chapter 4 resource masters -...

Chapter 4Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

44

© 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 4.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

acute triangle

base angles

congruence transformation

kuhn·GROO·uhns

congruent triangles

coordinate proof

corollary

equiangular triangle

equilateral triangle

exterior angle

(continued on the next page)

! " " # " " $

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

flow proof

included angle

included side

isosceles triangle

obtuse triangle

remote interior angles

right triangle

scalene triangle

SKAY·leen

vertex angle

! " # " $

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

44

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 4. 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 4.1Angle Sum Theorem

Theorem 4.2Third Angle Theorem

Theorem 4.3Exterior Angle Theorem

Theorem 4.4

Theorem 4.5Angle-Angle-Side Congruence (AAS)

Theorem 4.6Leg-Leg Congruence (LL)

Theorem 4.7Hypotenuse-Angle Congruence (HA)

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 4.8Leg-Angle Congruence (LA)

Theorem 4.9Isosceles Triangle Theorem

Theorem 4.10

Postulate 4.1Side-Side-Side Congruence (SSS)

Postulate 4.2Side-Angle-Side Congruence (SAS)

Postulate 4.3Angle-Side-Angle Congruence (ASA)

Postulate 3.4Hypotenuse-Leg Congruence(HL)

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Study Guide and InterventionClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

© Glencoe/McGraw-Hill 183 Glencoe Geometry

Less

on

4-1

Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.

• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.

• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.

• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.

• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.

Classify each triangle.

a.

All three angles are congruent, so all three angles have measure 60°.The triangle is an equiangular triangle.

b.

The triangle has one angle that is obtuse. It is an obtuse triangle.

c.

The triangle has one right angle. It is a right triangle.

Classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

4. 5. 6.60!

28! 92!F D

B

45!

45!90!X Y

W

65! 65!

50!

U V

T

60! 60!

60!

Q

R S

120!

30! 30!N O

P

67!

90! 23!

K

L M

90!

60! 30!

G

H J

25!35!

120!

D F

E

60!

A

B C

ExampleExample

ExercisesExercises


Classify Triangles by Sides You can classify a triangle by the measures of its sides.Equal numbers of hash marks indicate congruent sides.

• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.

• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.

• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.

Classify each triangle.

a. b. c.

Two sides are congruent. All three sides are The triangle has no pairThe triangle is an congruent. The triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle.

Classify each triangle as equilateral, isosceles, or scalene.

1. 2. 3.

4. 5. 6.

7. Find the measure of each side of equilateral !RST with RS ! 2x " 2, ST ! 3x,and TR ! 5x # 4.

8. Find the measure of each side of isosceles !ABC with AB ! BC if AB ! 4y,BC ! 3y " 2, and AC ! 3y.

9. Find the measure of each side of !ABC with vertices A(#1, 5), B(6, 1), and C(2, #6).Classify the triangle.

D E

F

x

x x8x

32x

32x

B

CA

UW

S

12 17

19Q O

MG

K I18

18 182

1

3!"G C

A

23 12

15X V

TN

R PL J

H

Study Guide and Intervention (continued)

Classifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

ExercisesExercises

Skills PracticeClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

4. 5. 6.

Identify the indicated type of triangles.

7. right 8. isosceles

9. scalene 10. obtuse

ALGEBRA Find x and the measure of each side of the triangle.

11. !ABC is equilateral with AB! 3x # 2, BC ! 2x " 4, and CA ! x " 10.

12. !DEF is isosceles, "D is the vertex angle, DE ! x " 7, DF ! 3x # 1, and EF ! 2x " 5.

Find the measures of the sides of !RST and classify each triangle by its sides.

13. R(0, 2), S(2, 5), T(4, 2)

14. R(1, 3), S(4, 7), T(5, 4)

E CD

A B


Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

Identify the indicated type of triangles if A!B! " A!D! " B!D! " D!C!, B!E! " E!D!, A!B! ! B!C!, and E!D! ! D!C!.

4. right 5. obtuse

6. scalene 7. isosceles

ALGEBRA Find x and the measure of each side of the triangle.

8. !FGH is equilateral with FG ! x " 5, GH ! 3x # 9, and FH ! 2x # 2.

9. !LMN is isosceles, "L is the vertex angle, LM ! 3x # 2, LN ! 2x " 1, and MN ! 5x # 2.

Find the measures of the sides of !KPL and classify each triangle by its sides.

10. K(#3, 2) P(2, 1), L(#2, #3)

11. K(5, #3), P(3, 4), L(#1, 1)

12. K(#2, #6), P(#4, 0), L(3, #1)

13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor.How many right angles are there?

A CD

EB

Practice Classifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Reading to Learn MathematicsClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Pre-Activity Why are triangles important in construction?

Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.

• Why are triangles used for braces in construction rather than other shapes?

• Why do you think that isosceles triangles are used more often thanscalene triangles in construction?

Reading the Lesson1. Supply the correct numbers to complete each sentence.

a. In an obtuse triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

b. In an acute triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

c. In a right triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

2. Determine whether each statement is always, sometimes, or never true.a. A right triangle is scalene.b. An obtuse triangle is isosceles.c. An equilateral triangle is a right triangle.d. An equilateral triangle is isosceles.e. An acute triangle is isosceles.f. A scalene triangle is obtuse.

3. Describe each triangle by as many of the following words as apply: acute, obtuse, right,scalene, isosceles, or equilateral.a. b. c.

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

definition of the same word. How is the use of the word acute, when used to describeacute pain, related to the use of the word acute when used to describe an acute angle oran acute triangle?

5

34

135!80!

70!

30!


Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the texteasier to understand.

Consider three points, A, B, and C on a coordinate grid.The y-coordinates of A and B are the same. The x-coordinate of B isgreater than the x-coordinate of A. Both coordinates of C are greaterthan the corresponding coordinates of B. Is triangle ABC acute, right,or obtuse?

To answer this question, first draw a sample triangle that fits the description.

Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B.

From the diagram you can see that triangle ABCmust be obtuse.

Answer each question. Draw a simple triangle on the grid above to help you.

1. Consider three points, R, S, and 2. Consider three noncollinear points,T on a coordinate grid. The J, K, and L on a coordinate grid. Thex-coordinates of R and S are the y-coordinates of J and K are thesame. The y-coordinate of T is same. The x-coordinates of K and Lbetween the y-coordinates of R are the same. Is triangle JKL acute,and S. The x-coordinate of T is less right, or obtuse?than the x-coordinate of R. Is angleR of triangle RST acute, right, or obtuse?

3. Consider three noncollinear points, 4. Consider three points, G, H, and ID, E, and F on a coordinate grid. on a coordinate grid. Points G and The x-coordinates of D and E are H are on the positive y-axis, andopposites. The y-coordinates of D and the y-coordinate of G is twice the E are the same. The x-coordinate of y-coordinate of H. Point I is on the F is 0. What kind of triangle must positive x-axis, and the x-coordinate!DEF be: scalene, isosceles, or of I is greater than the y-coordinateequilateral? of G. Is triangle GHI scalene,

isosceles, or equilateral?

BA

Q

x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

Study Guide and InterventionAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

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4-2

Angle Sum Theorem If the measures of two angles of a triangle are known,the measure of the third angle can always be found.

Angle Sum The sum of the measures of the angles of a triangle is 180.Theorem In the figure at the right, m"A " m"B " m"C ! 180.

CA

B

Find m"T.

m"R " m"S " m"T ! 180 Angle SumTheorem

25 " 35 " m"T ! 180 Substitution

60 " m"T ! 180 Add.

m"T ! 120 Subtract 60from each side.

35!

25!R T

S

Find the missing angle measures.

m"1 " m"A " m"B ! 180 Angle Sum Theorem

m"1 " 58 " 90 ! 180 Substitution

m"1 " 148 ! 180 Add.

m"1 ! 32 Subtract 148 fromeach side.

m"2 ! 32 Vertical angles arecongruent.

m"3 " m"2 " m"E ! 180 Angle Sum Theorem

m"3 " 32 " 108 ! 180 Substitution

m"3 " 140 ! 180 Add.

m"3 ! 40 Subtract 140 fromeach side.

58!

90!

108!

12 3

E

DAC

B

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the measure of each numbered angle.

1. 2.

3. 4.

5. 6. 20!

152!

DG

A

130!60!

1 2

S

R

T W

Q

O

NM

P58!

66!

50!

321

V

W T

U

30!

60!

2

1

S

Q R30!

1

90!

62!

1 N

M

P


Exterior Angle Theorem At each vertex of a triangle, the angle formed by one sideand an extension of the other side is called an exterior angle of the triangle. For eachexterior angle of a triangle, the remote interior angles are the interior angles that are notadjacent to that exterior angle. In the diagram below, "B and "A are the remote interiorangles for exterior "DCB.

Exterior AngleThe measure of an exterior angle of a triangle is equal to

Theoremthe sum of the measures of the two remote interior angles.m"1 ! m"A " m"B

AC

B

D1


Angles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Find m"1.

m"1 ! m"R " m"S Exterior Angle Theorem

! 60 " 80 Substitution

! 140 Add.

R T

S

60!

80!

1

Find x.

m"PQS ! m"R " m"S Exterior Angle Theorem

78 ! 55 " x Substitution

23 ! x Subtract 55 from each side.

S R

Q

P

55!

78!

x!


ExercisesExercises

Find the measure of each numbered angle.

1. 2.

3. 4.

Find x.

5. 6. E

FGH

58!

x !

x !B

A

DC

95!

2x ! 145!

U T

SRV

35! 36!

80!

13

2

POQ

N M60!

60!3 2

1

B C D

A

25!

35!

12Y Z W

X

65!

50!

1

Skills PracticeAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2


1. 2.

Find the measure of each angle.

3. m"1

4. m"2

5. m"3


6. m"1

7. m"2

8. m"3


9. m"1

10. m"2

11. m"3

12. m"4

13. m"5


14. m"1

15. m"2 63!

1

2D

A C

B

80!

60!

40!

105!

1 4 52

3

150!55!

70!

1 2

3

85! 55!

40!

1 2

3

146!

TIGERS80!

73!



1. 2.


3. m"1

4. m"2

5. m"3


6. m"1

7. m"4

8. m"3

9. m"2

10. m"5

11. m"6

Find the measure of each angle if "BAD and "BDC are right angles and m"ABC " 84.

12. m"1

13. m"2

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m"1.

145!1

64!1

2A

BC

D

118!36!

68!

70!

65!

82!

1

2

3 4

5

6

58!

39!

35!

12

3

40! 55!

72!

?

Practice Angles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Reading to Learn MathematicsAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Pre-Activity How are the angles of triangles used to make kites?


The frame of the simplest kind of kite divides the kite into four triangles.Describe these four triangles and how they are related to each other.

Reading the Lesson

1. Refer to the figure.

a. Name the three interior angles of the triangle. (Use threeletters to name each angle.)

b. Name three exterior angles of the triangle. (Use three lettersto name each angle.)

c. Name the remote interior angles of "EAB.

d. Find the measure of each angle without using a protractor.

i. "DBC ii. "ABC iii. "ACF iv. "EAB

2. Indicate whether each statement is true or false. If the statement is false, replace theunderlined word or number with a word or number that will make the statement true.

a. The acute angles of a right triangle are .

b. The sum of the measures of the angles of any triangle is .

c. A triangle can have at most one right angle or angle.

d. If two angles of one triangle are congruent to two angles of another triangle, then thethird angles of the triangles are .

e. The measure of an exterior angle of a triangle is equal to the of themeasures of the two remote interior angles.

f. If the measures of two angles of a triangle are 62 and 93, then the measure of thethird angle is .

g. An angle of a triangle forms a linear pair with an interior angle of thetriangle.

Helping You Remember

3. Many students remember mathematical ideas and facts more easily if they see themdemonstrated visually rather than having them stated in words. Describe a visual wayto demonstrate the Angle Sum Theorem.

exterior

35

difference

congruent

acute

100

supplementary

39!

23!

EA B D

CF


Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.

In triangle ABC, m"A, is twice m"B, and m"Cis 8 more than m"B. What is the measure of each angle?

Write and solve an equation. Let x ! m"B.m"A " m"B " m"C ! 180

2x " x " (x " 8) ! 1804x " 8 ! 180

4x ! 172x ! 43

So, m"A ! 2(43) or 86, m"B ! 43, and m"C ! 43 " 8 or 51.

Solve each problem.

1. In triangle DEF, m"E is three times 2. In triangle RST, m"T is 5 more than m"D, and m"F is 9 less than m"E. m"R, and m"S is 10 less than m"T.What is the measure of each angle? What is the measure of each angle?

3. In triangle JKL, m"K is four times 4. In triangle XYZ, m"Z is 2 more than twicem"J, and m"L is five times m"J. m"X, and m"Y is 7 less than twice m"X.What is the measure of each angle? What is the measure of each angle?

5. In triangle GHI, m"H is 20 more than 6. In triangle MNO, m"M is equal to m"N,m"G, and m"G is 8 more than m"I. and m"O is 5 more than three times What is the measure of each angle? m"N. What is the measure of each angle?

7. In triangle STU, m"U is half m"T, 8. In triangle PQR, m"P is equal to and m"S is 30 more than m"T. What m"Q, and m"R is 24 less than m"P.is the measure of each angle? What is the measure of each angle?

9. Write your own problems about measures of triangles.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

ExampleExample

Study Guide and InterventionCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3

Corresponding Parts of Congruent TrianglesTriangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, !ABC # !RST.

If !XYZ " !RST, name the pairs of congruent angles and congruent sides."X # "R, "Y # "S, "Z # "TX$Y$ # R$S$, X$Z$ # R$T$, Y$Z$ # S$T$

Identify the congruent triangles in each figure.

1. 2. 3.

Name the corresponding congruent angles and sides for the congruent triangles.

4. 5. 6. R

T

U S

B D

CA

F G L K

JE

K

J

L

MC

D

A

B

CA

B

LJ

K

Y

X ZT

SR

AC

BR

T

S

ExampleExample

ExercisesExercises


Identify Congruence Transformations If two triangles are congruent, you canslide, flip, or turn one of the triangles and they will still be congruent. These are calledcongruence transformations because they do not change the size or shape of the figure.It is common to use prime symbols to distinguish between an original !ABC and atransformed !A$B$C$.

Name the congruence transformation that produces !A#B#C# from !ABC.The congruence transformation is a slide."A # "A$; "B # "B$; "C #"C$;A$B$ # A$$$B$$$; A$C$ # A$$$C$$$; B$C$ # B$$$C$$$

Describe the congruence transformation between the two triangles as a slide, aflip, or a turn. Then name the congruent triangles.

1. 2.

3. 4.

5. 6.

x

y

OM

N

P

N$

P$x

y

OA$ B$

C$

A B

C

x

y

OB$B

A

Cx

y

O

Q$ P$

P

Q

x

y

ON$

M$P$

N

MP

x

y

OR

S$T$

S

T

x

y

O

A$

B$B

C$A C


Congruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

ExampleExample

ExercisesExercises

Skills PracticeCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3


1. 2.

3. 4.

Name the congruent angles and sides for each pair of congruent triangles.

5. !ABC # !FGH

6. !PQR # !STU

Verify that each of the following transformations preserves congruence, and namethe congruence transformation.

7. !ABC # !A$B$C$ 8. !DEF # !D$E$F $

x

y

OD$

E$

F$D

E

F

x

y

OA$

B$

C$

A

B

C

D

E

G

FRP

Q

S

WY

X C

AB

L

P

J

S

V

T



1. 2.

Name the congruent angles and sides for each pair of congruent triangles.

3. !GKP # !LMN

4. !ANC # !RBV

Verify that each of the following transformations preserves congruence, and namethe congruence transformation.

5. !PST # !P$S$T$ 6. !LMN # !L$M$N$

QUILTING For Exercises 7 and 8, refer to the quilt design.

7. Indicate the triangles that appear to be congruent.

8. Name the congruent angles and congruent sides of a pair of congruent triangles.

B

A

I

E

FH G

C D

x

y

O

M$

N$L$

M

NL

x

y

O

S$

T$P$

S

TP

MN

L

P

QD

R

SCA

B

Practice Congruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Reading to Learn MathematicsCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3

Pre-Activity Why are triangles used in bridges?


In the bridge shown in the photograph in your textbook, diagonal braceswere used to divide squares into two isosceles right triangles. Why do youthink these braces are used on the bridge?

Reading the Lesson1. If !RST # !UWV, complete each pair of congruent parts.

"R # # "W "T #

R$T$ # # U$W$ # W$V$

2. Identify the congruent triangles in each diagram.a. b.

c. d.

3. Determine whether each statement says that congruence of triangles is reflexive,symmetric, or transitive.a. If the first of two triangles is congruent to the second triangle, then the second

triangle is congruent to the first.b. If there are three triangles for which the first is congruent to the second and the second

is congruent to the third, then the first triangle is congruent to the third.c. Every triangle is congruent to itself.

Helping You Remember4. A good way to remember something is to explain it to someone else. Your classmate Ben is

having trouble writing congruence statements for triangles because he thinks he has tomatch up three pairs of sides and three pairs of angles. How can you help him understandhow to write correct congruence statements more easily?

R T

US

V

N O P

QM

S

P R

Q

CA

B

D


Transformations in The Coordinate PlaneThe following statement tells one way to map preimage points to image points in the coordinate plane.

(x, y) ! (x " 6, y # 3)

This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x " 6, y # 3).”With this transformation, for example, (3, 5) is mapped to (3 " 6, 5 # 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.

1. Does the transformation above appear to be a congruence transformation? Explain youranswer.

Draw the transformation image for each figure. Then tell whether thetransformation is or is not a congruence transformation.

2. (x, y) ! (x # 4, y) 3. (x, y) ! (x " 8, y " 7)

4. (x, y) ! (#x , #y) 5. (x, y) ! %#%12%x, y&

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C X

Z

Y

O

(x, y ) ! (x $ 6, y % 3)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Study Guide and InterventionProving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4


Less

on

4-4

SSS Postulate You know that two triangles are congruent if corresponding sides arecongruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate letsyou show that two triangles are congruent if you know only that the sides of one triangleare congruent to the sides of the second triangle.

SSS PostulateIf the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Write a two-column proof.Given: A$B$ # D$B$ and C is the midpoint of A$D$.Prove: !ABC # !DBC

Statements Reasons

1. A$B$ # D$B$ 1. Given2. C is the midpoint of A$D$. 2. Given3. A$C$ # D$C$ 3. Definition of midpoint4. B$C$ # B$C$ 4. Reflexive Property of #5. !ABC # !DBC 5. SSS Postulate

Write a two-column proof.

B

CDA

ExampleExample

ExercisesExercises

1.

