prove by using these statements and the

Substitution Property?

1. The Substitution Property says that if , then can be substituted for in any equation or inequality.

2. Since we know we can use the Substitution Property to substitute 10 for in the first equation.

3. This would give the equation:

4. Using the Subtraction Property of Equality we can subtract 2 from both sides to give the equation:

5. Using the Division Property of Equality we can divide both sides by 4 to find .

6. We are able to prove using the given statements and the Substitution Property.

3


1.

2.

3.

Answers to additional

practice:

1.

a. Given

b. Multiplication

Property of

Equality

c. Addition Property

of Equality

d. Division Property

of Equality

2. C

3. C

4




Identify the hypothesis and conclusion of a conditional statement.

Give counterexamples for false conditional statements.

Restate conditional statements in if-then form.

Vocabulary used in this lesson includes: conditional statement, hypothesis, conclusion, and counterexample.

Standards for Mathematical Practice to be demonstrated are: make use of structure, reason abstractly, critique the

reasoning of others, and construct viable arguments.



EXAMPLE:

Write the statement in if-then form: All 90o angles are right angles.

1. First, identify the hypothesis and the conclusion. When you rewrite the statement in if-then form, you may

need to reword the hypothesis or conclusion.

a. A conditional statement is a statement of logic that combines two statements or facts and can be

written in if-then form. The part of the statement that follows “if” is the hypothesis, and the part

that follows “then” is the conclusion.

2. Hypothesis: The measure of an angle is 90o.

3. Conclusion: It is a right angle.

4. Next, use the hypothesis and conclusion to write the conditional statement as an if-then statement. After

the “if” you write the hypothesis and after the “then” you write the conclusion.

a. If the measure of an angle is 90o, then it is a right angle.


EXAMPLE:

Which of the following is a counterexample of this statement?

"All mammals have legs.”

A. Whales are mammals that do not have legs

B. Cats are mammals that have legs

C. People only have two legs

D. Chairs are not mammals and have legs

A counterexample is an example that can be found for which the hypothesis is true, but the conclusion is false.

So the hypothesis in this statement would be “if an animal is a mammal” and we want this to remain true, but our

conclusion, “the animal will have legs” needs to be false. The counterexample would be “A” because the hypothesis of

the statement remains true, a whale is a mammal, and the conclusion is false, a whale does not have legs.

5



EXAMPLE

Identify the hypothesis and the conclusion of the statement:

If today is Friday, then tomorrow is Saturday.

The letter p and q are often used to represent the hypothesis and conclusion, respectively, in a conditional

statement. The basic form of an if-then statement would then be, “if p, then q”. So the hypothesis is

represented by “p” and follows the word if. The conclusion is represented by “q” and follows the word then.

If today is Friday, then tomorrow is Saturday.

p (hypothesis) q (conclusion)

Hypothesis: today is Friday

Conclusion: tomorrow is Saturday


EXAMPLE:

Christina says that is a counterexample that shows the following conditional statement is false. Is Christina

correct? Explain.

If , then .

Christina is correct because the hypothesis of the conditional is true, but the conclusion is false.

Since , this counterexample shows that the conditional statement is false. A counterexample must

show that the conclusion can be false when the hypothesis is true. In Christina’s example, the hypothesis is true

and the conclusion is false. Once you have found one counterexample for a condition statement the conditional

statement is false.

Tip: The words if and then are

not included when giving the

hypothesis and conclusion.

6


1. Write each statement in if-then form.

a. The only time I wake up early is when I set my alarm clock.

b. I eat breakfast at a restaurant only if it is a weekend.

c. An obtuse angle has a measure between 90° and 180°.

2. State or describe a counterexample for each conditional statement.

a. If then .

b. If three points A, B, and C are collinear, then B is between A and C.

3.


1.

a. If I wake up early, then I set

my alarm clock.

b. If I eat breakfast at a

restaurant, then it is a

weekend.

c. If an angle is obtuse, then its

measure is between 90° and

180°.

2.

a. A counterexample is

.

b. Two counterexamples are a

line with A between B and C

and a line with C between A

and B.

3. D

7




Write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement.

Write and interpret biconditional statements.

Identify logically equivalent statements.

Vocabulary used in this lesson includes: converse, inverse, contrapositive, interchange, negate, truth values, and

biconditional statement.

Standards for Mathematical Practice to be demonstrated are: make use of structure, critique the reasoning of others,

and reason abstractly.