Given: A$B$ # X$Y$, A$C$ # X$Z$, B$C$ # Y$Z$Prove: !ABC # !XYZ

Statements Reasons

1.A!B! " X!Y! 1.Given

2.!ABC " !XYZ 2. SSS Post.

2.

Given: R$S$ # U$T$, R$T$ # U$S$Prove: !RST # !UTS

Statements Reasons

1.R!S! " U!T! 1. Given

2.S!T! " T!S! 2. Refl. Prop.3.!RST " !UTS 3. SSS Post.

T U

R S

B Y

CA XZ


SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.

SAS PostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

For each diagram, determine which pairs of triangles can beproved congruent by the SAS Postulate.

a. b. c.

In !ABC, the angle is not The right angles are The included angles, "1 “included” by the sides A$B$ congruent and they are the and "2, are congruent and A$C$. So the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the SAS Postulate. !DEF # !JGH by the for two parallel lines.

SAS Postulate. !PSR # !RQP by the SAS Postulate.

For each figure, determine which pairs of triangles can be proved congruent bythe SAS Postulate.

1. 2. 3.

4. 5. 6.

J H

GF

K

C

BA

D

V

T

W

M

P L

N

M

X

W Z

YQT

P

UN M

R

P Q

S

1

2 R

D H

F E

G

J

A

B C

X

Y Z


Proving Congruence—SSS, SAS

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ExampleExample

ExercisesExercises

Skills PracticeProving Congruence—SSS, SAS

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Determine whether !ABC " !KLM given the coordinates of the vertices. Explain.

1. A(#3, 3), B(#1, 3), C(#3, 1), K(1, 4), L(3, 4), M(1, 6)

2. A(#4, #2), B(#4, 1), C(#1, #1), K(0, #2), L(0, 1), M(4, 1)

3. Write a flow proof.Given: P$R$ # D$E$, P$T$ # D$F$

"R # "E, "T # "FProve: !PRT # !DEF

Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.

4. 5. 6.

PR " DEGiven

PT " DF Given

"R " "E Given

"P " "D Third AngleTheorem

!PRT " !DEF SAS

"T " "F Given

T

R

P

F

E

D


Determine whether !DEF " !PQR given the coordinates of the vertices. Explain.

1. D(#6, 1), E(1, 2), F(#1, #4), P(0, 5), Q(7, 6), R(5, 0)

2. D(#7, #3), E(#4, #1), F(#2, #5), P(2, #2), Q(5, #4), R(0, #5)

3. Write a flow proof.Given: R$S$ # T$S$

V is the midpoint of R$T$.Prove: !RSV # !TSV

Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.

4. 5. 6.

7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in thediagram. How does he know that the lengths A$B$ and AB are equal?

A$B$

A B

C

RS " TSGiven

SV " SVReflexiveProperty

RV " VTDefinitionof midpoint

V is themidpoint of RT. Given

!RSV " !TSV SSS

S

R

V

T

Practice Proving Congruence—SSS, SAS

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Reading to Learn MathematicsProving Congruence—SSS, SAS

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Pre-Activity How do land surveyors use congruent triangles?


Why do you think that land surveyors would use congruent right trianglesrather than other congruent triangles to establish property boundaries?

Reading the Lesson


a. Name the sides of !LMN for which "L is the included angle.

b. Name the sides of !LMN for which "N is the included angle.

c. Name the sides of !LMN for which "M is the included angle.

2. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate that you would use. If not, write not possible.a. b.

c. E$H$ and D$G$ bisect each other. d.


3. Find three words that explain what it means to say that two triangles are congruent andthat can help you recall the meaning of the SSS Postulate.

GE

F

HD

R

T

SU

G

FD

E

C

A

DB

L

N

M


Congruent Parts of Regular Polygonal RegionsCongruent figures are figures that have exactly the same size and shape. There are manyways to divide regular polygonal regions into congruent parts. Three ways to divide anequilateral triangular region are shown. You can verify that the parts are congruent bytracing one part, then rotating, sliding, or reflecting that part on top of the other parts.

1. Divide each square into four congruent parts. Use three different ways.

2. Divide each pentagon into five congruent parts. Use three different ways.

3. Divide each hexagon into six congruent parts. Use three different ways.

4. What hints might you give another student who is trying to divide figures like those into congruent parts?

Enrichment

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Study Guide and InterventionProving Congruence—ASA, AAS

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ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two trianglesare congruent.

ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Find the missing congruent parts so that the triangles can beproved congruent by the ASA Postulate. Then write the triangle congruence.

a.

Two pairs of corresponding angles are congruent, "A # "D and "C # "F. If theincluded sides A$C$ and D$F$ are congruent, then !ABC # !DEF by the ASA Postulate.

b.

"R # "Y and S$R$ # X$Y$. If "S # "X, then !RST# !YXW by the ASA Postulate.

What corresponding parts must be congruent in order to prove that the trianglesare congruent by the ASA Postulate? Write the triangle congruence statement.

1. 2. 3.

4. 5. 6.

A C

B

E

D

S

V

UR T

D

A B

C

D CEA

B

YW

X

ZE A

BD

C

R T W Y

S X

A C

B

D F

E

ExampleExample

ExercisesExercises


AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.

AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding twoangles and side of a second triangle, then the two triangles are congruent.

You now have five ways to show that two triangles are congruent.• definition of triangle congruence • ASA Postulate• SSS Postulate • AAS Theorem• SAS Postulate

In the diagram, "BCA " "DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Postulate?A$C$ # A$C$ by the Reflexive Property of congruence. The congruent angles cannot be "1 and "2, because A$C$ would be the included side.If "B # "D, then !ABC # !ADC by the AAS Theorem.

In Exercises 1 and 2, draw and label !ABC and !DEF. Indicate which additionalpair of corresponding parts needs to be congruent for the triangles to becongruent by the AAS Theorem.

1. "A # "D; "B # "E 2. BC # EF; "A # "D

3. Write a flow proof.Given: "S # "U; T$R$ bisects "STU.Prove: "SRT # "URT

Given

Given

RT " RT Refl. Prop. of "

Def.of " bisectorTR bisects "STU.

!SRT " !URT

"STR " "UTR

AAS"SRT " "URTCPCTC

"S " "U

S

R T

U

BA

C

ED

F

CA

B

FD

E

D

C12A

B


Proving Congruence—ASA, AAS

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ExampleExample

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Write a flow proof.

1. Given: "N # "LJ$K$ # M$K$

Prove: !JKN # !MKL

2. Given: A$B$ # C$B$"A # "CD$B$ bisects "ABC.

Prove: A$D$ # C$D$

3. Write a paragraph proof.

Given: D$E$ || F$G$"E # "G

Prove: !DFG # !FDE

FG

D E

"A " "C

GivenAB " CB

Given CPCTCAD " CD

DB bisects "ABC. Given

!ABD " !CBDASA

"ABD " "CBDDef. of " bisector

A C

B

D

"N " "LGiven

JK " MK Given

"JKN " "MKL Vertical # are ".

!JKN " !MKLAAS

N

J

M

K L


1. Write a flow proof.Given: S is the midpoint of Q$T$.

Q$R$ || T$U$Prove: !QSR # !TSU

2. Write a paragraph proof.

Given: "D # "FG$E$ bisects "DEF.

Prove: D$G$ # F$G$

ARCHITECTURE For Exercises 3 and 4, use the following information.An architect used the window design in the diagram when remodeling an art studio. A$B$ and C$B$ each measure 3 feet.

3. Suppose D is the midpoint of A$C$. Determine whether !ABD # !CBD.Justify your answer.

4. Suppose "A # "C. Determine whether !ABD # !CBD. Justify your answer.

D

B

A C

D

G

F

E

"Q " "T

Given

QR || TU Given

Def.of midpoint

Alt. Int. # are ".

QS " TS S is the midpoint of QT.

!QSR " !TSUASA

"QSR " "TSUVertical # are ".

UQ S

RT

Practice Proving Congruence—ASA, AAS

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Reading to Learn MathematicsProving Congruence—ASA, AAS

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Pre-Activity How are congruent triangles used in construction?Read the introduction to Lesson 4-5 at the top of page 207 in your textbook.Which of the triangles in the photograph in your textbook appear to becongruent?

Reading the Lesson1. Explain in your own words the difference between how the ASA Postulate and the AAS

Theorem are used to prove that two triangles are congruent.

2. Which of the following conditions are sufficient to prove that two triangles are congruent?A. Two sides of one triangle are congruent to two sides of the other triangle.B. The three sides of one triangles are congruent to the three sides of the other triangle.C. The three angles of one triangle are congruent to the three angles of the other triangle.D. All six corresponding parts of two triangles are congruent.E. Two angles and the included side of one triangle are congruent to two sides and the

included angle of the other triangle.F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a

nonincluded angle of the other triangle.G. Two angles and a nonincluded side of one triangle are congruent to two angles and

the corresponding nonincluded side of the other triangle.H. Two sides and the included angle of one triangle are congruent to two sides and the

included angle of the other triangle.I. Two angles and a nonincluded side of one triangle are congruent to two angles and a

nonincluded side of the other triangle.

3. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate or theorem that you would use. If not, write not possible.a. b. T is the midpoint of R$U$.

Helping You Remember4. A good way to remember mathematical ideas is to summarize them in a general statement.

If you want to prove triangles congruent by using three pairs of corresponding parts,what is a good way to remember which combinations of parts will work?

R

S

T

U

V

A DCB

E


Congruent Triangles in the Coordinate PlaneIf you know the coordinates of the vertices of two triangles in the coordinateplane, you can often decide whether the two triangles are congruent. Theremay be more than one way to do this.

1. Consider ! ABD and !CDB whose vertices have coordinates A(0, 0),B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what youknow about congruent triangles and the coordinate plane to show that ! ABD # !CDB. You may wish to make a sketch to help get you started.

2. Consider !PQR and !KLM whose vertices are the following points.

P(1, 2) Q(3, 6) R(6, 5)K(#2, 1) L(#6, 3) M(#5, 6)

Briefly describe how you can show that !PQR # !KLM.

3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.

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Study Guide and InterventionIsosceles Triangles

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Properties of Isosceles Triangles An isosceles triangle has two congruent sides.The angle formed by these sides is called the vertex angle. The other two angles are calledbase angles. You can prove a theorem and its converse about isosceles triangles.

• If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem)

• If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If A$B$ # C$B$, then "A # "C.

If "A # "C, then A$B$ # C$B$.

A

BC

Find x.

BC ! BA, so m"A ! m"C. Isos. Triangle Theorem

5x # 10 ! 4x " 5 Substitution

x # 10 ! 5 Subtract 4x from each side.

x ! 15 Add 10 to each side.

B

A

C (4x $ 5)!

(5x % 10)!

Find x.

m"S ! m"T, soSR ! TR. Converse of Isos. ! Thm.

3x # 13 ! 2x Substitution

3x ! 2x " 13 Add 13 to each side.

x ! 13 Subtract 2x from each side.

R T

S

3x % 13

2x


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. Write a two-column proof.Given: "1 # "2Prove: A$B$ # C$B$

Statements Reasons

B

A C D

E

1 32

R S

T3x!

x!D

BG

L3x!

30!D

T Q

PK

(6x $ 6)! 2x!

W

Y Z3x!

S

V

T 3x % 6

2x $ 6R

P

Q2x !

40!


Properties of Equilateral Triangles An equilateral triangle has three congruentsides. The Isosceles Triangle Theorem can be used to prove two properties of equilateraltriangles.

1. A triangle is equilateral if and only if it is equiangular.2. Each angle of an equilateral triangle measures 60°.

Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle.Proof:Statements Reasons

1. !ABC is equilateral; P$Q$ || B$C$. 1. Given2. m"A ! m"B ! m"C ! 60 2. Each " of an equilateral ! measures 60°.3. "1 # "B, "2 # "C 3. If || lines, then corres. "s are #.4. m"1 ! 60, m"2 ! 60 4. Substitution5. !APQ is equilateral. 5. If a ! is equiangular, then it is equilateral.

Find x.

1. 2. 3.

4. 5. 6.

7. Write a two-column proof.Given: !ABC is equilateral; "1 # "2.Prove: "ADB # "CDB

Proof:

Statements Reasons

A

D

C

B12

R

O

HM 60!4x!

X

Z Y4x % 4

3x $ 8 60!

P Q

LV R60!

4x 40

L

N

M

K

!KLM is equilateral.

3x!G

J H

6x % 5 5x

D

F E6x!

A

B

P Q

C

1 2


Isosceles Triangles

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ExampleExample

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Refer to the figure.

1. If A$C$ # A$D$, name two congruent angles.

2. If B$E$ # B$C$, name two congruent angles.

3. If "EBA # "EAB, name two congruent segments.

4. If "CED # "CDE, name two congruent segments.

!ABF is isosceles, !CDF is equilateral, and m"AFD " 150.Find each measure.

5. m"CFD 6. m"AFB

7. m"ABF 8. m"A

In the figure, P!L! " R!L! and L!R! " B!R!.

9. If m"RLP ! 100, find m"BRL.

10. If m"LPR ! 34, find m"B.

11. Write a two-column proof.

Given: C$D$ # C$G$D$E$ # G$F$

Prove: C$E$ # C$F$

DE

FG

C

R P

BL

D

C

F

B

35!

A E

D

C

B

A E


Refer to the figure.

1. If R$V$ # R$T$, name two congruent angles.

2. If R$S$ # S$V$, name two congruent angles.

3. If "SRT # "STR, name two congruent segments.

4. If "STV # "SVT, name two congruent segments.

Triangles GHM and HJM are isosceles, with G!H! " M!H!and H!J! " M!J!. Triangle KLM is equilateral, and m"HMK " 50.Find each measure.

5. m"KML 6. m"HMG 7. m"GHM

8. If m"HJM ! 145, find m"MHJ.

9. If m"G ! 67, find m"GHM.

10. Write a two-column proof.

Given: D$E$ || B$C$"1 # "2

Prove: A$B$ # A$C$

11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle. Lincoln Hawks

E

D B

C

A1

23

4

G

M

LK

J

H

U

R

TV

S

Practice Isosceles Triangles

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Reading to Learn MathematicsIsosceles Triangles

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Pre-Activity How are triangles used in art?


• Why do you think that isosceles and equilateral triangles are used moreoften than scalene triangles in art?

• Why might isosceles right triangles be used in art?

Reading the Lesson1. Refer to the figure.

a. What kind of triangle is !QRS?

b. Name the legs of !QRS.

c. Name the base of !QRS.

d. Name the vertex angle of !QRS.

e. Name the base angles of !QRS.

2. Determine whether each statement is always, sometimes, or never true.

a. If a triangle has three congruent sides, then it has three congruent angles.

b. If a triangle is isosceles, then it is equilateral.

c. If a right triangle is isosceles, then it is equilateral.

d. The largest angle of an isosceles triangle is obtuse.

e. If a right triangle has a 45° angle, then it is isosceles.

f. If an isosceles triangle has three acute angles, then it is equilateral.

g. The vertex angle of an isosceles triangle is the largest angle of the triangle.

3. Give the measures of the three angles of each triangle.

a. an equilateral triangle

b. an isosceles right triangle

c. an isosceles triangle in which the measure of the vertex angle is 70

d. an isosceles triangle in which the measure of a base angle is 70

e. an isosceles triangle in which the measure of the vertex angle is twice the measure ofone of the base angles

Helping You Remember4. If a theorem and its converse are both true, you can often remember them most easily by

combining them into an “if-and-only-if” statement. Write such a statement for the IsoscelesTriangle Theorem and its converse.

R

Q

S


Triangle ChallengesSome problems include diagrams. If you are not sure how to solve theproblem, begin by using the given information. Find the measures of as manyangles as you can, writing each measure on the diagram. This may give youmore clues to the solution.

Enrichment

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1. Given: BE ! BF, "BFG # "BEF #"BED, m"BFE ! 82 andABFG and BCDE each haveopposite sides parallel andcongruent.

Find m"ABC.

3. Given: m"UZY ! 90, m"ZWX ! 45,!YZU # !VWX, UVXY is asquare (all sides congruent, allangles right angles).

Find m"WZY.

2. Given: AC ! AD, and A$B$#B$D$,m"DAC ! 44 andC$E$ bisects "ACD.

Find m"DEC.

4. Given: m"N ! 120, J$N$ # M$N$,!JNM # !KLM.

Find m"JKM.J

K

L

MN

A

D C

BE

U VW

XYZ

A

G DF E

CB

Study Guide and InterventionTriangles and Coordinate Proof

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Position and Label Triangles A coordinate proof uses points, distances, and slopes toprove geometric properties. The first step in writing a coordinate proof is to place a figure onthe coordinate plane and label the vertices. Use the following guidelines.

1. Use the origin as a vertex or center of the figure.2. Place at least one side of the polygon on an axis.3. Keep the figure in the first quadrant if possible.4. Use coordinates that make the computations as simple as possible.

Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis.Start with R(0, 0). If RT is a, then another vertex is T(a, 0).

For vertex S, the x-coordinate is %a2%. Use b for the y-coordinate,

so the vertex is S%%a2%, b&.

Find the missing coordinates of each triangle.

1. 2. 3.

Position and label each triangle on the coordinate plane.

4. isosceles triangle 5. isosceles right !DEF 6. equilateral triangle !EQI!RST with base R$S$ with legs e units long with vertex Q(0, a) and4a units long sides 2b units long

x

y

I(b, 0)E(–b, 0)

Q(0, a)

x

y

E(e, 0)

F(e, e)

D(0, 0)x

y T(2a, b)

R(0, 0) S(4a, 0)

x

y

G(2g, 0)

F(?, b)

E(?, ?)x

y

S(2a, 0)

T(?, ?)

R(0, 0)x

y

B(2p, 0)

C(?, q)

A(0, 0)

x

y

T(a, 0)R(0, 0)

S#a–2, b$

ExercisesExercises

ExampleExample


Write Coordinate Proofs Coordinate proofs can be used to prove theorems and toverify properties. Many coordinate proofs use the Distance Formula, Slope Formula, orMidpoint Theorem.

Prove that a segment from the vertex angle of an isosceles triangle to the midpoint of the base is perpendicular to the base.First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(#a, 0), and S(0, c). Then U(0, 0) is the midpoint of R$T$.

Given: Isosceles !RST; U is the midpoint of base R$T$.Prove: S$U$ " R$T$

Proof:U is the midpoint of R$T$ so the coordinates of U are %%#a

2" a%, %

0 "2

0%& ! (0, 0). Thus S$U$ lies on

the y-axis, and !RST was placed so R$T$ lies on the x-axis. The axes are perpendicular, so S$U$ " R$T$.