This example is a model to help solve Practice problems 11-13.

EXAMPLE:

Write the converse, inverse, and contrapositive of the following conditional statement.

If it thunders then it is raining.

1. Identify the hypothesis (p) and conclusion (q)

a. Hypothesis (p) – it thunders

b. Conclusion (q) – it is raining

2. Converse: If q, then p. Interchange the hypothesis and conclusion

a. If it is raining, then it thunders.

3. Inverse: If not p, then not q. Negate the hypothesis and negate the conclusion.

a. If it is not thundering then it is not raining.

4. Contrapositive: If not q, then not p. Interchange and negate both the hypothesis and conclusion. Or you could

think about it as negating the converse.

a. If it is not raining then it is not thundering.


EXAMPLE:

Write the definition of coplanar lines as a biconditional statement.

A biconditional statement is a conditional statement and its converse. It typically includes the words “if and only if”

as in a definition. For a biconditional to be a true statement, it must be true both “forward and backward.” Writing a

biconditional statement is equivalent to writing a conditional statement and its converse.

Conditional: If three lines are coplanar, then they lie in the same plane.

Converse: If three lines lie in the same plane, then they are coplanar.

Biconditional: Three lines are coplanar if and only if they lie in the same plane.

8



EXAMPLE

Give an example of a false statement that has a true converse.

You will want a statement that has a false conditional, but the converse is true when you interchange the hypothesis

and conclusion.

Statement: If a polygon has four sides, then the figure is a square. (false)

Converse: If a polygon is a square, then it has four sides. (true)


EXAMPLE:

What conclusions can be drawn from the given statements?

Given: (1) If you exercise regularly, then you have a healthy body.

(2) You do not have a healthy body.

Conclusions: If you do not have a healthy body then you do not exercise regularly.

If you exercise regularly, then you have a healthy body. The first statement is the conditional statement

and the second statement is the beginning of the contrapositive. Since the conditional and the

contrapositive are logically equivalent then the negation of the hypothesis must be true if the negation

of the conclusion is true.

9


1.

2.

3.


1.

a. Inverse: If it is not raining, then I

do not stay indoors;

Contrapositive: If I do not stay

indoors, then it is not raining.

b. Inverse: If I do not have a

hammer, then I do not hammer

in the morning; Contrapositive: If

I do not hammer in the morning,

then I do not have a hammer.

2. If people have the same ZIP code,

then they live in the same

neighborhood. If people live in the

same neighborhood, then they have

the same ZIP code.

3. D

1



Main Ideas for success in lesson # here:

Apply the Segment Addition Postulate to find lengths of segments.

Use the definition of midpoint to find lengths of segments.

Use a ruler to measure the length of a line segment.

Vocabulary used in this lesson includes: axiom, postulate, distance along a line, ruler postulate, segment addition

postulate, midpoint, congruent ( ), and bisect.

Standards for Mathematical Practice to be demonstrated are: attend to precision, reason abstractly, reason

quantitatively, and use appropriate tools strategically.



EXAMPLE:

Given: Point O is between points D and G, , , and . Find the value of .

1. First, draw a picture to help visualize the information given. Draw a segment where D and G are the

endpoints and O is between them.

2. Next, add in the information given about each part of the segment.

3. The Segment Addition Postulate tells us that if point O is between points D and G then . So

we will use this to write an equation and solve for .

Input the values of each segment into the equation.

Combine like terms

Subtract 18 from both sides

Divide both sides by 6

4. The value of is 7.

2



EXAMPLE

If M is the midpoint of , and , find the value of and the length of .

1. First, draw a picture to help visualize the information given. Draw a segment where P and Q are the

endpoints and M is in the middle of them.

2. Next, add in the information given about each part of the segment.

3. Because M is the midpoint of , you know that and . Therefore

4. Now use the Segment Addition Postulate to write an equation where . Use this equation to

solve for .

Input the values of each segment into the equation.

Combine like terms

Subtract from both sides


Divide 4 on both sides

The value of is 10.

5. Once we have the value of , we need to find the length of . Substitute 10 for in the expression for

.

has a length of 58 units.

3



EXAMPLE:

Point D is between points A and B. The distance between points A and D is

of AB. What is the coordinate of point D?

1. First we need to find the distance between A and B. Point A is at and point B is at .

2. Since the distance between A and D is

of AB, we need to find

of AB.