Prove that the segments joining the midpoints of the sides of a right triangle forma right triangle.

C(2a, 0)

B(0, 2b)

P

Q

R

A(0, 0)

x

y

T(a, 0)U(0, 0)R(–a, 0)

S(0, c)


Triangles and Coordinate Proof

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ExampleExample

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Skills PracticeTriangles and Coordinate Proof

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1. right !FGH with legs 2. isosceles !KLP with 3. isosceles !AND witha units and b units base K$P$ 6b units long base A$D$ 5a long


4. 5. 6.

7. 8. 9.

10. Write a coordinate proof to prove that in an isosceles right triangle, the segment fromthe vertex of the right angle to the midpoint of the hypotenuse is perpendicular to thehypotenuse.

Given: isosceles right !ABC with "ABC the right angle and M the midpoint of A$C$Prove: B$M$ " A$C$

C(2a, 0)

A(0, 2a)

M

B(0, 0)

x

y

U(a, 0)

T(?, ?)

S(–a, 0)x

y

P(7b, 0)

R(?, ?)

N(0, 0)x

y

Q(?, ?)

R(2a, b)

P(0, 0)

x

y

N(3b, 0)

M(?, ?)

O(0, 0)x

y

Y(2b, 0)

Z(?, ?)

X(0, 0)x

y

B(2a, 0)

A(0, ?)

C(0, 0)

x

yN #5–

2a, b$

A(0, 0) D(5a, 0)x

y L(3b, c)

K(0, 0) P(6b, 0)x

yF(0, a)

G(0, 0) H(b, 0)



1. equilateral !SWY with 2. isosceles !BLP with 3. isosceles right !DGJsides %

14%a long base B$L$ 3b units long with hypotenuse D$J$ and

legs 2a units long


4. 5. 6.

NEIGHBORHOODS For Exercises 7 and 8, use the following information.Karina lives 6 miles east and 4 miles north of her high school. After school she works parttime at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.

7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall areat the vertices of a right triangle.

Given: !SKMProve: !SKM is a right triangle.

8. Find the distance between the mall and Karina’s home.

x

y

S(0, 0)

K(6, 4)

M(–2, 3)

x

y

P(2b, 0)

M(0, ?)

N(?, 0)x

y

C(?, 0)

E(0, ?)

B(–3a, 0)x

y S(?, ?)

J(0, 0) R#1–3b, 0$

x

yD(0, 2a)

G(0, 0) J(2a, 0)x

yP #3–

2b, c$

B(0, 0) L(3b, 0)x

y Y #1–8a, b$

W #1–4a, 0$S(0, 0)

Practice Triangles and Coordinate Proof

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Reading to Learn MathematicsTriangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7


Less

on

4-7

Pre-Activity How can the coordinate plane be useful in proofs?


From the coordinates of A, B, and C in the drawing in your textbook, whatdo you know about !ABC?

Reading the Lesson

1. Find the missing coordinates of each triangle.a. b.


a. Find the slope of S$R$ and the slope of S$T$.

b. Find the product of the slopes of S$R$ and S$T$. What does this tell you about S$R$ and S$T$?

c. What does your answer from part b tell you about !RST ?

d. Find SR and ST. What does this tell you about S$R$ and S$T$?

e. What does your answer from part d tell you about !RST?

f. Combine your answers from parts c and e to describe !RST as completely as possible.

g. Find m"SRT and m"STR.

h. Find m"OSR and m"OST.


3. Many students find it easier to remember mathematical formulas if they can put theminto words in a compact way. How can you use this approach to remember the slope andmidpoint formulas easily?

x

y

S(0, a)

R(–a, 0) T(a, 0)O(0, 0)

x

y

F(?, ?)E(?, a)

D(?, ?)x

y

T(a, ?)

R(?, b)

S(?, ?)


How Many Triangles?Each puzzle below contains many triangles. Count them carefully.Some triangles overlap other triangles.

How many triangles are there in each figure?

1. 2. 3.

4. 5. 6.

How many triangles can you form by joining points on each circle? List the vertices of each triangle.

7. 8.

8. 9. QR

P

U

S

TV

J K

O

L

MN

E F

I

GH

B

C

DA

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

Chapter 4 Test, Form 144


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. How would this triangle be classified by angles? A. acute B. equiangularC. obtuse D. right

2. What is the value of x if !ABC is equilateral?

A. !8 B. !"18"

C. "12" D. 2

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question.

3. What is m"2?A. 50 B. 70 C. 110 D. 120

4. What is m"4?A. 10 B. 60 C. 100 D. 120

5. What are the congruent triangles in the diagram? A. !ABC ! !EBD B. !ABE ! !CBDC. !AEB ! !CBD D. !ABE ! !CDB

6. If !CJW ! !AGS, m"A # 50, m"J # 45,and m"S # 16x $ 5, what is x? A. 17.5 B. 11.875C. 6 D. 5

7. Which postulate can be used to prove the triangles congruent? A. SSS B. SAS C. ASA D. AAS

8. What reason should be given for statement 5 in the proof?Given: D"B" is the perpendicular bisector of A"C".Prove: !ADB ! !CDB

Statements Reasons1. DB is the perpendicular bisector of A"C". 1. Given2. A"B" ! C"B" 2. Midpoint Theorem3. "ABD ! "CBD 3. ! line; all right # are !.4. D"B" ! D"B" 4. Reflexive Property5. !ADB ! !CDB 5.

A. SSS B. AAS C. ASA D. SAS

?

A C

D

B

C W

J

(16x ! 5)"

45"

A S

G

50"

A

E

CB

D

A C

B

10x # 5

6x ! 37.5x

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE

70"

2 13 460"

40"


Chapter 4 Test, Form 1 (continued)44

9.

10.

11.

12.

13.

Use the proof for Questions 9–10 and write the letter for the correct answer in the blank at theright of each question.

Given: L is the midpoint of J"M"; J"K" || N"M".Prove: !JKL ! !MNLStatements Reasons

1. L is the midpoint of J"M". 1. Given2. J"L" ! M"L" 2. Definition of midpoint3. J"K" || M"N" 3. Given4. "JKL ! "MNL 4. Alt. int. # are !.5. "JLK ! "MLN 5.6. !JKL ! !MNL 6.

9. What is the reason for "JLK ! "MLN?A. definition of midpointB. corresponding anglesC. vertical anglesD. alternate interior angles

10. What is the reason for !JKL ! !MNL?A. AAS B. ASA C. SAS D. SSS

Use the figure for Questions 11–12 and write the letter for the correct answer in the blank at the right of each question.

11. If !LMN is isosceles and T is the midpoint of L"N",which postulate can be used to prove !MLT ! !MNT?A. AAA B. AAS C. SAS D. ABC

12. If !MLT ! !MNT, what is used to prove "1 ! "2?A. CPCTCB. definition of isosceles triangleC. definition of perpendicularD. definition of angle bisector

13. What are the missing coordinates of this triangle?A. (2a, 2c) B. (2a, 0)C. (0, 2a) D. (a, 2c)

Bonus What is the classification by sides of a triangle with coordinates A(5, 0), B(0, 5), and C(!5, 0)?

x

yM(a, c)

N(?, ?)L(0, 0)

1 2

M

TL N

(Question 10)(Question 9)

N

K

L MJ

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2A44


Ass

essm

ents


1. What is the length of the sides of this equilateral triangle?A. 42 B. 30C. 15 D. 12

2. What is the classification of !ABC with vertices A(4, 1), B(2, !1), and C(!2, !1) by its sides?A. equilateral B. isosceles C. scalene D. right

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.

3. What is m"1?A. 40 B. 50 C. 70 D. 90

4. What is m"3?A. 40 B. 70 C. 90 D. 110

5. If !DJL ! !EGS, which segment in !EGS corresponds to D"L"?A. E"G" B. E"S" C. G"S" D. G"E"

6. Which triangles are congruent in the figure?A. !KLJ ! !MNL B. !JLK ! !NLMC. !JKL ! !LMN D. !JKL ! !MNL

Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question.

Given: R"J" || E"I"; R"I" bisects J"E".Prove: !RJN ! !IENStatements Reasons

1. R"J" || I"E" 1. Given2. "RJN ! "IEN 2.3. R"I" bisects J"E". 3. Given4. J"N" ! E"N" 4. Definition of bisector5. "RNJ ! "INE 5. Vert. # are !.6. !RJN ! !IEN 6.

7. What is the reason for statement 2 in the proof?A. Isosceles Triangle Theorem B. same side interior anglesC. corresponding angles D. Alternate Interior Angle Theorem

8. What is the reason for statement 6?A. ASA B. AAS C. SAS D. SSS

(Question 8)

(Question 7)

E N

R

J

I

LK

NJ

M

70"

50"1

2

3

6x # 3

9x # 123x ! 61.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2A (continued)44

9.

10.

11.

12.

13.

9. If !ABC is isosceles and A"E" ! F"C", which theorem or postulate can be used to prove !AEB ! !CFB? A. SSS B. SASC. ASA D. AAS


Given: D"A" || Y"N"; D"A" ! Y"N"Prove: "NDY ! "DNAStatements Reasons

1. D"A" || Y"N" 1. Given2. "ADN ! "YND 2. Alt. int. # are !.3. D"A" ! Y"N" 3. Given4. D"N" ! D"N" 4. Reflexive Property5. !NDY ! !DNA 5.6. "NDY ! "DNA 6.

10. What is the reason for statement 5?A. ASA B. AASC. SAS D. SSS

11. What is the reason for statement 6?A. Alt. int. "s are !. B. CPCTCC. Corr. angles are !. D. Isosceles Triangle Theorem

12. What is the classification of a triangle with vertices A(3, 3), B(6, !2), C(0, !2)by its sides?A. isosceles B. scaleneC. equilateral D. right

13. What are the missing coordinates of the triangle?A. (!2b, 0) B. (0, 2b)C. (!c, 0) D. (0, !c)

Bonus Name the coordinates of points A and C in isosceles right !ABC if point Cis in the second quadrant.

x

yB(0, a)

A(?, ?)

x

y(0, c)

(?, ?) (2b, 0)

(Question 11)(Question 10)

D A

NY

A C

B

E F

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2B44


Ass

essm

ents


1. What is the length of the sides of this equilateral triangle? A. 2.5 B. 5C. 15 D. 20

2. What is the classification of !ABC with vertices A(0, 0), B(4, 3), and C(4, !3)by its sides?A. equilateral B. isosceles C. scalene D. right

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.

3. What is m"1?A. 120 B. 90 C. 60 D. 30

4. What is m"2?A. 120 B. 90 C. 60 D. 30

5. If !TGS ! !KEL, which angle in !KEL corresponds to "T?A. "L B. "E C. "K D. "A

6. Which triangles are congruent in the figure?A. !HMN ! !HGN B. !HMN ! !NGHC. !NMH ! !NGH D. !MNH ! !HGN


Given: A"B" || C"D"; A"C" bisects B"D".Prove: !ABE ! !CDEStatements Reasons1. A"C" bisects B"D". 1. Given2. B"E" ! D"E" 2.3. A"B" || C"D" 3. Given4. "ABE ! "CDE 4. Alt. int. # are !.5. 5.Vert. # are !.6. !ABE ! !CDE 5. ASA

7. What is the reason for statement 2?A. Definition of bisector B. Midpoint TheoremC. Given D. Alternate Interior Angle Theorem

8. What is the statement for reason 5?A. "BEA ! "DEC B. "ABE ! "CDEC. "EAB ! "ECD D. "BEC ! "DEA

(Question 8)

(Question 7)

AB

CD

E

H G

NM

120"12

7x # 15

3x ! 54x1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2B (continued)44

9.

10.

11.

12.

13.

9. If A"F" ! D"E", A"B" ! F"C" and A"B" || F"C", which theorem or postulate can be used to prove !ABE ! !FCD?A. AAS B. ASAC. SAS D. SSS


Given: E"G" ! I"A"; "EGA ! "IAGProve: "GEN ! "AINStatements Reasons

1. E"G" ! I"A" 1. Given

2. "EGA ! "IAG 2. Given

3. G"A" ! G"A" 3. Reflexive Property

4. !EGA ! !IAG 4.

5. "GEN ! "AIN 5.

10. What is the reason for statement 4?A. SSS B. ASA C. SAS D. AAS

11. What is the reason for statement 5?A. Alt. int. # are !. B. Same Side Interior AnglesC. Corr. angles are !. D. CPCTC

12. What is the classification of a triangle with vertices A(!3, !1), B(!2, 2),C(3, 1) by its sides?A. scalene B. isoscelesC. equilateral D. right

13. What are the missing coordinates of the triangle?A. (a, 0) B. (b, 0)C. (c, 0) D. (0, c)

Bonus Find x in the triangle.(5x ! 60)" (2x ! 51)"

(30 # 10x)"(43 # 2x)"

x

y

(?, ?)

(a, 0)(#a, 0)

(Question 11)

(Question 10)

G A

N

E I

A DF E

CB

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2C44


Ass

essm

ents

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, and AC if !ABC is equilateral.

3. Find the measure of the sides of the triangle if the vertices of!EFG are E(!3, 3), F(1, !1), and G(!3, !5). Then classify thetriangle by its sides.


4. m"1

5. m"2

6. m"3

7. Identify the congruent triangles and name their correspondingcongruent angles.

8. Verify that !ABC ! !A%B%C%preserves congruence, assuming that corresponding angles arecongruent.

x

y

O

A%

B%

B

C%

AC

A

D F

GB

C

110" 2

1

3

A C

B

8x

7x ! 310x # 6

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2C (continued)44

9. ABCD is a quadrilateral with A"B" ! C"D" and A"B" || C"D". Name the postulate that could be used to prove !BAC ! !DCA. Choose from SSS, SAS, ASA, and AAS.

10. !KLM is an isosceles triangle and "1 ! "2. Name the theorem that could be used to determine "LKP ! "LMN. Then name thepostulate that could be used to prove !LKP ! !LMN. Choose from SSS, SAS, ASA, and AAS.

11. Use the figure to find m"1.

12. Find x.

13. Position and label isosceles !ABC with base A"B" b units longon the coordinate plane.

14. C"P" joins point C in isosceles right !ABC to the midpoint P, of A"B".Name the coordinates of P. Thendetermine the relationship between A"B" and C"P".

Bonus Without finding any other angles or sides congruent,which pair of triangles can be proved to be congruent bythe HL Theorem?

A

B

C D

E

F X

Y

Z M

N

O

x

yA(0, b)

B(b, 0)C(0, 0)

(10x ! 20)"15x "

(18x # 12)"

40"

190"

1

P1 2

N MK

L

A

D

B

C

NAME DATE PERIOD

9.

10.

11.

12.

13.

14.

B:

C(b–2, c)

B(b, 0)A

Chapter 4 Test, Form 2D44


Ass

essm

ents

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, AC if !ABC is isosceles.

3. Find the measure of the sides of the triangle if the vertices of!EFG are E(1, 4), F(5, 1), and G(2, !3). Then classify thetriangle by its sides.


4. m"1

5. m"2

6. m"3

7. Identify the congruent triangles and name their corresponding congruentangles.

8. Verify that !JKL ! !J%K%L%preserves congruence, assuming that corresponding angles arecongruent.

9. In quadrilateral JKLM, J"K" ! L"K"and M"K" bisects "LKJ. Name thepostulate that could be used to prove !MKL ! !MKJ. Choose from SSS, SAS, ASA, and AAS.

M

K

J

L

x

y

O

K%J%

K

L%

J

L

DB

EC

FA

70"1 3280"

A C

B

2x ! 20

9x # 5

5x ! 5

1.

2.

3.

4.

5.

6.

7.

8.

9.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2D (continued)44

10. !ABC is an isosceles triangle with B"D" ! A"C". Name the theorem that could be used to determine "A ! "C. Then name the postulate that could be used to prove !BDA ! !BDC. Choose from SSS, SAS, ASA, and AAS.

11. Use the figure to find m"1.

12. Find x.

13. Position and label equilateral !KLM with side lengths 3a units long on the coordinate plane.

14. M"N" joins the midpoint of A"B" and the midpoint of A"C" in !ABC. Findthe coordinates of M and N, and theslopes of M"N" and B"C".

Bonus Without finding any other angles or sides congruent,which pair of triangles can be proved to be congruent by the LL Theorem?

A

B

C D

E

F X

Y

Z M

N

O

x

y

B(a, 0)

C(0, b)

M(?, ?)

N(?, ?)

A(0, 0)

(6x ! 4)"

(18x # 8)"

80"1

B D

C

A

NAME DATE PERIOD

10.

11.

12.

13.

14.

B:

M(3a, 0)

L(1.5a, b)

K(0, 0)

Chapter 4 Test, Form 344


Ass

essm

ents

1. If !ABC is isosceles, "B is the vertex angle, AB # 20x ! 2,BC # 12x $ 30, and AC # 25x, find x and the measure of eachside of the triangle.

2. Given A(0, 4), B(5, 4), and C(!3, !2), find the measure of thesides of the triangle. Then classify the triangle by its sides and angles.

Use the figure to answer Questions 3–5.

3. Find x.

4. m"1, if m"1 # 4x $ 10.

5. m"2

6. Verify that the following preserves congruence, assuming thatcorresponding angles are congruent. !ABC is reflected overthe x-axis as follows.A(!1, 1) " A%(!1, !1)B(4, 2) " B%(4, !2)C(1, 5) " C%(1, !5)Verify !ABC ! !A%B%C%.

7. Determine whether !GHI ! !JKL, given G(1, 2), H(5, 4),I(3, 6) and J(!4, !5), K(0, !3), L(!2, !1). Explain.

8. In the figure, A"C" ! F"D", A"B" || D"E",and A"C" || F"D". Name the postulate that could be used to prove !ABC ! !DEC. Choose from SSS,SAS, ASA, and AAS.

D

E

A

BF

C

(8x # 30)"(3x # 10)"2

1

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 3 (continued)44

For Questions 9 and 10, complete this two-column proof.

Given: !ABC is an isosceles triangle with base A"C".D is the midpoint of A"C".

Prove: B"D" bisects "ABC.

Statements Reasons1. !ABC is isosceles 1. Given

with base A"C".

2. A"B" ! C"B" 2. Def. of isosceles triangle.

3. "A ! "C 3.

4. D is the midpoint of A"C". 4. Given

5. A"D" ! C"D" 5. Midpoint Theorem

6. !ABD ! !CBD 6.

7. "1 ! "2 7. CPCTC

8. B"D" bisects "ABC. 8. Def. of angle bisector

11. Find x.

12. Position and label isosceles !ABC with base A"B" (a $ b) unitslong on a coordinate plane

Bonus In the figure, !ABC is isosceles, !ADC is equilateral,!AEC is isosceles, and the measures of "9, "1, and "3are all equal. Find the measures of the nine numberedangles.