This tells us that point D is 9 units away from A.

3. To find the coordinate of D we need to add 9 units to the coordinate of A.

the coordinate of D is 3


EXAMPLE

Explain how to use a ruler to measure the length of a line segment.

To measure a certain line segment, place the ruler with its edge along the line segment such that the zero mark

of the ruler coincides with an endpoint. Now we read the mark on the ruler which is against the other endpoint.

That mark would be length of the line segment. If the line segment does not start at the “zero” on the ruler, to

find the length of the segment we either count the number of units between the ends of the line segment or

take the absolute value of the difference between the endpoints.


EXAMPLE:

Compare and contrast a postulate and a theorem.

A postulate is understood as true without proof, while a theorem must be proven. A theorem is a proposition that

can be deduced from postulates. Usually postulates provide the starting point for the proof of a theorem. We

make a series of logical arguments using postulates to prove a theorem.

4


1. Suppose point T is between points R and V on a line. If RT = 6.3 units and RV = 13.1 units, then what is TV?

A. 2.5 units

B. 6.8 units

C. 7.8 units

D. 19.4 units

2. Suppose P is between M and N.

a. If , , and , what is the value of ?

b. If , , and , what is the value of ?

3. Use the centimeter ruler shown.

a. What is the length of

b. What number on the ruler represents the midpoint of

4. Points P, M, and T are on a line and PT - PM = MT. Which point is between the other two? Explain your answer.


1. B

2.

a. 5

b. 3

3.

a. 9cm

b. 19.5

4. M is between P and T. Starting

with PT - PM = MT, add PM to

each side to get PT = MT = PM or

PT = PM + MT. That equation

satisfies the situation that M is a

point between P and T.

5



Main Ideas for success in lesson #here:

Apply the Angle Addition Postulate to find angle measures.

Use the definition of angle bisector to find angle measures.

Use a protractor to measure the degree of an angle.

Vocabulary used in this lesson includes: protractor postulate, angle addition postulate, bisector of an angle, adjacent

angles, congruent angles, vertical angles, perpendicular ( ), complementary, and supplementary.

Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, express regularity in

repeated reasoning, construct viable arguments, and critique the reasoning of others Capitalize the first letter



EXAMPLE:

Point K is in the interior of m∠ABC, m∠ABC = , m∠ABK = 42°, and m∠KBC = . What is m∠ABC?

1. First, draw an angle to help visualize the information given. Draw an angle whose vertex is B and the rays

creating the angle are and as shown below. Point K needs to be in the interior of ∠ABC as shown below.

2. Next, add in the information given about each part of the angle.

3. The Angle Addition Postulate tells us that ∠ ∠ ∠ .

∠ ∠ ∠ Angle Addition Postulate

Substitute the values of each

angle into the equation

Combine Like Terms




This example is a model to help solve Practice problems.

∠

∠

∠

∠

4. Once you have the value

of , substitute it into

the equation of ∠ABC.

The measure of angle ABC

is 110.

6



EXAMPLE

bisects ∠ . If ∠ and ∠ , then what is ∠ ?

1. First, draw an angle to help visualize the information

given. Draw an angle whose vertex is Q and the rays

creating the angle are and as shown below.

bisects the angle which means it goes through the

middle of the angle, creating two congruent angles.

2. Next, add in the information given about each part of

the angle.

3. Because bisects ∠ , you know that

∠ ∠ and ∠ ∠ . Therefore

∠ Add this to the diagram. Include

that the congruent symbol means =.

4. Use the Angle Addition Postulate

to write an equation where

∠ ∠ ∠ .


angle and solve for y.

∠ ∠ ∠ Angle Addition Postulate


angle into the equation

Combine Like Terms


Add 2 to both sides


5. Substitute the value of y into the equation for ∠

∠

6.

7



EXAMPLE

∠C and ∠D are supplementary. If ∠ and ∠ , what is the measure of each angle?

1. If ∠C and ∠D are supplementary this means that the sum of their angle measures is 180o. Write an equation

where ∠C + ∠D = 180 and solve for .

∠ ∠ Supplementary angles are angles whose measure sums to 180

Substitute the values of each angle into the equation

Combine like terms



The value of is 43.

2. Substitute the value of into the equations for ∠C and ∠D.

∠ ∠

∠ ∠

∠ ∠

∠

3. Check to make sure your answers give supplementary angles; they add to 180.


EXAMPLE:

Kevin knows that point B is in the interior of ∠ and he knows that ∠ ∠ . What can Kevin conclude from

this information? Explain. Make a sketch that supports your answer.

1. Draw an angle with point F being the vertex and

the rays creating the angle are and .

2. Add so that B is in the interior of the angle

and ∠ ∠

3. Since B is in the interior of ∠ and ∠ ∠ , we can conclude that is the angle bisector of

∠ because the bisector of an angle is a ray that divides the angle into two congruent adjacent

angles, ∠ ∠ .

8


1. Make sense of problems. Suppose that bisects ∠MPN. What conclusion can you make?

2. Suppose bisects ∠CAR. If ∠ and ∠ , what is ∠ ?

3. ∠D and ∠E are complementary. If ∠ and ∠ , what is ?


practice:

1. m∠MPQ = m∠QPN

2. 35°

3. 11

9



Main Ideas for success in lesson# here:

Use the Pythagorean Theorem to derive the Distance Formula.

Use the Distance Formula to find the distance between two points on the coordinate plane.

Vocabulary used in this lesson includes: derive, Pythagorean Theorem, hypotenuse, and Venn diagram.

Standards for Mathematical Practice to be demonstrated are: model with mathematics, attend to precision, express

regularity in repeated reasoning, reason abstractly, and reason quantitatively Capitalize the first letter



EXAMPLE:

Find the distance between the points with the given coordinates.

and

1. The distance d between the points (x1, y1) and (x2, y2) is given by

.

2. Label the points as follows and Therefore, ,

and .

Hint: is read as “x sub one” and it refers to the first x-value.

3. Substitute the values into the Distance Formula.

Distance Formula

, and Substitute into formula

4. Simplify the equation.

Use order of operations to simplify the equation.

Simplify inside the parenthesis first.

Square and before you add explain what

squaring is

Add 16 and 49 before you take the square root

explain square root

**justify why the answer is reasonable for your

example, modeling 23 separately, you may need an

additional pg

The distance between and is which is about 8.06 units.


EXAMPLE:

Besides the Distance Formula, what other method could you use to find the distance between two points in the

coordinate plane?

The structure of the Distance Formula and the Pythagorean Theorem are nearly identical. When using the

Pythagorean Theorem it involves the relationship between the lengths of the legs and the hypotenuse of a

right triangle.

10



EXAMPLE

Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 5). What is the perimeter of the triangle?

1. First, we need to find the length of each side of the triangle. Use the Distance Formula to find these lengths.

2. Use the Distance Formula to find the length of AB.

Distance Formula

Substitute into formula:

, and



Square and before you add refer to ex#_ for how to

square #’s


Take the square root of 9. The length of AB is 3 units.

3. Use the Distance Formula to find the length of AC. Refer to ex#_for finding sq. roots

Distance Formula


, and



Square and before you add refer to ex#_...


Take the square root of 16. The length of AC is 4 units.

4. Use the Distance Formula to find the length of CB.

Distance Formula

Substitute into formula

, and



Square and before you add

Add 9 and 16 before you take the square root Refer to

ex#_for finding sq. roots

Take the square root of 25. The length of CB is 5 units.

5. The perimeter of a triangle is the sum of the lengths of its 3 sides. In this case we will add .

Perimeter = units

11



EXAMPLE

Use the Distance Formula to show that .

1. If , this means that the length of AB is the same

as the length of CD. To find the length of the segments we

use the Distance Formula.

2. Use the Distance Formula to find the length of AB.

Distance Formula


, and



Square and before you add Refer to ex# for …


Take the square root of 26. Refer to ex#_for

finding sq. roots

The length of AB is about 5.1 units.

3. Use the Distance Formula to find the length of CD.

Distance Formula


, and



Square and before you add


Take the square root of 26.

The length of AB is about 5.1 units.

4. So, by the definition of congruent segments, because AB=CD.

12


1. Which expression represents the distance between points and ?

2. The coordinates of the vertices of triangle are , and . Find the perimeter of the triangle.

3. The coordinates of the vertices of a triangle are , and .

a. Find AB.

b. Find BC.

c. Find AC.

d. Bases on the lengths of the sides, what kind of triangle is ?


practice:

1. D

2.

3.

13



Main Ideas for success in lesson# here:

Use inductive reasoning to determine the Midpoint Formula.

Use the Midpoint Formula to find the coordinates of the midpoint of a segment on the coordinate plane.