987

1

65

23

4B

C

DE

A

(15x ! 15)"(21x # 3)"

(17x ! 9)"

(Question 10)

(Question 9)

21

A C

B

D

NAME DATE PERIOD

9.

10.

11.

12.

B:

B(a ! b, 0)

C(a ! b, c)2

A(0, 0)

Chapter 4 Open-Ended Assessment44


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1.

a. Classify the triangle by its angles and sides.

b. Show the steps needed to solve for x.

2. a. Describe how to determine whether a triangle with coordinates A(1, 4),B(1, !1), and C(!4, 4) is an equilateral triangle.

b. Is the triangle equilateral? Explain.

3. Explain how to find m"1 and m"2 in the figure.

4.

a. State the theorem or postulate that can be used to prove that thetriangles are congruent.

b. List their corresponding congruent angles and sides.

5.

Given: A"B" || D"E", A"D" bisects B"E".Prove: !ABC ! !DEC by using the ASA postulate.

A

C

B

E D

J

DL

E

G

S

C

BD E

A 58"

40" 62"

1 2

(9x ! 4)"

(20x # 10)"

NAME DATE PERIOD

SCORE


Chapter 4 Vocabulary Test/Review44

Choose from the terms above to complete each sentence.

1. A triangle that is equilateral is also called a(n) .

2. A(n) has at least one obtuse angle.

3. The sum of the is equivalent to the exterior angle of atriangle.

4. The angles of an isosceles triangle are congruent.

5. A triangle with different measures for each side is classified asa(n) .

6. A organizes a series of statements in logical orderwritten in boxes and uses arrows to indicate the order of thestatements.

7. A triangle that is translated, reflected or rotated and preservesits shape, is said to be a(n) .

8. The ASA postulate involves two corresponding angles andtheir corresponding .

9. A uses figures in the coordinate plane and algebra toprove geometric concepts.

10. The is formed by the congruent legs of an isoscelestriangle.

In your own words—

11. corollary

12. congruent triangles

13. acute triangle

?

?

?

?

?

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

acute trianglebase anglescongruence

transformationscongruent triangles

coordinate proofcorollaryequiangular triangleequilateral triangleexterior angle

flow proofincluded angleincluded sideisosceles triangleobtuse triangle

remote interior anglesright trianglescalene trianglevertex angle

NAME DATE PERIOD

SCORE

Chapter 4 Quiz (Lessons 4–1 and 4–2)

44


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. Use a protractor to classify the triangle by its angles and sides.

2. STANDARDIZED TEST PRACTICE What is the best classification of this triangle by its angles and sides?A. acute isosceles B. right isoscelesC. obtuse isosceles D. obtuse equilateral

3. If !ABC is an isosceles triangle, "B is the vertex angle,AB # 6x $ 3, BC # 8x ! 1, and AC # 10x ! 10, find x and themeasures of each side of the triangle.

4. If A(1, 5), B(3, !2), and C(!3, 0), find the measures of thesides of !ABC. Then classify the triangle by its sides.

Find the measure of each angle in the figure.

5. m"1 6. m"2

7. m"3 8. m"4

9. m"5 10. m"6

70"65"

1

2 34

56 107"

43"


44

1.

2.

3.

4.

1. Identify the congruent triangles in the figure.

2. STANDARDIZED TEST PRACTICE If !JGO ! !RWI, whichangle corresponds to "I?A. "J B. "R C. "G D. "O

3. Verify that the following preserves congruence assuming that correspondingangles are congruent. !ABC ! !A%B%C%

4. In quadrilateral EFGH, F"G" ! H"E", and F"G" || H"E". Name the postulate that could be used to prove !EHF ! !GFH. Choosefrom SSS, SAS, ASA, and AAS.

E H

GF

x

y

OC%

BB%

AC

A%

K

MLN

NAME DATE PERIOD

SCORE



44

1.

2.

3.

4.

For Questions 1 and 2, complete the two-column proof bysupplying the missing information for each correspondinglocation.

Given: "Z ! "C; A"K" bisects "ZKC.Prove: !AKZ ! !AKCStatements Reasons

1. "Z ! "C; A"K" bisects "ZKC. 1. Given2. "ZKA ! "CKA 2.3. A"K" ! A"K" 3. Reflexive Property4. !AKZ ! !AKC 4.

Refer to the figure for Questions 3 and 4.

3. Find m"1. 4. Find m"2.21

(Question 2)

(Question 1)

A

K

Z C

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Chapter 4 Quiz (Lesson 4–7)

44

1.

2.

3.

4.

5.

1. Find the missing coordinates.

Position and label each triangle on a coordinate plane.

2. Right !DJL with hypotenuse D"J"; LJ # "12"DL and D"L" is

a units long.

3. isosceles !EGS with base E"S" "12"b units long

For Questions 4 and 5, complete the coordinate proof bysupplying the missing information for each correspondinglocation.

Given: !ABC with A(!1, 1), B(5, 1), and C(2, 6).Prove: !ABC is isosceles.By the Distance Formula the lengths of the three sides are asfollows: . Since , !ABC is isosceles.(Question 5)(Question 4)

x

y

C(?, ?)

I(?, ?)

M(#b, 0)

S(1–2b, 0)

G(1–4b, c)

E(0, 0)

J(a–2, 0)

D(0, a)

L(0, 0)

NAME DATE PERIOD

SCORE

Chapter 4 Mid-Chapter Test (Lessons 4–1 through 4–3)

44


Ass

essm

ents

1. What is the best classification for this triangle?A. acute scaleneB. obtuse equilateralC. acute isoscelesD. obtuse isosceles


2. What is m"1?A. 50 B. 60C. 100 D. 105

3. What is m"2?A. 40 B. 50C. 60 D. 100

4. If !SJL ! !DMT, which segment in !DMT corresponds to L"S" in !SJL?A. D"T" B. T"D"C. M"D" D. M"T"

50"

50"

60"

2

1

5.

6.

7.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

Part II

5. Find the measures of the sides of !ABC and classify it by itssides. A(1, 3), B(5, !2), and C(0, !4)

6. In !ABC and !A%B%C%, "A ! "A%,"B ! "B%, and "C ! "C%. Find the lengths needed to prove !ABC ! !A%B%C%.

7. What information would you need to know about P"O" and L"N" for !LMPto be congruent to !NMO by SSS? P O

N

L

M

x

y

O

C%

B

B%A

CA%

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 4 Cumulative Review(Chapters 1–4)

44

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

1. Name the geometric figure that is modeled by the second handof a clock. (Lesson 1-1)

2. Find the precision for a measurement of 36 inches. (Lesson 1-2)

For Questions 3–5, use the number line.

3. Find BC. (Lesson 1-3)

4. Find the coordinate of the midpoint of A"D". (Lesson 1-3)

5. If B is the midpoint of a segment having one endpoint at E,what is the coordinate of its other endpoint? (Lesson 1-3)

For Questions 6 and 7, determine whether each statementis always, sometimes, or never true. Explain your answer.(Lesson 2-5)

6. If D"E" ! E"F", then E is the midpoint of D"F".

7. If points A and B lie in plane Q , then AB!"# lies in Q .

8. Find the slope of a line parallel to x # 2. (Lesson 3-3)

9. Find the distance between y # !9 and y # !5. (Lesson 3-6)

For Questions 10–12, use the figure.

10. Name the segment that represents the distance from F to AD!"#. (Lesson 3-6)

11. Classify !ADC. (Lesson 4-1)

12. Find m"ACD. (Lesson 4-2)

13. Name the corresponding congruent angles and sides for !PQR ! !HGB. (Lesson 4-3)

14. If "QRP ! "SRT, and R is the midpoint of P"T", which theorem or postulate can beused to prove !QRP ! !SRT? Choose from SSS, SAS, ASA, and AAS. (Lesson 4-5)

15. Name the missing coordinates of !GEF. (Lesson 4-7)

x

y

F(2b, ?)D(0, 0) G(?, ?)

E(?, ?)

Q S

P TR

50"30"

85"

ED

A CB

F

#5 #4 #3 #2 #1#10 #9 #8 #7 #6 0 1 2 3 4 5 6

B ECA D

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–4)


1. If m"1 # 5x ! 4, and m"2 # 52 ! 9y, which values for x and ywould make "1 and "2 complementary? (Lesson 1-5)

A. x # 2, y # 12 B. x # 12, y # 2

C. x # 27, y # "13" D. x # "

13", y # 27

2. Which is not a polygon? (Lesson 1-6)

E. F. G. H.

3. Complete the statement so that its conditional and its converseare true.If "1 ! "2, then "1 and "2 . (Lesson 2-3)

A. are supplementary. B. are complementary.C. have the same measure. D. are alternate interior angles.

4. Complete this proof. (Lesson 2-7)

Given: U"V" ! V"W"V"W" ! W"X"

Prove: UV # WXProof:Statements Reasons

1. U"V" ! V"W"; V"W" ! W"X" 1. Given2. UV # VW; VW # WX 2.3. UV # WX 3. Transitive Property

E. Definition of congruent segmentsF. Substitution PropertyG. Segment Addition PostulateH. Symmetric Property

5. Which equation has a slope of "13" and a y-intercept of !2? (Lesson 3-4)

A. y # "13"x $ 2 B. y # "

13"x ! 2

C. y # 2x ! "13" D. y # !2x $ "

13"

6. Classify !DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1)

E. acute F. equiangular G. obtuse H. right

7. Which postulate or theorem can be used to prove !ABD ! !CBD? (Lesson 4-4)

A. SAS B. SSSC. ASA D. AAS

A CB

D

?

U V

XW

?

NAME DATE PERIOD

SCORE 44

Part 1: Multiple ChoiceInstructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7. A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

8. What is the y-coordinate of the midpoint ofA(12, 6) and B(!15, !6)? (Lesson 1-3)

9. If m"1 # 112, find m"10.(Lesson 3-2)

10. If J"K" || L"M", then "4 must be supplementary to" . (Lesson 3-5)

11. Find PR if !PQR is isosceles, "Q is the vertexangle, PQ # 4x ! 8, QR # x $ 7, and PR # 6x ! 12. (Lesson 4-1)

?

6

7

4

5H

L

M

J

K

!k

m

n

1 23

9 1011 12

45 6

7 8

NAME DATE PERIOD

44

Part 2: Grid InInstructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short ResponseInstructions: Show your work or explain in words how you found your answer.

8. 9.

10. 11.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

12. The perimeter of a regular pentagon is 14.5 feet. If each sidelength of the pentagon is doubled, what is the new perimeter?(Lesson 1-6)

13. Make a conjecture about the next number in the sequence 5, 7,11, 17, 25. (Lesson 2-1)

14. Find m"PQR. (Lesson 4-2)

15. If PQ # QS, QS # SR, and m"R # 20, find m"PSQ.(Lesson 4-6) P RS

Q

63" 125" 10"P S

Q

RT

12.

13.

14.

15.

0 6 8

6 1 8

Standardized Test PracticeStudent Record Sheet (Use with pages 232–233 of the Student Edition.)

44

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 DCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 12 and 14, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

9 12 14

10

11

12 (grid in)

13

14 (grid in)

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 15–16 on the back of this paper.


Stu

dy

Gu

ide

and I

nte

rven

tion

Cla

ssify

ing

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill18

3G

lenc

oe G

eom

etry

Lesson 4-1

Cla

ssif

y Tr

ian

gle

s b

y A

ng

les

One

way

to

clas

sify

a t

rian

gle

is b

y th

e m

easu

res

of it

s an

gles

.

•If

one

of th

e an

gles

of a

tria

ngle

is a

n ob

tuse

ang

le, t

hen

the

tria

ngle

is a

n ob

tuse

tri

angl

e.

•If

one

of th

e an

gles

of a

tria

ngle

is a

rig

ht a

ngle

, the

n th

e tr

iang

le is

a r

ight

tri

angl

e.

•If

all t

hree

of th

e an

gles

of a

tria

ngle

are

acu

te a

ngle

s, th

en th

e tr

iang

le is

an

acut

e tr

iang

le.

•If

all t

hree

ang

les

of a

n ac

ute

tria

ngle

are

con

grue

nt, t

hen

the

tria

ngle

is a

n eq

uian

gula

r tr

iang

le.

Cla

ssif

y ea

ch t

rian

gle.

a.

All

thre

e an

gles

are

con

grue

nt,s

o al

l thr

ee a

ngle

s ha

ve m

easu

re 6

0°.

The

tri

angl

e is

an

equi

angu

lar

tria

ngle

.

b.

The

tri

angl

e ha

s on

e an

gle

that

is o

btus

e.It

is a

n ob

tuse

tri

angl

e.

c.

The

tri

angl

e ha

s on

e ri

ght

angl

e.It

is a

rig

ht t

rian

gle.

Cla

ssif

y ea

ch t

rian

gle

as a

cute

,equ

ian

gula

r,ob

tuse

,or

righ

t.

1.2.

3.

righ

tob

tuse

equi

angu

lar

4.5.

6.

acut

eri

ght

obtu

se

60!

28!

92!

FDB

45!

45!

90!

XY

W

65!

65!

50!

UV

T

60!

60!

60!

Q

RS

120!

30!

30!

NO

P

67!

90!

23!

K LM

90!

60!

30!

G

HJ

25!

35!

120!

DF

E

60!A

BC

Exam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

oe/M

cGra

w-H

ill18

4G

lenc

oe G

eom

etry

Cla

ssif

y Tr

ian

gle

s b

y Si

des

You

can

clas

sify

a t

rian

gle

by t

he m

easu

res

of it

s si

des.

Equ

al n

umbe

rs o

f ha

sh m

arks

indi

cate

con

grue

nt s

ides

.

•If

all t

hree

side

s of

a tr

iang

le a

re c

ongr

uent

, the

n th

e tr

iang

le is

an

equi

late

ral t

rian

gle.

•If

at le

ast t

wo

side

s of

a tr

iang

le a

re c

ongr

uent

, the

n th

e tr

iang

le is

an

isos

cele

s tr

iang

le.

•If

no tw

osi

des

of a

tria

ngle

are

con

grue

nt, t

hen

the

tria

ngle

is a

sca

lene

tri

angl

e.

Cla

ssif

y ea

ch t

rian

gle.

a.b.

c.

Tw

o si

des

are

cong

ruen

t.A

ll th

ree

side

s ar

e T

he t

rian

gle

has

no p

air

The

tri

angl

e is

an

cong

ruen

t.T

he t

rian

gle

of c

ongr

uent

sid

es.I

t is

is

osce

les

tria

ngle

.is

an

equi

late

ral t

rian

gle.

a sc

alen

e tr

iang

le.

Cla

ssif

y ea

ch t

rian

gle

as e

quil

ate

ral,

isos

cele

s,or

sca

len

e.

1.2.

3.

scal

ene

equi

late

ral

scal

ene

4.5.

6.

isos

cele

sis

osce

les

equi

late

ral

7.F

ind

the

mea

sure

of

each

sid

e of

equ

ilate

ral !

RS

Tw

ith

RS

!2x

"2,

ST

!3x

,an

d T

R!

5x#

4.2

8.F

ind

the

mea

sure

of

each

sid

e of

isos

cele

s !

AB

Cw

ith

AB

!B

Cif

AB

!4y

,B

C!

3y"

2,an

d A

C!

3y.

AB

"B

C"

8,A

C"

6

9.F

ind

the

mea

sure

of

each

sid

e of

!A

BC

wit

h ve

rtic

es A

(#1,

5),B

(6,1

),an

d C

(2,#

6).

Cla

ssif

y th

e tr

iang

le.

AB

"B

C"

%65!

,AC

"%

130

!;!

AB

Cis

isos

cele

s.

DE

Fx

xx

8x32

x

32x

B CA

UW

S

1217

19Q

O

MG

KI

18

1818

21

3!

"G

CA

2312

15X

V

TN

RP

LJ

H

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Cla

ssify

ing

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-1

4-1

Exam

ple

Exam

ple

Exercis

esExercis

es

Answers (Lesson 4-1)


An

swer

s

Skil

ls P

ract

ice

Cla

ssify

ing

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-1

4-1

©G

lenc

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w-H

ill18

5G

lenc

oe G

eom

etry

Lesson 4-1

Use

a p

rotr

acto

r to

cla

ssif

y ea

ch t

rian

gle

as a

cute

,equ

ian

gula

r,ob

tuse

,or

righ

t.

1.2.

3.

equi

angu

lar

obtu

seri

ght

4.5.

6.

acut

eob

tuse

acut

e

Iden

tify

th

e in

dic

ated

typ

e of

tri

angl

es.

7.ri

ght

8.is

osce

les

!A

BE

,!B

CE

!B

CD

,!B

DE

9.sc

alen

e10

.obt

use

!A

BE

,!B

CE

!B

DE

ALG

EBR

AF

ind

xan

d t

he

mea

sure

of

each

sid

e of

th

e tr

ian

gle.

11.!

AB

Cis

equ

ilate

ral w

ith

AB

!3x

#2,

BC

!2x

"4,

and

CA

!x

"10

.x

"6,

AB

"16

,BC

"16

,CA

"16

12.!

DE

Fis

isos

cele

s,"

Dis

the

ver

tex

angl

e,D

E!

x"

7,D

F!

3x#

1,an

d E

F!

2x"

5.x

"4,

DE

"11

,DF

"11

,EF

"13

Fin

d t

he

mea

sure

s of

th

e si

des

of

!R

ST

and

cla

ssif

y ea

ch t

rian

gle

by i

ts s

ides

.

13.R

(0,2

),S

(2,5

),T

(4,2

)

RS

"%

13!,S

T"

%13!

,RT

"4;

isos

cele

s

14.R

(1,3

),S

(4,7

),T

(5,4

)

RS

"5,

ST

"%

10!,R

T"

%17!

;sca

lene

EC

D

AB

©G

lenc

oe/M

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6G

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oe G

eom

etry

Use

a p

rotr

acto

r to

cla

ssif

y ea

ch t

rian

gle

as a

cute

,equ

ian

gula

r,ob

tuse

,or

righ

t.

1.2.

3.

obtu

seac

ute

righ

t

Iden

tify

th

e in

dic

ated

typ

e of

tri

angl

es i

f A !

B!"

A!D!

"B!

D!"

D!C!

,B!E!

"E!

D!,A!

B!!

B!C!

,an

d E!

D!!

D!C!

.

4.ri

ght

5.ob

tuse

!A

BC

,!C

DE

!B

ED

,!B

DC

6.sc

alen

e7.

isos

cele

s!

AB

C,!

CD

E!