Vocabulary used in this lesson includes: midpoint.

Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, make use of structure,

reason quantitatively, and make sense of problems capitalize the first letter


This example is a model to help solve Practice problems 9, 10, 11 and 12.

EXAMPLE:

Find the coordinates of the midpoint of the segment with the given endpoints.

and

The midpoint of a segment is the point that divides the segment into two congruent segments. In order to find

the midpoint of a line segment you use the Midpoint Formula:

. You are finding the average of

the x-coordinates and the average of the y-coordinates.

1. Label the points as follows and Therefore, ,

and .

Hint: is read as “x sub one” and it refers to the first x-value.

2. Substitute the values into the Midpoint Formula and simplify.

Midpoint Formula

and Substitute into the formula.

Simplify the numerator.

Simplify the fractions, if possible.

3. The midpoint is located at

14



EXAMPLE

Find and explain the errors that were made in the following calculation of the coordinates of a midpoint. Then fix the

errors and determine the correct answer.

Find the coordinates of the midpoint M of the segment with endpoints X(-4, 2) and Y(0, 8).

The midpoint can be found by averaging the x-coordinates of the two different points and averaging the y-coordinates

of the two different points. The mistakes made was instead of adding the two x-coordinates and adding the two y-

coordinates, the second coordinate was subtracted from the first. The midpoint can be found correctly by changing

the subtraction sign to an addition sign and simplifying.

The correct coordinates of the midpoint are .


EXAMPLE:

AB is graphed on a coordinate plane. Explain how you would determine the coordinates of the point on the segment

that is

of the distance from A to B.

One way to find the coordinates would be to find the midpoint M of AB. Below is an example of a line segment and

its midpoint graphed on a coordinate plane.

Next, you could find the midpoint MB since that would be

of the distance from A to B. So the coordinates of the

point on the segment that is ¾ the distance from A to B would be located halfway between the midpoint of AB and B.

Point D is between points A and B. The distance between points A and D is

of AB. What is the coordinate of point D?

1. First we need to find the distance between A and B. Point A is at -6 and point B is at 6.

2. Since the distance between A and D is

of AB, we need to find

of AB.

15


1. has endpoints and . Find the coordinates of the midpoint of .

2. Which expression represents the midpoint of the line segment with endpoints and ?

3. For the coordinates (5, 8) and (9, 14), one is an endpoint of a line segment and the other is the midpoint. How

many possibilities are there for the other endpoint? Find each one. Explain your method.


1. (4.5, 3.5)

2. C

3.

1




Students use definitions, properties, and theorems to justify a statement.

Write two-column proofs to prove theorems about lines and angles.

Vocabulary used in this lesson includes: properties, postulates, and midpoint of a line segment

Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and

Critique the reasoning of others


Lines CF, DH, and EA intersect at point B. Use this figure for Items 4–8. Write the definition, postulate, or property that

justifies each statement.


EXAMPLE:

If ∠6 is supplementary to ∠ABF, then m∠6 + m∠ABF = 180°. (meaning that two angles have a value that adds up to 180

degrees.)

Definition of supplementary angles (meaning that two angles have a value that adds up to 180 degrees.)


EXAMPLE:

If ∠5 ≅ ∠4, then bisects ∠ABG.

Definition of angle bisector (meaning a ray or line that has split an angle into two equal parts.)


EXAMPLE

DB + BH = DH

Segment Addition Postulate (meaning that you add two segments together to get a larger segment)


EXAMPLE:

If ∠CBH is a right angle, then .

Definition of perpendicular lines (meaning two lines that intersect at a 90 degree or right angle)

2

Practice Support for Lesson Continued:


EXAMPLE

If m∠5 = m∠4, then m∠5 + m∠6 = m∠4 + m∠6.

Addition Property of Equality (meaning you can replace an equal angle in an addition problem and get the same total

angle value.)


EXAMPLE:

Write a statement related to the figure above that can be justified by the Angle Addition Postulate. (meaning the

addition of two angles to get a larger angle value.)

m∠CBH = m∠CBA + m∠ABH

Because the two angles combined complete the angle CBH.

3


1. Use the diagram shown.

Write a statement that can be justified by each of the following:

a. definition of angle bisector

b. Angle Addition Postulate

2. Use the diagram shown.

4




Complete two-column proofs to prove theorems about segments.

Complete two-column proofs to prove theorems about angles.