AB

D,!

BE

D,!

BD

C

AL

GE

BR

AF

ind

xan

d t

he

mea

sure

of

each

sid

e of

th

e tr

ian

gle.

8.!

FG

His

equ

ilate

ral w

ith

FG

!x

"5,

GH

!3x

#9,

and

FH

!2x

#2.

x"

7,FG

"12

,GH

"12

,FH

"12

9.!

LM

Nis

isos

cele

s,"

Lis

the

ver

tex

angl

e,L

M!

3x#

2,L

N!

2x"

1,an

d M

N!

5x#

2.x

"3,

LM"

7,LN

"7,

MN

"13

Fin

d t

he

mea

sure

s of

th

e si

des

of

!K

PL

and

cla

ssif

y ea

ch t

rian

gle

by i

ts s

ides

.

10.K

(#3,

2) P

(2,1

),L

(#2,

#3)

KP

"%

26!,P

L"

4%2!,

LK"

%26!

;iso

scel

es

11.K

(5,#

3),P

(3,4

),L

(#1,

1)K

P"

%53!

,PL

"5,

LK"

2%13!

;sca

lene

12.K

(#2,

#6)

,P(#

4,0)

,L(3

,#1)

KP

"2%

10!,P

L"

5%2!,

LK"

5%2!;

isos

cele

s

13.D

ESIG

ND

iana

ent

ered

the

des

ign

at t

he r

ight

in a

logo

con

test

sp

onso

red

by a

wild

life

envi

ronm

enta

l gro

up.U

se a

pro

trac

tor.

How

man

y ri

ght

angl

es a

re t

here

?5

AC

DEB

Pra

ctic

e (A

vera

ge)

Cla

ssify

ing

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-1

4-1



Rea

din

g t

o L

earn

Math

emati

csC

lass

ifyin

g Tr

iang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-1

4-1

©G

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7G

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eom

etry

Lesson 4-1

Pre-

Act

ivit

yW

hy

are

tria

ngl

es i

mp

orta

nt

in c

onst

ruct

ion

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

1 at

the

top

of

page

178

in y

our

text

book

.

•W

hy a

re t

rian

gles

use

d fo

r br

aces

in c

onst

ruct

ion

rath

er t

han

othe

r sh

apes

?S

ampl

e an

swer

:Tri

angl

es li

e in

a p

lane

and

are

rig

id s

hape

s.•

Why

do

you

thin

k th

at is

osce

les

tria

ngle

s ar

e us

ed m

ore

ofte

n th

ansc

alen

e tr

iang

les

in c

onst

ruct

ion?

Sam

ple

answ

er:I

sosc

eles

tria

ngle

s ar

e sy

mm

etri

cal.

Rea

din

g t

he

Less

on

1.Su

pply

the

cor

rect

num

bers

to

com

plet

e ea

ch s

ente

nce.

a.In

an

obtu

se t

rian

gle,

ther

e ar

e ac

ute

angl

e(s)

,ri

ght

angl

e(s)

,and

obtu

se a

ngle

(s).

b.In

an

acut

e tr

iang

le,t

here

are

ac

ute

angl

e(s)

,ri

ght

angl

e(s)

,and

obtu

se a

ngle

(s).

c.In

a r

ight

tri

angl

e,th

ere

are

acut

e an

gle(

s),

righ

t an

gle(

s),a

nd

obtu

se a

ngle

(s).

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.

A r

ight

tri

angl

e is

sca

lene

.so

met

imes

b.A

n ob

tuse

tri

angl

e is

isos

cele

s.so

met

imes

c.A

n eq

uila

tera

l tri

angl

e is

a r

ight

tri

angl

e.ne

ver

d.A

n eq

uila

tera

l tri

angl

e is

isos

cele

s.al

way

se.

An

acut

e tr

iang

le is

isos

cele

s.so

met

imes

f.A

sca

lene

tri

angl

e is

obt

use.

som

etim

es

3.D

escr

ibe

each

tri

angl

e by

as

man

y of

the

fol

low

ing

wor

ds a

s ap

ply:

acut

e,ob

tuse

,rig

ht,

scal

ene,

isos

cele

s,or

equ

ilat

eral

.a.

b.c.

acut

e,sc

alen

eob

tuse

,iso

scel

esri

ght,

scal

ene

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

nonm

athe

mat

ical

defi

niti

on o

f th

e sa

me

wor

d.H

ow is

the

use

of

the

wor

d ac

ute,

whe

n us

ed t

o de

scri

beac

ute

pain

,rel

ated

to

the

use

of t

he w

ord

acut

ew

hen

used

to

desc

ribe

an

acut

e an

gle

oran

acu

te t

rian

gle?

Sam

ple

answ

er:B

oth

are

rela

ted

to t

he m

eani

ng o

f ac

ute

as s

harp

.An

acut

e pa

inis

a s

harp

pai

n,an

d an

acu

te a

ngle

can

beth

ough

t of

as

an a

ngle

with

a s

harp

poi

nt.I

n an

acu

te tr

iang

leal

l of

the

angl

es a

re a

cute

.

5

34

135!

80!70!

30!

01

20

03

10

2

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hem

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you

read

geo

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ry,y

ou m

ay n

eed

to d

raw

a d

iagr

am t

o m

ake

the

text

easi

er t

o un

ders

tand

.

Con

sid

er t

hre

e p

oin

ts,A

,B,a

nd

Con

a c

oord

inat

e gr

id.

Th

e y-

coor

din

ates

of

Aan

d B

are

the

sam

e.T

he

x-co

ord

inat

e of

Bis

grea

ter

than

th

e x-

coor

din

ate

of A

.Bot

h c

oord

inat

es o

f C

are

grea

ter

than

th

e co

rres

pon

din

g co

ord

inat

es o

f B

.Is

tria

ngl

e A

BC

acu

te,r

igh

t,or

obt

use

?

To a

nsw

er t

his

ques

tion

,fir

st d

raw

a s

ampl

e tr

iang

le

that

fit

s th

e de

scri

ptio

n.

Side

AB

mus

t be

a h

oriz

onta

l seg

men

t be

caus

e th

e y-

coor

dina

tes

are

the

sam

e.Po

int

Cm

ust

be lo

cate

d to

the

rig

ht a

nd u

p fr

om p

oint

B.

Fro

m t

he d

iagr

am y

ou c

an s

ee t

hat

tria

ngle

AB

Cm

ust

be o

btus

e.

An

swer

eac

h q

ues

tion

.Dra

w a

sim

ple

tri

angl

e on

th

e gr

id a

bove

to

hel

p y

ou.

1.C

onsi

der

thre

e po

ints

,R,S

,and

2.

Con

side

r th

ree

nonc

ollin

ear

poin

ts,

Ton

a c

oord

inat

e gr

id.T

he

J,K

,and

Lon

a c

oord

inat

e gr

id.T

hex-

coor

dina

tes

of R

and

Sar

e th

ey-

coor

dina

tes

of J

and

Kar

e th

esa

me.

The

y-c

oord

inat

e of

Tis

sam

e.T

he x

-coo

rdin

ates

of K

and

Lbe

twee

n th

e y-

coor

dina

tes

of R

are

the

sam

e.Is

tri

angl

e JK

L a

cute

,an

d S

.The

x-c

oord

inat

e of

Tis

less

righ

t,or

obt

use?

righ

tth

an t

he x

-coo

rdin

ate

of R

.Is

angl

eR

of t

rian

gle

RS

T a

cute

,rig

ht,o

r ob

tuse

?ac

ute

3.C

onsi

der

thre

e no

ncol

linea

r po

ints

,4.

Con

side

r th

ree

poin

ts,G

,H,a

nd I

D,E

,and

Fon

a c

oord

inat

e gr

id.

on a

coo

rdin

ate

grid

.Poi

nts

Gan

d T

he x

-coo

rdin

ates

of D

and

Ear

eH

are

on t

he p

osit

ive

y-ax

is,a

ndop

posi

tes.

The

y-c

oord

inat

es o

f Dan

dth

e y-

coor

dina

te o

f Gis

tw

ice

the

Ear

e th

e sa

me.

The

x-c

oord

inat

e of

y-co

ordi

nate

of H

.Poi

nt I

is o

n th

e F

is 0

.Wha

t ki

nd o

f tr

iang

le m

ust

posi

tive

x-a

xis,

and

the

x-co

ordi

nate

!D

EF

be:

scal

ene,

isos

cele

s,or

of I

is g

reat

er t

han

the

y-co

ordi

nate

equi

late

ral?

isos

cele

sof

G.I

s tr

iang

le G

HI

scal

ene,

isos

cele

s,or

equ

ilate

ral?

scal

eneB

A

Q x

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

PE

RIO

D__

___

4-1

4-1

Exam

ple

Exam

ple



An

swer

s

Stu

dy

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ide

and I

nte

rven

tion

Ang

les

of T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

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ATE

____

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RIO

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An

gle

Su

m T

heo

rem

If t

he m

easu

res

of t

wo

angl

es o

f a

tria

ngle

are

kno

wn,

the

mea

sure

of

the

thir

d an

gle

can

alw

ays

be f

ound

.

Ang

le S

umT

he s

um o

f the

mea

sure

s of

the

angl

es o

f a tr

iang

le is

180

.Th

eore

mIn

the

figur

e at

the

right

, m"

A"

m"

B"

m"

C!

180.

CA

B

Fin

d m

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.

m"

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m"

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m"

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le S

umT

heor

em

25 "

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m"

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Sub

stitu

tion

60 "

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Add

.

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trac

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from

eac

h si

de.

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d t

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le S

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each

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tex

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tri

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e an

gle

form

ed b

y on

e si

dean

d an

ext

ensi

on o

f th

e ot

her

side

is c

alle

d an

ext

erio

r an

gle

of t

he t

rian

gle.

For

each

exte

rior

ang

le o

f a

tria

ngle

,the

rem

ote

inte

rior

an

gles

are

the

inte

rior

ang

les

that

are

not

adja

cent

to

that

ext

erio

r an

gle.

In t

he d

iagr

am b

elow

,"B

and

"A

are

the

rem

ote

inte

rior

angl

es f

or e

xter

ior

"D

CB

.

Ext

erio

r A

ngle

The

mea

sure

of a

n ex

terio

r an

gle

of a

tria

ngle

is e

qual

to

Theo

rem

the

sum

of t

he m

easu

res

of th

e tw

o re

mot

e in

terio

r an

gles

.m

"1

!m

"A

"m

"B

AC

B

D1

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dy

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ide

and I

nte

rven

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(con

tinu

ed)

Ang

les

of T

rian

gles

NA

ME

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4-2

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1



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les

of T

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gles

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ME

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4-2

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Lesson 4-2

Fin

d t

he

mis

sin

g an

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mea

sure

s.

1.27

2.17

,17

Fin

d t

he

mea

sure

of

each

an

gle.

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55

4.m

"2

55

5.m

"3

70

Fin

d t

he

mea

sure

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"1

125

7.m

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8.m

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95

Fin

d t

he

mea

sure

of

each

an

gle.

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140

10.m

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40

11.m

"3

65

12.m

"4

75

13.m

"5

115

Fin

d t

he

mea

sure

of

each

an

gle.

14.m

"1

27

15.m

"2

2763

!

1

2D

AC

B

80!

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52

3

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3

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3

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d t

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sure

of

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gle.

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97

4.m

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5.m

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62

Fin

d t

he

mea

sure

of

each

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gle.

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104

7.m

"4

45

8.m

"3

65

9.m

"2

79

10.m

"5

73

11.m

"6

147

Fin

d t

he

mea

sure

of

each

an

gle

if "

BA

Dan

d

"B

DC

are

righ

t an

gles

an

d m

"A

BC

"84

.

12.m

"1

26

13.m

"2

32

14.C

ON

STR

UC

TIO

NT

he d

iagr

am s

how

s an

ex

ampl

e of

the

Pra

tt T

russ

use

d in

bri

dge

cons

truc

tion

.Use

the

dia

gram

to

find

m"

1.55

145!

1

64!

1

2AB

C

D

118!

36!

68!

70! 65

! 82!

1

2

34

5

6

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3

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Pra

ctic

e (A

vera

ge)

Ang

les

of T

rian

gles

NA

ME

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ATE

____

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4-2

4-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csA

ngle

s of

Tri

angl

es

NA

ME

____

____

____

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____

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____

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ATE

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4-2

4-2

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Lesson 4-2

Pre-

Act

ivit

yH

ow a

re t

he

angl

es o

f tr

ian

gles

use

d t

o m

ake

kit

es?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

2 at

the

top

of

page

185

in y

our

text

book

.

The

fra

me

of t

he s

impl

est

kind

of

kite

div

ides

the

kit

e in

to f

our

tria

ngle

s.D

escr

ibe

thes

e fo

ur t

rian

gles

and

how

the

y ar

e re

late

d to

eac

h ot

her.

Sam

ple

answ

er:T

here

are

tw

o pa

irs

of r

ight

tri

angl

es t

hat

have

the

sam

e si

ze a

nd s

hape

.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.N

ame

the

thre

e in

teri

or a

ngle

s of

the

tri

angl

e.(U

se t

hree

lett

ers

to n

ame

each

ang

le.)

"B

AC

,"A

BC

,"B

CA

b.N

ame

thre

e ex

teri

or a

ngle

s of

the

tri

angl

e.(U

se t

hree

lett

ers

to n

ame

each

ang

le.)

"E

AB

,"D

BC

,"FC

Ac.

Nam

e th

e re

mot

e in

teri

or a

ngle

s of

"E

AB

."

AB

C,"

BC

Ad.

Fin

d th

e m

easu

re o

f ea

ch a

ngle

wit

hout

usi

ng a

pro

trac

tor.

i."

DB

C62

ii.

"A

BC

118

iii.

"A

CF

157

iv.

"E

AB

141

2.In

dica

te w

heth

er e

ach

stat

emen

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,rep

lace

the

unde

rlin

ed w

ord

or n

umbe

r w

ith

a w

ord

or n

umbe

r th

at w

ill m

ake

the

stat

emen

t tr

ue.

a.T

he a

cute

ang

les

of a

rig

ht t

rian

gle

are

.fa

lse;

com

plem

enta

ryb.

The

sum

of

the

mea

sure

s of

the

ang

les

of a

ny t

rian

gle

is

.fa

lse;

180

c.A

tri

angl

e ca

n ha

ve a

t m

ost

one

righ

t an

gle

or

angl

e.fa

lse;

obtu

sed.

If t

wo

angl

es o

f on

e tr

iang

le a

re c

ongr

uent

to

two

angl

es o

f an

othe

r tr

iang

le,t

hen

the

thir

d an

gles

of

the

tria

ngle

s ar

e .

true

e.T

he m

easu

re o

f an

ext

erio

r an

gle

of a

tri

angl

e is

equ

al t

o th

e of

the

mea

sure

s of

the

tw

o re

mot

e in

teri

or a

ngle

s.fa

lse;

sum

f.If

the

mea

sure

s of

tw

o an

gles

of

a tr

iang

le a

re 6

2 an

d 93

,the

n th

e m

easu

re o

f th

eth

ird

angl

e is

.

fals

e;25

g.A

n an

gle

of a

tri

angl

e fo

rms

a lin

ear

pair

wit

h an

inte

rior

ang

le o

f th

etr

iang

le.

true

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stud

ents

rem

embe

r m

athe

mat

ical

idea

s an

d fa

cts

mor

e ea

sily

if t

hey

see

them

dem

onst

rate

d vi

sual

ly r

athe

r th

an h

avin

g th

em s

tate

d in

wor

ds.D

escr

ibe

a vi

sual

way

to d

emon

stra

te t

he A

ngle

Sum

The

orem

.S

ampl

e an

swer

:Cut

off

the

ang

les

of a

tri

angl

e an

d pl

ace

them

si

de-b

y-si

de o

n on

e si

de o

f a

line

so t

hat

thei

r ve

rtic

es m

eet

at a

com

mon

poin

t.Th

e re

sult

will

sho

w t

hree

ang

les

who

se m

easu

res

add

up t

o 18

0.

exte

rior

35

diff

eren

ce

cong

ruen

t

acut

e

100

supp

lem

enta

ry

39!

23!

EA

BD

CF

©G

lenc

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cGra

w-H

ill19

4G

lenc

oe G

eom

etry

Find

ing

Ang

le M

easu

res

in T

rian

gles

You

can

use

alge

bra

to s

olve

pro

blem

s in

volv

ing

tria

ngle

s.

In t

rian

gle

AB

C,m

"A

,is

twic

e m

"B

,an

d m

"C

is 8

mor

e th

an m

"B

.Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

Wri

te a

nd s

olve

an

equa

tion

.Let

x!

m"

B.

m"

A"

m"

B"

m"

C!

180

2x"

x"

(x"

8)!

180

4x"

8!

180

4x!

172

x!

43So

,m"

A!

2(43

)or

86,m

"B

!43

,and

m"

C!

43"

8or

51.

Sol

ve e

ach

pro

blem

.

1.In

tri

angl

e D

EF

,m"

E is

thr

ee t

imes

2.In

tri

angl

e R

ST

,m"

Tis

5 m

ore

than

m

"D

,and

m"

Fis

9 le

ss t

han

m"

E.

m"

R,a

nd m

"S

is 1

0 le

ss t

han

m"

T.

Wha

t is

the

mea

sure

of

each

ang

le?

Wha

t is

the

mea

sure

of

each

ang

le?

m"

D"

27,m

"E

"81

,m"

F"

72m

"R

"60

,m"

S"

55,m

"T

"65

3.In

tri

angl

e JK

L,m

"K

is f

our

tim

es4.

In t

rian

gle

XY

Z,m

"Z

is 2

mor

e th

an t

wic

em

"J,

and

m"

Lis

fiv

e ti

mes

m"

J.m

"X

,and

m"

Yis

7 le

ss t

han

twic

e m

"X

.W

hat

is t

he m

easu

re o

f ea

ch a

ngle

?W

hat

is t

he m

easu

re o

f ea

ch a

ngle

?

m"

J"

18,m

"K

"72

,m"

L"

90m

"X

"37

,m"

Y"

67,m

"Z

"76

5.In

tri

angl

e G

HI,

m"

H is

20

mor

e th

an6.

In t

rian

gle

MN

O,m

"M

is e

qual

to

m"

N,

m"

G,a

nd m

"G

is 8

mor

e th

an m

"I.

and

m"

Ois

5 m

ore

than

thr

ee t

imes

W

hat

is t

he m

easu

re o

f ea

ch a

ngle

?m

"N

.Wha

t is

the

mea

sure

of

each

ang

le?

m"

G"

56,m

"H

"76

,m"

I"48

m"

M"

m"

N"

35,m

"O

"11

0

7.In

tri

angl

e S

TU

,m"

U is

hal

f m

"T

,8.

In t

rian

gle

PQ

R,m

"P

is e

qual

to

and

m"

Sis

30

mor

e th

an m

"T

.Wha

tm

"Q

,and

m"

Ris

24

less

tha

n m

"P.

is t

he m

easu

re o

f ea

ch a

ngle

?W

hat

is t

he m

easu

re o

f ea

ch a

ngle

?

m"

S"

90,m

"T

"60

,m"

U"

30m

"P

"m

"Q

"68

,m"

R"

44

9.W

rite

you

r ow

n pr

oble

ms

abou

t m

easu

res

of t

rian

gles

.S

ee s

tude

nts’

wor

k.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-2

4-2

Exam

ple

Exam

ple



Stu

dy

Gu

ide

and I

nte

rven

tion

Con

grue

nt T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

5G

lenc

oe G

eom

etry

Lesson 4-3

Co

rres

po

nd

ing

Par

ts o

f C

on

gru

ent

Tria

ng

les

Tri

angl

es t

hat

have

the

sam

e si

ze a

nd s

ame

shap

e ar

e co

ngr

uen

t tr

ian

gles

.Tw

o tr

iang

les

are

cong

ruen

t if

and

on

ly if

all

thre

e pa

irs

of c

orre

spon

ding

ang

les

are

cong

ruen

t an

d al

l thr

ee p

airs

of

corr

espo

ndin

g si

des

are

cong

ruen

t.In

th

e fi

gure

,!A

BC

#!

RS

T.

If !

XY

Z"

!R

ST

,nam

e th

e p

airs

of

con

gru

ent

angl

es a

nd

con

gru

ent

sid

es.

"X

#"

R,"

Y#

"S

,"Z

#"

TX $

Y$#

R$S$,

X$Z$

#R$

T$,Y$

Z$#

S$T$

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

3.

!A

BC

"!

JKL

!A

BC

"!

DC

B!

JKM

"!

LMK

Nam

e th

e co

rres

pon

din

g co

ngr

uen

t an

gles

an

d s

ides

for

th

e co

ngr

uen

t tr

ian

gles

.

4.5.

6.

"E

""

J;"

F"

"K

;"

A"

"D

;"

R"

"T;

"G

""

L;E!F!

"J!K!

;"

AB

C"

"D

CB

;"

RS

U"

"TS

U;

E!G!"

J!L!;F!

G!"

K!L!

"A

CB

""

DB

C;

"R

US

""

TUS

;A!

B!"

D!C!

;A!C!

"D!

B!;

R!U!

"T!U!

;R!S!

"T!S!

;B!

C!"

C!B!

S!U!"

S!U!

R TUS

BD

CA

FG

LK J

E

K J

L MC

D

A

B

CA

B

LJK

Y

XZ

T

SR

AC

BR

TS

Exam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

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cGra

w-H

ill19

6G

lenc

oe G

eom

etry

Iden

tify

Co

ng

ruen

ce T

ran

sfo

rmat

ion

sIf

tw

o tr

iang

les

are

cong

ruen

t,yo

u ca

nsl

ide,

flip

,or

turn

one

of

the

tria

ngle

s an

d th

ey w

ill s

till

be c

ongr

uent

.The

se a

re c

alle

dco

ngr

uen

ce t

ran

sfor

mat

ion

sbe

caus

e th

ey d

o no

t ch

ange

the

siz

e or

sha

pe o

f th

e fi

gure

.It

is c

omm

on t

o us

e pr

ime

sym

bols

to

dist

ingu

ish

betw

een

an o

rigi

nal !

AB

Can

d a

tran

sfor

med

!A

$B$C

$.

Nam

e th

e co

ngr

uen

ce t

ran

sfor

mat

ion

th

at p

rod

uce

s !

A#B

#C#

from

!A

BC

.T

he c

ongr

uenc

e tr

ansf

orm

atio

n is

a s

lide.

"A

#"

A$;

"B

#"

B$;

"C

#"

C$;

A $B$

#A$

$$B$$$;

A$C$

#A$

$$C$$$;

B$C$

#B$

$$C$$$

Des

crib

e th

e co

ngr

uen

ce t

ran

sfor

mat

ion

bet

wee

n t

he

two

tria

ngl

es a

s a

slid

e,a

flip

,or

a tu

rn.T

hen

nam

e th

e co

ngr

uen

t tr

ian

gles

.

1.2.

flip;

!R

ST

"!

RS

#T#

slid

e;!

MN

P"

!M

#N#P

#

3.4.

turn

;!O

PQ

"!

OP

#Q#

flip;

!A

BC

"!

AB

#C

5.6.

slid

e;!

AB

C"

!A

#B#C

#tu

rn;!

MN

P"

!M

N#P

#

x

y

OM

N

P

N$

P$

x

y

OA

$B

$

C$

ABC

x

y

OB

$B

A Cx

y

O

Q$

P$

P Q

x

y

ON

$

M$

P$

N MP

x

y

OR

S$

T$

S

T

x

y

O

A$

B$

B

C$

AC

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Con

grue

nt T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Con

grue

nt T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

7G

lenc

oe G

eom

etry

Lesson 4-3

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

!JP

L"

!TV

S!

AB

C"

!W

XY

3.4.

!P

QR

"!

PS

R!

DE

F"

!D

GF

Nam

e th

e co

ngr

uen

t an

gles

an

d s

ides

for

eac

h p

air

of c

ongr

uen

t tr

ian

gles

.

5.!

AB

C#

!F

GH

"A

""

F,"

B"

"G

,"C

""

H;A!

B!"

F!G!,B!

C!"

G!H!

,A!C!

"F!H!

6.!

PQ

R#

!S

TU

"P

""

S,"

Q"

"T,

"R

""

U;P!

Q!"

S!T!,Q!

R!"

T!U!,P!

R!"

S!U!

Ver

ify

that

eac

h o

f th

e fo

llow

ing

tran

sfor

mat

ion

s p

rese

rves

con

gru

ence

,an

d n

ame

the

con

gru

ence

tra

nsf

orm

atio

n.

7.!

AB

C#

!A

$B$C

$8.

!D

EF

#!

D$E

$F$

AB

"2%

2!,A

#B#

"2%

2!,D

E"

4,D

#E#

"4,

EF

"5,

BC

"2%

2!,B

#C#

"2%

2!,E

#F#

"5,

DF

" 3

,D#F

#"

3,

AC

"4,

A#C

#"

4,"

A"

"A

#,"

D"

"D

#,"

E"

"E

#,

"B

""

B#,

"C

""

C#;

slid

e"

F"

"F

#;fli

p

x

y

OD

$

E$

F$

DE

F

x

y

OA$

B$

C$

A

B

C

D

E G

FR

P

Q S

WY

XC

AB

L

P

J

S

V

T

©G

lenc

oe/M

cGra

w-H

ill19

8G

lenc

oe G

eom

etry

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

!A

BC

"!

DR

S!

LMN

"!

QP

N

Nam

e th

e co

ngr

uen

t an

gles

an

d s

ides

for

eac

h p

air

of c

ongr

uen

t tr

ian

gles

.

3.!

GK

P#

!L

MN

"G

""

L,"

K"

"M

,"P

""

N;G!

K!"

L!M!,K!

P!"

M!N!

,G!P!

"L!N!

4.!

AN

C#

!R

BV

"A

""

R,"

N"

"B

,"C

""

V;A!

N!"

R!B!

,N!C!

"B!

V!,A!

C!"

R!V!

Ver

ify

that

eac

h o

f th

e fo

llow

ing

tran

sfor

mat

ion

s p

rese

rves

con

gru

ence

,an

d n

ame

the

con

gru

ence

tra

nsf

orm

atio

n.

5.!

PS

T#

!P

$S$T

$6.

!L

MN

#!

L$M

$N$

PS

"%

13!,P

#S#

"%

13!,

LM"

2%2!,

L#M

#"

2%2!,

ST

"%

5!,S

#T#

"%

5!,P

T"

%10!

,M

N"

%29!

,M#N

#"

%29!

,P

#T#

"%

10!,"

P"

"P

#,LN

"7,

L#N

#"

7,"

L"

"L#

,"

S"

"S

#,"

T"

!T

#;fli

p"

M"

"M

#,"

N"

"N

#;fli

p

QU

ILTI

NG

For

Exe

rcis

es 7

an

d 8

,ref

er t

o th

e qu

ilt

des

ign

.

7.In

dica

te t

he t

rian

gles

tha

t ap

pear

to

be c

ongr

uent

.!

AB

I"!

EB

F,!

CB

D"

!H

BG

8.N

ame

the

cong

ruen

t an

gles

and

con

grue

nt s

ides

of

a pa

ir o

f co

ngru

ent

tria

ngle

s.S

ampl

e an

swer

:"A

""

E,"

AB

I""

EB

F,"

I""

F;A!

B!"

E!B!,B!

I!"B!

F!,A!

I!"E!F!

B

A I

E FH

G

CD

x

y

O

M$

N$

L$

M

NL

x

y

O

S$

T$

P$

S

TP

MN

L

P

QD

R

SC

A

BPra

ctic

e (A

vera

ge)

Con

grue

nt T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3


©G

lenc

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cGra

w-H

ill20

0G

lenc

oe G

eom

etry

Tran

sfor

mat

ions

in T

he C

oord

inat

e P

lane

The

fol

low

ing

stat

emen

t te

lls o

ne w

ay t

o m

ap p

reim

age

poin

ts t

o im

age

poin

ts in

the

coo

rdin

ate

plan

e.

(x,y

) !(x

"6,

y#

3)

Thi

s ca

n be

rea

d,“T

he p

oint

wit

h co

ordi

nate

s (x

,y)

is

map

ped

to t

he p

oint

wit

h co

ordi

nate

s (x

"6,

y#

3).”

Wit

h th

is t

rans

form

atio

n,fo

r ex

ampl

e,(3

,5)

is m

appe

d to

(3

"6,

5 #

3) o

r (9

,2).

The

fig

ure

show

s ho

w t

he t

rian

gle

AB

Cis

map

ped

to t

rian

gle

XY

Z.

1.D

oes

the

tran

sfor

mat

ion

abov

e ap

pear

to

be a

con

grue

nce

tran

sfor

mat

ion?

Exp

lain

you

ran

swer

.Ye

s;th

e tr

ansf

orm

atio

n sl

ides

the

fig

ure

to t

he lo

wer

rig

ht w

ithou

tch

angi

ng it

s si

ze o

r sh

ape.

Dra

w t

he

tran

sfor

mat

ion

im

age

for

each

fig

ure

.Th

en t

ell

wh

eth

er t

he

tran

sfor

mat

ion

is

or i

s n

ot a

con

gru

ence

tra

nsf

orm

atio

n.

2.(x

,y) !

(x#

4,y)

yes

3.(x

,y) !

(x"

8,y

"7)

yes

4.(x

,y) !

(#x

,#y)

yes

5.(x

,y) !

%#%1 2% x

,y&

no

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

CX

ZY

O

(x, y

) ! (x

$ 6

, y %

3)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3


Rea

din

g t

o L

earn

Math

emati

csC

ongr

uent

Tri

angl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

9G

lenc

oe G

eom

etry

Lesson 4-3

Pre-

Act

ivit

yW

hy

are

tria

ngl

es u

sed

in

bri

dge

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

3 at

the

top

of

page

192

in y

our

text

book

.

In t

he b

ridg

e sh

own

in t

he p

hoto

grap

h in

you

r te

xtbo

ok,d

iago

nal b

race

sw

ere

used

to

divi

de s

quar

es in

to t

wo

isos

cele

s ri

ght

tria

ngle

s.W

hy d

o yo

uth

ink

thes

e br

aces

are

use

d on

the

bri

dge?

Sam

ple

answ

er:T

hedi

agon

al b

race

s m

ake

the

stru

ctur

e st

rong

er a

nd p

reve

nt it

from

bei

ng d

efor

med

whe

n it

has

to w

ithst

and

a he

avy

load

.

Rea

din

g t

he

Less

on

1.If

!R

ST

#!

UW

V,c

ompl

ete

each

pai

r of

con

grue

nt p

arts

.

"R

##

"W

"T

#

R$T$

##

U$W$

#W$

V$

2.Id

enti

fy t

he c

ongr

uent

tri

angl

es in

eac

h di

agra

m.

a.!

AB

C"

!A

DC

b.!

PQ

S"

!R

QS

c.d.

!M

NO

"!

QP

O!

RTV

"!

US

V3.

Det

erm

ine

whe

ther

eac

h st

atem

ent

says

tha

t co

ngru

ence

of

tria

ngle

s is

ref

lexi

ve,

sym

met

ric,

or t

rans

itiv

e.a.

If t

he f

irst

of

two

tria

ngle

s is

con

grue

nt t

o th

e se

cond

tri

angl

e,th

en t

he s

econ

dtr

iang

le is

con

grue

nt t

o th

e fi

rst.

sym

met

ric

b.If

the

re a

re t

hree

tri

angl

es fo

r w

hich

the

firs

t is

con

grue

nt t

o th

e se

cond

and

the

sec

ond

is c

ongr

uent

to

the

thir

d,th

en t

he f

irst

tri

angl

e is

con

grue

nt t

o th

e th

ird.

tran

sitiv

ec.

Eve

ry t

rian

gle

is c

ongr

uent

to

itse

lf.re

flexi

ve

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Yo

ur c

lass

mat

e B

en is

havi

ng t

roub

le w

riti

ng c

ongr

uenc

e st

atem

ents

for

tri

angl

es b

ecau

se h

e th

inks

he

has

tom

atch

up

thre

e pa

irs

of s

ides

and

thr

ee p

airs

of a

ngle

s.H

ow c

an y

ou h

elp

him

und

erst

and

how

to

wri

te c

orre

ct c

ongr

uenc

e st

atem

ents

mor

e ea

sily

?S

ampl

e an

swer

:Wri

te t

heth

ree

vert

ices

of

one

tria

ngle

in a

ny o

rder

.The

n w

rite

the

cor

resp

ondi

ngve

rtic

es o

f th

e se

cond

tri

angl

e in

the

sam

e or

der.

If th

e an

gles

are

wri

tten

in t

he c

orre

ct c

orre

spon

denc

e,th

e si

des

will

aut

omat

ical

ly b

e in

the

corr

ect

corr

espo

nden

ce a

lso.

RT

US

V

NO

P

QM

S

PR

Q

CA

B D

S!T!R!

S!U!

V!

"V

"S

"U



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Pro

ving

Con

grue

nce—

SS

S,S

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

1G

lenc

oe G

eom

etry

Lesson 4-4

SSS

Post

ula

teYo

u kn

ow t

hat

two

tria

ngle

s ar

e co

ngru

ent

if c

orre

spon

ding

sid

es a

reco

ngru

ent

and

corr

espo

ndin

g an

gles

are

con

grue

nt.T

he S

ide-

Side

-Sid

e (S

SS)

Post

ulat

e le

tsyo

u sh

ow t

hat

two

tria

ngle

s ar

e co

ngru

ent

if y

ou k

now

onl

y th

at t

he s

ides

of

one

tria

ngle

are

cong

ruen

t to

the

sid

es o

f th

e se

cond

tri

angl

e.

SS

S P

ostu

late

If th

e si

des

of o

ne tr

iang

le a

re c

ongr

uent

to th

e si

des

of a

sec

ond

tria

ngle

, th

en th

e tr

iang

les

are

cong

ruen

t.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:A $

B$#

D$B$

and

Cis

the

mid

poin

t of

A$D$

.P

rove

:!A

BC

#!

DB

C

Stat

emen

tsR

easo

ns

1.A$

B$#

D$B$

1.G

iven

2.C

is t

he m

idpo

int

of A$

D$.

2.G

iven

3.A$

C$#

D$C$

3.D

efin

itio

n of

mid

poin

t4.

B$C$

#B$

C$4.

Ref

lexi

ve P

rope

rty

of #

5.!

AB

C#

!D

BC

5.SS

S Po

stul

ate

Wri

te a

tw

o-co

lum

n p

roof

.

B CD

A

Exam

ple

Exam

ple

Exercis

esExercis

es

1.

Giv

en:A$

B$#

X$Y$

,A$C$

#X$

Z$,B$

C$#

Y$Z$

Pro

ve:!

AB

C#

!X

YZ

Stat

emen

tsR

easo

ns

1.A!

B!"

X!Y!1.

Giv

enA!

C!"

X!Z!B!

C!"

Y!Z!2.

!A

BC

"!

XY

Z2.

SS

S P

ost.

2.

Giv

en:R$

S$#

U$T$

,R$T$

#U$

S$P

rove

:!R

ST

#!

UT

S

Stat

emen

tsR

easo

ns

1.R!

S!"

U!T!

1.G

iven

R!T!

"U!

S!2.

S!T!"

T!S!2.

Ref

l.P

rop.

3.!

RS

T"

!U

TS3.

SS

S P

ost.

TU

RS

BY

CA

XZ

©G

lenc

oe/M

cGra

w-H

ill20

2G

lenc

oe G

eom

etry

SAS

Post

ula

teA

noth

er w

ay t

o sh

ow t

hat

two

tria

ngle

s ar

e co

ngru

ent

is t

o us

e th

e Si

de-A

ngle

-Sid

e (S

AS)

Pos

tula

te.

SA

S P

ostu

late

If tw

o si

des

and

the

incl

uded

ang

le o

f one

tria

ngle

are

con

grue

nt to

two

side

s an

d th

e in

clud

ed a

ngle

of a

noth

er tr

iang

le, t

hen

the

tria

ngle

s ar

e co

ngru

ent.

For

eac

h d

iagr

am,d

eter

min

e w

hic

h p

airs

of

tria

ngl

es c

an b

ep

rove

d c

ongr

uen

t by

th

e S

AS

Pos

tula

te.

a.b.

c.

In !

AB

C,t

he a

ngle

is n

ot

The

rig

ht a

ngle

s ar

e T

he in

clud

ed a

ngle

s,"

1 “i

nclu

ded”

by t

he s

ides

A$B$

cong

ruen

t an

d th

ey a

re t

hean

d "

2,ar

e co

ngru

ent

an

d A $

C$.S

o th

e tr

iang

les

incl

uded

ang

les

for

the

beca

use

they

are

ca

nnot

be

prov

ed c

ongr

uent

co

ngru

ent

side

s.al

tern

ate

inte

rior

ang

les

by t

he S

AS

Post

ulat

e.!

DE

F#

!JG

Hby

the

fo

r tw

o pa

ralle

l lin

es.

SAS

Post

ulat

e.!