Vocabulary used in this lesson includes: Vertical angles theorem, and Two Column Proofs

Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and

Attend to precision.



EXAMPLE:

Given: BAC, CAD, DAE, EAF, Ray AC bisects BAD, Ray AE bisects DAF.

Prove:

m BAC, + m EAF = m CAE

Statement Reason

Ray AC bisects BAD

Ray AE bisects DAF

1. Given

m DAE = m EAF

m BAC = m CAD

2. Bisected angles are equal in

measure. (Angle bisector)

m BAC + m EAF = m

CAD + m DAE

3. Angle Addition

(referred to in 6-1)

m CAD + m DAE = m

CAE

4. Angle Addition

m BAC + m EAF = m

CAE

5. Substitution

5


This example is a model to help solve Practice problems 5. (Keep in mind that the proof is about the use of the

segment addition postulate not the actual value of the segments.)

EXAMPLE:

Given: A-B-C-D on

AD

Prove:

AB + BC + CD = AD

Statement

Reason

Given: A-B-C-D on AD

1. Given

AB + BD = AD 2. Segment Addition

BC + CD = BD 3. Segment Addition

AB + BC + CD = AD 4. Substitution


EXAMPLE

6



EXAMPLE:

Statements Reasons

1.

1. Given

2.

2. Each angle has one unique angle bisector.

3.

3. An angle bisector is a ray whose endpoint is the vertex of the angle and which divides the angle into two congruent angles.

4.

4. Reflexive Property. A quantity is congruent to itself.

5.

5. SAS - If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

6.

6. CPCTC - Corresponding parts of congruent triangles are congruent.


EXAMPLE

How do you know that the triangle is item 7 is a Isosceles triangle?

Since Isosceles triangles are by definition triangles that have two sides and two angle’s congruent, and the proof asked us

to prove two angles congruent we know that the triangle must be isosceles.

7

Additional Practice:

1.

2.

Additional Practice Answers:

1. D

2. a. Definition of angle Bisector

b. Given

c. Substitution

d. ∠ is supplementary to ∠ .

e. Definition of supplementary

angles.

3. a. m∠1 =37; m∠PTR =53

b. Angle Addition Postulate

c. m∠2 =16

d. Subtraction Property of Equality

3.

1




Make conjectures about the angles formed by a pair of parallel lines and a transversal.

Students need to prove theorems about these angles.

Vocabulary used in this lesson includes: parallel, transversal, corresponding angles, same-side interior angles,

alternate interior angles, confirms, and means parallel.

Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Construct

viable arguments, Express regularity in repeated reasoning, and Use appropriate tools strategically


In the diagram, a || b (meaning that line a is parallel to line b, also meaning that the lines run side by side and do not

touch.)Use the diagram for Items 15–20. Determine whether each statement in Items 15–18 is true or false. Justify your

response with the appropriate postulate or theorem.


EXAMPLE:

7 is supplementary to 6 .

True; Same-Side Interior Angles Theorem (meaning that the angles are located on the inside of the two lines and are on

the same side of the transversal c).


EXAMPLE:

1

True, Corresponding Angles Theorem (meaning that the angles are in the same general location in there pod of four

angles grouping)


EXAMPLE:

2 supplementary

False, Vertical angles like 2 and 6 are congruent.

2

Practice Support for Lesson Continued:


EXAMPLE:

1

True, alternate interior angles theorem


EXAMPLE:

If m 1 = 2x − 20, and m 4 = 4x + 5, what is m 1? What is m 4?

m 1 + m 4=180 same side exterior angles are supplementary (as stated in ex 15)

(2x - 20) + (4x + 5)=180 Substitution

6x - 15=180 Combine like terms

6x – 15 + 15=180 + 15 Addition property of equality

6x=195 Add

=

Division property of equality

X=32.5

m 1=2x - 20 so 2(32.5) - 20= 45 and

m 4=4x + 5 so 4(32.5) + 5=135

finally m 1=45 and m 4=135


EXAMPLE:

Based on your answer to example 19, what are the measures of the other numbered angles in the diagram? Explain your

reasoning.

m 3 = m 5 = m 7 = 45°; m 2 = m 6 = m 8 = 135°.

Sample explanation: 1 3 and 4 2 by the Corresponding Angles Postulate.

5 3, 6 4, 7 1, and 8 2 by the Vertical Angles Theorem.

3


1.

2.

3.


1.

2.

3.