PS

R#

!R

QP

by t

he

SAS

Post

ulat

e.

For

eac

h f

igu

re,d

eter

min

e w

hic

h p

airs

of

tria

ngl

es c

an b

e p

rove

d c

ongr

uen

t by

the

SA

S P

ostu

late

.

1.2.

3.

!TR

U"

!P

MN

by t

he

"X

QY

and

"W

QZ

are

"M

PL

""

NP

LS

AS

Pos

tula

te.

not

the

incl

uded

ang

les

beca

use

both

are

fo

r th

e co

ngru

ent

righ

t an

gles

.se

gmen

ts.T

he t

rian

gles

!

MP

L"

!N

PL

by

are

not

cong

ruen

t by

th

e S

AS

Pos

tula

te.

the

SA

S P

ostu

late

.

4.5.

6.

The

tria

ngle

s ca

nnot

"

D"

"B

beca

use

The

cong

ruen

t be

pro

ved

cong

ruen

t bo

th a

re r

ight

ang

les.

angl

es a

re t

he

by t

he S

AS

Pos

tula

te.

The

two

tria

ngle

s ar

e in

clud

ed a

ngle

s fo

r co

ngru

ent

by t

he S

AS

th

e co

ngru

ent

side

s.P

ostu

late

.!

FJH

"!

GH

Jby

the

SA

S P

ostu

late

.

JH

GF

K

CBA D

V

T

W

M

PL

N M

X

WZY

QT

P

UN

MR

PQ

S

1

2R

DH

FE

G

J

A BC

X YZ

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Pro

ving

Con

grue

nce—

SS

S,S

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

Pro

ving

Con

grue

nce—

SS

S,S

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

3G

lenc

oe G

eom

etry

Lesson 4-4

Det

erm

ine

wh

eth

er !

AB

C"

!K

LM

give

n t

he

coor

din

ates

of

the

vert

ices

.Exp

lain

.

1.A

(#3,

3),B

(#1,

3),C

(#3,

1),K

(1,4

),L

(3,4

),M

(1,6

)

AB

"2,

KL

"2,

BC

"2%

2!,LM

"2%

2!,A

C"

2,K

M"

2.Th

e co

rres

pond

ing

side

s ha

ve t

he s

ame

mea

sure

and

are

con

grue

nt,

so !

AB

C"

!K

LMby

SS

S.

2.A

(#4,

#2)

,B(#

4,1)

,C(#

1,#

1),K

(0,#

2),L

(0,1

),M

(4,1

)

AB

"3,

KL

"3,

BC

"%

13!,L

M"

4,A

C"

%10!

,KM

"5.

The

corr

espo

ndin

g si

des

are

not

cong

ruen

t,so

!A

BC

is n

ot

cong

ruen

t to

!K

LM.

3.W

rite

a f

low

pro

of.

Giv

en:

P $R$#

D$E$

,P$T$

#D$

F$"

R#

"E

,"T

#"

FP

rove

:!

PR

T#

!D

EF

Pro

of:

Det

erm

ine

wh

ich

pos

tula

te c

an b

e u

sed

to

pro

ve t

hat

th

e tr

ian

gles

are

con

gru

ent.

If i

t is

not

pos

sibl

e to

pro

ve t

hat

th

ey a

re c

ongr

uen

t,w

rite

not

pos

sibl

e.

4.5.

6.

SS

SS

AS

not

poss

ible

PR "

DE

Give

n

PT "

DF

Give

n

"R

" "

E Gi

ven

"P

" "

D

Third

Ang

leTh

eore

m

!PR

T "

!D

EF

SAS

"T

" "

F Gi

ven

T

R

P

F

E

D

©G

lenc

oe/M

cGra

w-H

ill20

4G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er !

DE

F"

!P

QR

give

n t

he

coor

din

ates

of

the

vert

ices

.Exp

lain

.

1.D

(#6,

1),E

(1,2

),F

(#1,

#4)

,P(0

,5),

Q(7

,6),

R(5

,0)

DE

"5%

2!,P

Q"

5%2!,

EF

"2%

10!,Q

R"

2%10!

,DF

"5%

2!,P

R"

5%2!.

!D

EF

"!

PQ

Rby

SS

S s

ince

cor

resp

ondi

ng s

ides

hav

e th

e sa

me

mea

sure

and

are

con

grue

nt.

2.D

(#7,

#3)

,E(#

4,#

1),F

(#2,

#5)

,P(2

,#2)

,Q(5

,#4)

,R(0

,#5)

DE

"%

13!,P

Q"

%13

,!

EF

"2%

5!,Q

R"

%26!

,DF

"%

29!,P

R"

%13!

.C

orre

spon

ding

sid

es a

re n

ot c

ongr

uent

,so

!D

EF

is n

ot c

ongr

uent

to

!P

QR

.

3.W

rite

a f

low

pro

of.

Giv

en:

R $S$

#T$

S$V

is t

he m

idpo

int

of R$

T$.

Pro

ve:

!R

SV

#!

TS

V

Pro

of:

Det

erm

ine

wh

ich

pos

tula

te c

an b

e u

sed

to

pro

ve t

hat

th

e tr

ian

gles

are

con

gru

ent.

If i

t is

not

pos

sibl

e to

pro

ve t

hat

th

ey a

re c

ongr

uen

t,w

rite

not

pos

sibl

e.

4.5.

6.

not

poss

ible

SA

S o

r S

SS

SS

S

7.IN

DIR

ECT

MEA

SUR

EMEN

TTo

mea

sure

the

wid

th o

f a

sink

hole

on

hi

s pr

oper

ty,H

arm

on m

arke

d of

f co

ngru

ent

tria

ngle

s as

sho

wn

in t

hedi

agra

m.H

ow d

oes

he k

now

tha

t th

e le

ngth

s A$

B$

and

AB

are

equa

l?S

ince

"A

CB

and

"A

#CB

#ar

e ve

rtic

al a

ngle

s,th

ey a

re

cong

ruen

t.In

the

fig

ure,

A!C!

"A!

#!C!an

d B!

C!"

B!#!C!

.So

!A

BC

"!

A#B

#Cby

SA

S.B

y C

PC

TC,t

he le

ngth

s A

#B#

and

AB

are

equa

l.A$

B$

AB

C

RS

" T

SGi

ven

SV "

SV

Refle

xive

Prop

erty

RV "

VT

Defin

ition

of m

idpo

int

V is

th

em

idp

oin

t o

f RT

. Gi

ven

!R

SV "

!TS

V

SSS

S

R V T

Pra

ctic

e (A

vera

ge)

Pro

ving

Con

grue

nce—

SS

S,S

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csP

rovi

ng C

ongr

uenc

e—S

SS

,SA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

5G

lenc

oe G

eom

etry

Lesson 4-4

Pre-

Act

ivit

yH

ow d

o la

nd

su

rvey

ors

use

con

gru

ent

tria

ngl

es?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

4 at

the

top

of

page

200

in y

our

text

book

.

Why

do

you

thin

k th

at la

nd s

urve

yors

wou

ld u

se c

ongr

uent

rig

ht t

rian

gles

rath

er t

han

othe

r co

ngru

ent

tria

ngle

s to

est

ablis

h pr

oper

ty b

ound

arie

s?S

ampl

e an

swer

:Lan

d is

usu

ally

div

ided

into

rec

tang

ular

lots

,so

the

ir b

ound

arie

s m

eet

at r

ight

ang

les.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.N

ame

the

side

s of

!L

MN

for

whi

ch "

Lis

the

incl

uded

ang

le.

L!M!,L!

N!b.

Nam

e th

e si

des

of !

LM

Nfo

r w

hich

"N

is t

he in

clud

ed a

ngle

.

N!L!,

N!M!

c.N

ame

the

side

s of

!L

MN

for

whi

ch "

Mis

the

incl

uded

ang

le.

M!L!,

M!N!

2.D

eter

min

e w

heth

er y

ou h

ave

enou

gh in

form

atio

n to

pro

ve t

hat

the

two

tria

ngle

s in

eac

hfi

gure

are

con

grue

nt.I

f so

,wri

te a

con

grue

nce

stat

emen

t an

d na

me

the

cong

ruen

cepo

stul

ate

that

you

wou

ld u

se.I

f no

t,w

rite

not

pos

sibl

e.a.

b.

!A

BD

"!

CB

D;S

AS

not

poss

ible

c.E$

H$an

d D$

G$bi

sect

eac

h ot

her.

d.

!D

EF

"!

GH

F;S

AS

!R

SU

#!

TSU

;SS

S

Hel

pin

g Y

ou

Rem

emb

er

3.F

ind

thre

e w

ords

tha

t ex

plai

n w

hat

it m

eans

to

say

that

tw

o tr

iang

les

are

cong

ruen

t an

dth

at c

an h

elp

you

reca

ll th

e m

eani

ng o

f th

e SS

S Po

stul

ate.

Sam

ple

answ

er:C

ongr

uent

tri

angl

es a

re t

rian

gles

tha

t ar

e th

e sa

me

size

and

shap

e,an

d th

e S

SS

Pos

tula

te e

nsur

es t

hat

two

tria

ngle

s w

ith t

hree

corr

espo

ndin

g si

des

cong

ruen

t w

ill b

e th

e sa

me

size

and

sha

pe.

GE

F

HD

R T

SU

G

FD

E

CA

DB

L

N

M

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eom

etry

Con

grue

nt P

arts

of R

egul

ar P

olyg

onal

Reg

ions

Con

grue

nt f

igur

es a

re f

igur

es t

hat

have

exa

ctly

the

sam

e si

ze a

nd s

hape

.The

re a

re m

any

way

s to

div

ide

regu

lar

poly

gona

l reg

ions

into

con

grue

nt p

arts

.Thr

ee w

ays

to d

ivid

e an

equi

late

ral t

rian

gula

r re

gion

are

sho

wn.

You

can

veri

fy t

hat

the

part

s ar

e co

ngru

ent

bytr

acin

g on

e pa

rt,t

hen

rota

ting

,slid

ing,

or r

efle

ctin

g th

at p

art

on t

op o

f th

e ot

her

part

s.

1.D

ivid

e ea

ch s

quar

e in

to f

our

cong

ruen

t pa

rts.

Use

thr

ee

diff

eren

t w

ays.

Sam

ple

answ

ers

are

show

n.

2.D

ivid

e ea

ch p

enta

gon

into

fiv

e co

ngru

ent

part

s.U

se t

hree

di

ffer

ent

way

s.S

ampl

e an

swer

s ar

e sh

own.

3.D

ivid

e ea

ch h

exag

on in

to s

ix c

ongr

uent

par

ts.U

se t

hree

di

ffer

ent

way

s.S

ampl

e an

swer

s ar

e sh

own.

4.W

hat

hint

s m

ight

you

giv

e an

othe

r st

uden

t w

ho is

try

ing

to d

ivid

e fi

gure

s lik

e th

ose

into

con

grue

nt p

arts

?S

ee s

tude

nts’

wor

k.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-4

4-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Pro

ving

Con

grue

nce—

AS

A,A

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5

©G

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eom

etry

Lesson 4-5

ASA

Po

stu

late

The

Ang

le-S

ide-

Ang

le (

ASA

) Po

stul

ate

lets

you

sho

w t

hat

two

tria

ngle

sar

e co

ngru

ent.

AS

A P

ostu

late

If tw

o an

gles

and

the

incl

uded

sid

e of

one

tria

ngle

are

con

grue

nt to

two

angl

es

and

the

incl

uded

sid

e of

ano

ther

tria

ngle

, the

n th

e tr

iang

les

are

cong

ruen

t.

Fin

d t

he

mis

sin

g co

ngr

uen

t p

arts

so

that

th

e tr

ian

gles

can

be

pro

ved

con

gru

ent

by t

he

AS

A P

ostu

late

.Th

en w

rite

th

e tr

ian

gle

con

gru

ence

.

a.

Tw

o pa

irs

of c

orre

spon

ding

ang

les

are

cong

ruen

t,"

A#

"D

and

"C

#"

F.I

f th

ein

clud

ed s

ides

A$C$

and

D$F$

are

cong

ruen

t,th

en !

AB

C#

!D

EF

by t

he A

SA P

ostu

late

.

b.

"R

#"

Yan

d S$R$

# X$

Y$.I

f "

S#

"X

,the

n !

RS

T#

!Y

XW

by t

he A

SA P

ostu

late

.

Wh

at c

orre

spon

din

g p

arts

mu

st b

e co

ngr

uen

t in

ord

er t

o p

rove

th

at t

he

tria

ngl

esar

e co

ngr

uen

t by

th

e A

SA

Pos

tula

te?

Wri

te t

he

tria

ngl

e co

ngr

uen

ce s

tate

men

t.

1.2.

3.

D!C!

"B!

C!;

W!Y!

"W!

Y!;"

AB

E"

"C

BD

;!

CD

E"

!C

BA

"X

YW

""

ZYW

;!

AB

E"

!C

BD

!W

XY

"!

WZY

4.5.

6.

B!D!

"D!

B!;

S!T!"

V!T!;

"A

CB

""

E;

"A

DB

""

CB

D;

!R

ST

"!

UV

T!

AB

C"

!C

DE

!A

BD

"!

CD

B

AC

B

E

D

S

V

UR

T

D

AB

C

DC

EA

B

YW

X ZE

A

BD

C

RT

WY

SX

AC

B

DF

E

Exam

ple

Exam

ple

Exercis

esExercis

es

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eom

etry

AA

S Th

eore

mA

noth

er w

ay t

o sh

ow t

hat

two

tria

ngle

s ar

e co

ngru

ent

is t

he A

ngle

-A

ngle

-Sid

e (A

AS)

The

orem

.

AA

S T

heor

emIf

two

angl

es a

nd a

non

incl

uded

sid

e of

one

tria

ngle

are

con

grue

nt to

the

corr

espo

ndin

g tw

oan

gles

and

sid

e of

a s

econ

d tr

iang

le, t

hen

the

two

tria

ngle

s ar

e co

ngru

ent.

You

now

hav

e fi

ve w

ays

to s

how

tha

t tw

o tr

iang

les

are

cong

ruen

t.•

defi

niti

on o

f tr

iang

le c

ongr

uenc

e•

ASA

Pos

tula

te•

SSS

Post

ulat

e•

AA

S T

heor

em•

SAS

Post

ulat

e In t

he

dia

gram

,"B

CA

""

DC

A.W

hic

h s

ides

ar

e co

ngr

uen

t? W

hic

h a

dd

itio

nal

pai

r of

cor

resp

ond

ing

par

ts

nee

ds

to b

e co

ngr

uen

t fo

r th

e tr

ian

gles

to

be c

ongr

uen

t by

th

e A

AS

Pos

tula

te?

A $C$

# A$

C$by

the

Ref

lexi

ve P

rope

rty

of c

ongr

uenc

e.T

he c

ongr

uent

an

gles

can

not

be "

1 an

d "

2,be

caus

e A $

C$w

ould

be

the

incl

uded

sid

e.If

"B

#"

D,t

hen

!A

BC

#!

AD

Cby

the

AA

S T

heor

em.

In E

xerc

ises

1 a

nd

2,d

raw

an

d l

abel

!A

BC

and

!D

EF

.In

dic

ate

wh

ich

ad

dit

ion

alp

air

of c

orre

spon

din

g p

arts

nee

ds

to b

e co

ngr

uen

t fo

r th

e tr

ian

gles

to

beco

ngr

uen

t by

th

e A

AS

Th

eore

m.

1."

A#

"D

;"B

#"

E2.

BC

#E

F;"

A#

"D

If B!

C!"

E!F!(o

r if

A!C!

"D!

F!),

If "

C"

"F

(or

if "

B"

"E

),th

en !

AB

C"

!D

EF

by t

he

then

!A

BC

"!

DE

F by

the

AA

S T

heor

em.

AA

S T

heor

em.

3.W

rite

a f

low

pro

of.

Giv

en:"

S#

"U

;T $R$

bise

cts

"S

TU

.P

rove

:"S

RT

#"

UR

T

Give

n

Give

n

RT "

RT

Re

fl. P

rop.

of "

Def.o

f " b

isec

tor

TR b

isec

ts "

STU

.

!SR

T "

!U

RT

"ST

R "

"U

TR

AAS

"SR

T "

"U

RTCP

CTC

"S

" "

U

S

RT

U

BA

C

ED

F

CA

B

FD

E

D

C1 2

A

B

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Pro

ving

Con

grue

nce—

AS

A,A

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Pro

ving

Con

grue

nce—

AS

A,A

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5

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lenc

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eom

etry

Lesson 4-5

Wri

te a

flo

w p

roof

.

1.G

iven

:"

N#

"L

J $K$#

M$K$

Pro

ve:

!JK

N#

!M

KL

Pro

of:

2.G

iven

:A$

B$#

C$B$

"A

#"

CD $

B$bi

sect

s "

AB

C.

Pro

ve:

A $D$

#C$

D$

Pro

of:

3.W

rite

a p

arag

raph

pro

of.

Giv

en:

D$E$

|| F$G$

"E

#"

GP

rove

:!

DF

G#

!F

DE

Pro

of:S

ince

it is

giv

en t

hat

D!E!

|| F!G!

,it

follo

ws

that

"E

DF

""

GFD

,be

caus

e al

t.in

t.#

are

".I

t is

giv

en t

hat

"E

""

G.B

y th

e R

efle

xive

P

rope

rty,

D!F!

"F!D!

.So

!D

FG"

!FD

Eby

AA

S.

FG

DE

"A

" "

C

Give

nA

B "

CB

Give

nCP

CTC

AD

" C

D

DB

bis

ects

"A

BC

. Gi

ven

!A

BD

" !

CB

DAS

A

"A

BD

" "

CB

DDe

f. of

" b

isec

tor

AC

B D

"N

" "

LGi

ven

JK "

MK

Gi

ven

"JK

N "

"M

KL

Verti

cal #

are

".

!JK

N "

!M

KL

AAS

N

J

M

KL

©G

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eom

etry

1.W

rite

a f

low

pro

of.

Giv

en:

Sis

the

mid

poin

t of

Q $T$

.Q $

R$|| T$

U$P

rove

:!

QS

R#

!T

SU

Sam

ple

proo

f:

2.W

rite

a p

arag

raph

pro

of.

Giv

en:

"D

#"

FG $

E$bi

sect

s "

DE

F.

Pro

ve:

D $G$

#F$G$

Pro

of:S

ince

it is

giv

en t

hat

G!E!

bise

cts

"D

EF,

"D

EG

""

FEG

by t

he

defin

ition

of

an a

ngle

bis

ecto

r.It

is g

iven

tha

t "

D"

"F.

By

the

Ref

lexi

ve P

rope

rty,

G!E!