4




Student need to develop theorems to show that lines are parallel.

Determine whether lines are parallel.

Vocabulary used in this lesson includes: Converse of a theorem or postulate.

Standards for Mathematical Practice to be demonstrated are: Make use of structure, Use appropriate tools

strategically, Reason abstractly, Attend to precision, and Model with mathematics


For Items 12–14, use the diagram to answer each question. Then justify your answer.


EXAMPLE:

Given that m 12= 152° and m 6 = 152°, is m || p?

Yes. I know that 12 6. I also know that 6 8 because these angles are vertical Angles that are created by the

intersection of two lines or line segments. So, by the Transitive Property (meaning if a = b, and b = c, then a = c), 12

8. 12 and 8 are congruent since they are corresponding angles, (meaning that the angles are in the same general

location in there pod of four angles grouping) so m || p by the Converse of the Corresponding Angles Postulate.


EXAMPLE:

Given that m 9 = 52° and m 5 = 56°, is m || n?

No. Sample justification: 9 and 5 are alternate interior angles formed by lines m and n and a transversal. Because

these angles are not congruent, line m is not parallel to line n.


EXAMPLE

Given that m 6 = 124° and m 3 = 52°, is n || p?

No. Sample justification:

6 and 8 are vertical angles, so they are congruent and m 8 = 124°.

3 and 9 are vertical angles, so they are congruent and m 9 = 52°.

8 and 9 are same-side interior angles formed by lines n and p and a transversal Because these angles are not

supplementary (124° + 52° ≠ 180°), line n is not parallel to line p.

5



EXAMPLE:

Two lines are cut by a transversal such that a pair of corresponding angles are right angles. Are the two lines parallel?

Explain.

Yes. Sample explanation: All right angles are congruent, so the corresponding angles are congruent. Thus, the lines are

parallel by the Converse of the Alternate Interior Angles Theorem


EXAMPLE:

Describe how a construction worker can determine whether two power lines painted on the ground are parallel. Assume

that the worker has a protractor, string, and two stakes.

The worker can place the stakes outside the lines and tie the string to them so that the string crosses the power lines like

a transversal. Then the worker can measure 2 corresponding angles formed by the power lines and the string. If the

corresponding angles are congruent, then the power lines are parallel.

6


1.

2.

3.


1.

2. 8

3.

7




Students need to develop theorems to show that lines are perpendicular.

Determine whether lines are perpendicular.

Vocabulary used in this lesson includes: perpendicular, perpendicular lines, Perpendicular Postulate, Parallel Postulate,

Perpendicular Transverse Theorem, and Perpendicular bisector.

Standards for Mathematical Practice to be demonstrated are: Make use of structure, Reason abstractly, Critique the

reasoning of others, and Use appropriate tools strategically.


In the diagram, ℓ || m, m 1 = 90°, and 5 is a right angle. Use the diagram for Items 9-11.


EXAMPLE:

Explain how you know that m ⊥ p.

4 measures 90°, so it is a right angle. Because lines m and p intersect to form a right angle, they are perpendicular.


EXAMPLE:

Show that m 10 = 90°

I know that ℓ || m and m ⊥ p, and ℓ ⊥ p, so line ℓ must also be perpendicular to line p. Perpendicular lines intersect to

form right angles, so 10 must be a right angle and measure 90°.


EXAMPLE:

Show that ℓ || n

2 and 6 are both right angles, so they are congruent. 2 and 6 are congruent corresponding angles formed by lines

ℓ and n and a transversal, so ℓ || n by the Converse of the Corresponding Angles Postulate

8



EXAMPLE:

The perpendicular bisector of intersects at point B. If =15 what is

Since =15 and it is the point of intersection of the perpendicular bisector (meaning intersecting a line at a right or 90

degree angle and then splitting that line in half,) the value is doubled and is 30.


EXAMPLE:

An angle formed by the intersection of two lines is Acute. Could the lines be perpendicular? Explain

No. An acute angle measures less than 90°, so it is not a right angle. Because one of the angles formed by the intersection

of the lines is not a right angle, the lines are not perpendicular.

9


1.

2.

3.

Additional Practice answers:

1. C

2.

3.

1




Make conjectures about the slopes of parallel and perpendicular lines.

Use slope to determine whether lines are parallel or perpendicular.

Vocabulary used in this lesson includes: Slope, Parallel, Perpendicular, Conjecture, and opposite reciprocals.

Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Express

regularity in repeated reasoning, and Model with mathematics



EXAMPLE:

Is ? (meaning is line TQ parallel to line SR?)

Yes, the slope of is -

which reduces to -2, and has a slope of -

which also reduces to -2. Since the slopes are

equal they are parallel.


EXAMPLE:

Is ? (Meaning is line QR meeting line SR at a right angle?)

Yes, The slope (the rise and run of the line) of is

and the slope of is -2. Since the slopes are opposite reciprocals

(meaning is the fraction flipped and the power reversed) they are perpendicular (meaning that the lines meet at a right or

90 degree angle)


EXAMPLE

Is ?

No, The slope of is -

and the slope of is -2. Since they are not opposite reciprocals (meaning is the fraction flipped

and the power reversed) the lines are not parallel.

2



EXAMPLE:

Line m passes through the origin (0,0) and the point (3, 4). What is the slope of a line parallel to line m?

If line m passes through the origin (0,0), and through (3,4) the slope would be

. Because:

Following the slope

formula learned in Algebra 1. Therefore the slope of the perpendicular line would have to be an opposite reciprocal.

That slope would be -

.


EXAMPLE

RST is a right angle, if has a slope of 5 what is the slope of ?

If the slope of one leg of the angle is 5 then the other leg would have to be perpendicular to create a right angle. SO

the slope would have to be opposite reciprocals (meaning is the fraction flipped and the power reversed.) so the

other leg (ST) would have a slope of

.

3


1.

2.

3.


1.

2. C

3.

4




Write the equation of a line that is parallel to a given line.

Write the equation of a line that is perpendicular to a given line.

Vocabulary used in this lesson includes: Slope intercept form, Y-intercept of a line, and point slope form

Standards for Mathematical Practice to be demonstrated are: Reason abstractly and quantitatively, construct viable

arguments, Model with mathematics, and Make sense of problems.



EXAMPLE:

What is the equation of a line parallel to y = −2x + 5 that passes through point (5,1)?

Since we are looking for a parallel line (meaning that the lines run side by side and do not touch) the slope must be an

equal slope. The given slope is -2, so the parallel slope is also -2. The next item is to find the y-intercept (the point that

the line crosses the y axis) Use the point slope form to write the correct equation in slope intercept form.

y - 1 = -2(x – 5) Point slope form

y – 1 = -2x + 10 Distributive Property

y – 1 + 1 = -2x +10 +1 Addition property of equality

y = -2x + 11 Simplify

So the final equation that is parallel and goes through (5,1) is y = -2x + 11.


EXAMPLE:

What is the equation of a line perpendicular to y=4x−3 that passes through point (-2,1)?

Since we are looking for a perpendicular line the slope must be an opposite reciprocal slope. The given slope is 4, so the

perpendicular slope is -

. The next item is to find the y-intercept. Use the point slope form to write the correct equation

in slope intercept form.

Point slope form

y - 1 = -

(x + 2) Substitute

y – 1 = -

x -

Distributive Property

y – 1 + 1 = -

x -

+1 Addition property of equality

y = -

x +

Simplify

So the final equation that is parallel and goes through (-2,1) is y = -

x +

.

5


This example is a model to help solve Practice problem 18. (use the diagram below for problem 18)

EXAMPLE

What is the equation of the line parallel to the given line that passes through point (2,2)

First you must determine the slope of the given line. The given line has a slope of 3. Now that we have the slope we can

find the equation in the same way we did in the previous questions. Using that we can find the equation to be y = 3x – 4.


EXAMPLE:

What is the equation of the line perpendicular to the given line that passes through point (2,2)?

Again the first thing is to detriment the slope of the give line, and the slope is 3 still. Now the perpendicular slope

would be the -

. Now that we have the slope we can find the equation in the same way we did in the previous

questions. Using that we can find the equation to be y = -

x +

.


EXAMPLE:

How could you check that your answer to Item 19 is reasonable?

You could graph the line and check that it passes through the point (2, 2) and that it appears to be perpendicular to line

given.


EXAMPLE:

∆DBA has vertices at the points A(−1, 4), B(0, 0), and D(4, 1). What kind of triangle is ∆DBA? Explain your answer.

The triangle has a right triangle. The reason is because the slope of is -4 and the slope of is -

. These two slopes

are opposite reciprocals.

6


1.

2.

3.

Answers to additional practice

1.

2.

3. D

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