"G!

E!.S

o !

DE

G"

!FE

Gby

AA

S.T

here

fore

D!

G!"

F!G!by

CP

CTC

.

AR

CH

ITEC

TUR

EF

or E

xerc

ises

3 a

nd

4,u

se t

he

foll

owin

g in

form

atio

n.

An

arch

itec

t us

ed t

he w

indo

w d

esig

n in

the

dia

gram

whe

n re

mod

elin

g an

art

stu

dio.

A $B$

and

C$B$

each

mea

sure

3 f

eet.

3.Su

ppos

e D

is t

he m

idpo

int

of A$

C$.D

eter

min

e w

heth

er !

AB

D#

!C

BD

.Ju

stif

y yo

ur a

nsw

er.

Sin

ce D

is t

he m

idpo

int

of A!

C!,A!

D!"

C!D!

by t

he d

efin

ition

of

mid

poin

t.A!

B!"

C!B!

by t

he d

efin

ition

of

cong

ruen

t se

gmen

ts.B

y th

e R

efle

xive

P

rope

rty,

B!D!

"B!

D!.S

o !

AB

D"

!C

BD

by S

SS

.

4.Su

ppos

e "

A#

"C

.Det

erm

ine

whe

ther

!A

BD

#!

CB

D.J

usti

fy y

our

answ

er.

We

are

give

n A!

B!"

C!B!

and

"A

""

C.B!

D!"

B!D!

by t

he R

efle

xive

P

rope

rty.

Sin

ce S

SA

can

not

be u

sed

to p

rove

tha

t tr

iang

les

are

cong

ruen

t,w

e ca

nnot

say

whe

ther

!A

BD

"!

CB

D.

DB

AC

D

G

F

E

"Q

" "

T

Give

n

QR

|| TU

Gi

ven

Def.o

f mid

poin

t

Alt.

Int.

# a

re "

.

QS

" T

S S

is t

he

mid

po

int

of

QT.

!Q

SR "

!TS

UAS

A

"Q

SR "

"TS

UVe

rtica

l # a

re "

.

UQ

S

RT

Pra

ctic

e (A

vera

ge)

Pro

ving

Con

grue

nce—

AS

A,A

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5



Rea

din

g t

o L

earn

Math

emati

csP

rovi

ng C

ongr

uenc

e—A

SA

,AA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5

©G

lenc

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1G

lenc

oe G

eom

etry

Lesson 4-5

Pre-

Act

ivit

yH

ow a

re c

ongr

uen

t tr

ian

gles

use

d i

n c

onst

ruct

ion

?R

ead

the

intr

oduc

tion

to

Les

son

4-5

at t

he t

op o

f pa

ge 2

07 in

you

r te

xtbo

ok.

Whi

ch o

f th

e tr

iang

les

in t

he p

hoto

grap

h in

you

r te

xtbo

ok a

ppea

r to

be

cong

ruen

t?S

ampl

e an

swer

:The

four

rig

ht tr

iang

les

are

cong

ruen

tto

eac

h ot

her.

The

two

obtu

se is

osce

les

tria

ngle

s ar

e co

ngru

ent

to e

ach

othe

r.R

ead

ing

th

e Le

sso

n1.

Exp

lain

in y

our

own

wor

ds t

he d

iffe

renc

e be

twee

n ho

w t

he A

SA P

ostu

late

and

the

AA

ST

heor

em a

re u

sed

to p

rove

tha

t tw

o tr

iang

les

are

cong

ruen

t.S

ampl

e an

swer

:In

AS

A,y

ou u

se t

wo

pair

s of

con

grue

nt a

ngle

s an

d th

ein

clud

edco

ngru

ent

side

s.In

AA

S,y

ou u

se t

wo

pair

s of

con

grue

nt a

ngle

san

d a

pair

of

noni

nclu

ded

cong

ruen

t si

des.

B,D

,E,G

,H2.

Whi

ch o

f the

follo

win

g co

ndit

ions

are

suf

ficie

nt t

o pr

ove

that

tw

o tr

iang

les

are

cong

ruen

t?A

.Tw

o si

des

of o

ne t

rian

gle

are

cong

ruen

t to

tw

o si

des

of t

he o

ther

tri

angl

e.B

.The

thr

ee s

ides

of

one

tria

ngle

s ar

e co

ngru

ent

to t

he t

hree

sid

es o

f th

e ot

her

tria

ngle

.C

.The

thr

ee a

ngle

s of

one

tri

angl

e ar

e co

ngru

ent

to t

he t

hree

ang

les

of t

he o

ther

tri

angl

e.D

.All

six

corr

espo

ndin

g pa

rts

of t

wo

tria

ngle

s ar

e co

ngru

ent.

E.T

wo

angl

es a

nd t

he in

clud

ed s

ide

of o

ne t

rian

gle

are

cong

ruen

t to

tw

o si

des

and

the

incl

uded

ang

le o

f th

e ot

her

tria

ngle

.F.

Tw

o si

des

and

a no

ninc

lude

d an

gle

of o

ne t

rian

gle

are

cong

ruen

t to

tw

o si

des

and

ano

ninc

lude

d an

gle

of t

he o

ther

tri

angl

e.G

.Tw

o an

gles

and

a n

onin

clud

ed s

ide

of o

ne t

rian

gle

are

cong

ruen

t to

tw

o an

gles

and

the

corr

espo

ndin

g no

ninc

lude

d si

de o

f th

e ot

her

tria

ngle

.H

.Tw

o si

des

and

the

incl

uded

ang

le o

f on

e tr

iang

le a

re c

ongr

uent

to

two

side

s an

d th

ein

clud

ed a

ngle

of

the

othe

r tr

iang

le.

I.T

wo

angl

es a

nd a

non

incl

uded

sid

e of

one

tri

angl

e ar

e co

ngru

ent

to t

wo

angl

es a

nd a

noni

nclu

ded

side

of

the

othe

r tr

iang

le.

3.D

eter

min

e w

heth

er y

ou h

ave

enou

gh in

form

atio

n to

pro

ve t

hat

the

two

tria

ngle

s in

eac

hfi

gure

are

con

grue

nt.I

f so

,wri

te a

con

grue

nce

stat

emen

t an

d na

me

the

cong

ruen

cepo

stul

ate

or t

heor

em t

hat

you

wou

ld u

se.I

f no

t,w

rite

not

pos

sibl

e.a.

!A

EB

"!

DE

C;A

AS

b.T

is t

he m

idpo

int

of R$

U$.

!R

ST

"!

UV

T;A

SA

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r m

athe

mat

ical

idea

s is

to

sum

mar

ize

them

in a

gen

eral

sta

tem

ent.

If y

ou w

ant

to p

rove

tri

angl

es c

ongr

uent

by

usin

g th

ree

pair

s of

cor

resp

ondi

ng p

arts

,w

hat

is a

goo

d w

ay t

o re

mem

ber

whi

ch c

ombi

nati

ons

of p

arts

will

wor

k?S

ampl

e an

swer

:At l

east

one

pai

r of

cor

resp

ondi

ng p

arts

mus

t be

side

s.If

you

use

two

pair

s of

sid

es a

nd o

ne p

air

of a

ngle

s,th

e an

gles

mus

t be

the

incl

uded

ang

les.

If yo

u us

e tw

o pa

irs

of a

ngle

s an

d on

e pa

ir o

f si

des,

then

the

sid

es m

ust

both

be

incl

uded

by

the

angl

es o

r m

ust

both

be

corr

espo

ndin

g no

ninc

lude

d si

des.

RS

T

U V

AD

CB

E

©G

lenc

oe/M

cGra

w-H

ill21

2G

lenc

oe G

eom

etry

Con

grue

nt T

rian

gles

in th

e C

oord

inat

e P

lane

If y

ou k

now

the

coo

rdin

ates

of

the

vert

ices

of

two

tria

ngle

s in

the

coo

rdin

ate

plan

e,yo

u ca

n of

ten

deci

de w

heth

er t

he t

wo

tria

ngle

s ar

e co

ngru

ent.

The

rem

ay b

e m

ore

than

one

way

to

do t

his.

1.C

onsi

der

!A

BD

and

!C

DB

who

se v

erti

ces

have

coo

rdin

ates

A(0

,0),

B(2

,5),

C(9

,5),

and

D(7

,0).

Bri

efly

des

crib

e ho

w y

ou c

an u

se w

hat

you

know

abo

ut c

ongr

uent

tri

angl

es a

nd t

he c

oord

inat

e pl

ane

to s

how

tha

t !

AB

D#

!C

DB

.You

may

wis

h to

mak

e a

sket

ch t

o he

lp g

et y

ou s

tart

ed.

Sam

ple

answ

er:S

how

tha

t th

e sl

opes

of

A !B !

and

C !D !

are

equa

l and

tha

t th

e sl

opes

of

A !D !

and

B !C !

are

equa

l.C

oncl

ude

that

A !B !

&C !

D !an

d B !

C ! &

A !D !

.Use

the

ang

le r

elat

ions

hips

for

para

llel l

ines

and

a t

rans

vers

al a

nd t

he f

act

that

B !D !

is a

com

-m

on s

ide

for

the

tria

ngle

s to

con

clud

e th

at

!A

BD

"!

CD

Bby

AS

A.

2.C

onsi

der

!P

QR

and

!K

LM

who

se v

erti

ces

are

the

follo

win

g po

ints

.

P(1

,2)

Q(3

,6)

R(6

,5)

K(#

2,1)

L(#

6,3)

M(#

5,6)

Bri

efly

des

crib

e ho

w y

ou c

an s

how

tha

t !

PQ

R#

!K

LM

.

Use

the

Dis

tanc

e Fo

rmul

a to

fin

d th

e le

ngth

s of

the

sid

es o

fbo

th t

rian

gles

.Con

clud

e th

at !

PQ

R"

!K

LMby

SS

S.

3.If

you

kno

w t

he c

oord

inat

es o

f al

l the

ver

tice

s of

tw

o tr

iang

les,

is it

al

way

spo

ssib

le t

o te

ll w

heth

er t

he t

rian

gles

are

con

grue

nt?

Exp

lain

.

Yes;

you

can

use

the

Dis

tanc

e Fo

rmul

a an

d S

SS

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-5

4-5



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Isos

cele

s Tr

iang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

3G

lenc

oe G

eom

etry

Lesson 4-6

Pro

per

ties

of

Iso

scel

es T

rian

gle

sA

n is

osce

les

tria

ngl

eha

s tw

o co

ngru

ent

side

s.T

he a

ngle

for

med

by

thes

e si

des

is c

alle

d th

e ve

rtex

an

gle.

The

oth

er t

wo

angl

es a

re c

alle

dba

se a

ngl

es.Y

ou c

an p

rove

a t

heor

em a

nd it

s co

nver

se a

bout

isos

cele

s tr

iang

les.

•If

tw

o si

des

of a

tri

angl

e ar

e co

ngru

ent,

then

the

ang

les

oppo

site

th

ose

side

s ar

e co

ngru

ent.

(Iso

scel

es T

rian

gle

Th

eore

m)

•If

tw

o an

gles

of

a tr

iang

le a

re c

ongr

uent

,the

n th

e si

des

oppo

site

th

ose

angl

es a

re c

ongr

uent

.If

A$B$#

C$B$,

then

"A

#"

C.

If "

A#

"C

, the

n A$B$

#C$

B$.

A

BC

Fin

d x

.

BC

!B

A,s

o m

"A

!m

"C

.Is

os. T

riang

le T

heor

em

5x#

10 !

4x"

5S

ubst

itutio

n

x#

10 !

5S

ubtr

act 4

xfr

om e

ach

side

.

x!

15A

dd 1

0 to

eac

h si

de.

B

AC( 4

x $

5) !

( 5x

% 1

0)!

Fin

d x

.

m"

S!

m"

T,s

oS

R!

TR

.C

onve

rse

of Is

os. !

Thm

.

3x#

13 !

2xS

ubst

itutio

n

3x!

2x"

13A

dd 1

3 to

eac

h si

de.

x!

13S

ubtr

act 2

xfr

om e

ach

side

.

RT

S 3x %

13

2x

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Fin

d x

.

1.35

2.12

3.15

4.12

5.20

6.36

7.W

rite

a t

wo-

colu

mn

proo

f.G

iven

:"1

#"

2P

rove

:A $B$

#C$

B$

Stat

emen

tsR

easo

ns

1."

1 "

"2

1.G

iven

2."

2 "

"3

2.Ve

rtic

al a

ngle

s ar

e co

ngru

ent.

3."

1 "

"3

3.Tr

ansi

tive

Pro

pert

y of

"4.

A!B!

"C!

B!4.

If tw

o an

gles

of

a tr

iang

le a

re "

,the

n th

e si

des

oppo

site

the

ang

les

are

".

B

AC

D

E

13

2

RS

T 3x!

x!D

BG

L3x

!

30!

D TQP

K

( 6x

$ 6

) !2x

!

W

YZ

3x!

S

V

T3x

% 6

2x $

6R

P

Q2x

!40

!

©G

lenc

oe/M

cGra

w-H

ill21

4G

lenc

oe G

eom

etry

Pro

per

ties

of

Equ

ilate

ral T

rian

gle

sA

n eq

uil

ater

al t

rian

gle

has

thre

e co

ngru

ent

side

s.T

he I

sosc

eles

Tri

angl

e T

heor

em c

an b

e us

ed t

o pr

ove

two

prop

erti

es o

f eq

uila

tera

ltr

iang

les.

1.A

tria

ngle

is e

quila

tera

l if a

nd o

nly

if it

is e

quia

ngul

ar.

2.E

ach

angl

e of

an

equi

late

ral t

riang

le m

easu

res

60°.

Pro

ve t

hat

if

a li

ne

is p

aral

lel

to o

ne

sid

e of

an

equ

ilat

eral

tri

angl

e,th

en i

t fo

rms

anot

her

equ

ilat

eral

tr

ian

gle.

Pro

of:

Stat

emen

tsR

easo

ns

1.!

AB

Cis

equ

ilate

ral;

P$Q$|| B$

C$.

1.G

iven

2.m

"A

!m

"B

!m

"C

!60

2.E

ach

"of

an

equi

late

ral !

mea

sure

s 60

°.3.

"1

#"

B,"

2 #

"C

3.If

||lin

es,t

hen

corr

es."

s ar

e #

.4.

m"

1 !

60,m

"2

!60

4.Su

bsti

tuti

on5.

!A

PQ

is e

quila

tera

l.5.

If a

!is

equ

iang

ular

,the

n it

is e

quila

tera

l.

Fin

d x

.

1.10

2.5

3.10

4.10

5.12

6.15

7.W

rite

a t

wo-

colu

mn

proo

f.G

iven

:!A

BC

is e

quila

tera

l;"

1 #

"2.

Pro

ve:"

AD

B#

"C

DB

Pro

of:

Stat

emen

tsR

easo

ns

1.!

AB

Cis

equ

ilate

ral.

1.G

iven

2.A!

B!"

C!B!

;"A

""

C2.

An

equi

late

ral !

has

"si

des

and

"an

gles

.3.

"1

""

23.

Giv

en4.

!A

BD

"!

CB

D4.

AS

A P

ostu

late

5."

AD

B"

"C

DB

5.C

PC

TC

A D C

B1 2

R O

HM

60!

4x!

X

ZY

4x %

4

3x $

860

!

PQ

LV

R60

!

4x40

L N M

K

!K

LM is

equ

ilate

ral.

3x!

G

JH

6x %

55 x

D

FE

6x!

A

B

PQ

C

12

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Isos

cele

s Tr

iang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

Isos

cele

s Tr

iang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

5G

lenc

oe G

eom

etry

Lesson 4-6

Ref

er t

o th

e fi

gure

.

1.If

A$C$

#A$

D$,n

ame

two

cong

ruen

t an

gles

."

AC

D"

"C

DA

2.If

B$E$

#B$

C$,n

ame

two

cong

ruen

t an

gles

."

BE

C"

"B

CE

3.If

"E

BA

#"

EA

B,n

ame

two

cong

ruen

t se

gmen

ts.

E!B!"

E!A!

4.If

"C

ED

#"

CD

E,n

ame

two

cong

ruen

t se

gmen

ts.

C!E!

"C!

D!

!A

BF

is i

sosc

eles

,!C

DF

is e

quil

ater

al,a

nd

m"

AF

D"

150.

Fin

d e

ach

mea

sure

.

5.m

"C

FD

606.

m"

AF

B55

7.m

"A

BF

708.

m"

A55

In t

he

figu

re,P!

L!"

R!L!

and

L!R!

"B!

R!.

9.If

m"

RL

P!

100,

find

m"

BR

L.

20

10.I

f m

"L

PR

!34

,fin

d m

"B

.68

11.W

rite

a t

wo-

colu

mn

proo

f.

Giv

en:

C$D$

#C $

G$D$

E$#

G$F$

Pro

ve:

C$E$

#C$

F$

Pro

of:

Sta

tem

ents

Rea

sons

1.C!

D!"

C!G!

1.G

iven

2."

D"

"G

2.If

2 si

des

of a

!ar

e "

,the

n th

e #

oppo

site

thos

e si

des

are

".

3.D!

E!"

G!F!

3.G

iven

4.!

CD

E"

!C

GF

4.S

AS

5.C!

E!"

C!F!

5.C

PC

TC

D E F G

CRP

BL

D

C F

B

35!

AE

D

C

B

AE

©G

lenc

oe/M

cGra

w-H

ill21

6G

lenc

oe G

eom

etry

Ref

er t

o th

e fi

gure

.

1.If

R$V$

#R$

T$,n

ame

two

cong

ruen

t an

gles

."

RTV

""

RV

T

2.If

R$S$

#S$V$

,nam

e tw

o co

ngru

ent

angl

es.

"S

VR

""

SR

V

3.If

"S

RT

#"

ST

R,n

ame

two

cong

ruen

t se

gmen

ts.

S!T!"

S!R!

4.If

"S

TV

#"

SV

T,n

ame

two

cong

ruen

t se

gmen

ts.

S!T!"

S!V!

Tri

angl

es G

HM

and

HJ

Mar

e is

osce

les,

wit

h G!

H!"

M!H!

and

H!J!

"M!

J!.T

rian

gle

KL

Mis

equ

ilat

eral

,an

d m

"H

MK

"50

.F

ind

eac

h m

easu

re.

5.m

"K

ML

606.

m"

HM

G70

7.m

"G

HM

40

8.If

m"

HJM

!14

5,fi

nd m

"M

HJ.

17.5

9.If

m"

G!

67,f

ind

m"

GH

M.

46

10.W

rite

a t

wo-

colu

mn

proo

f.

Giv

en:

D$E$

|| B$C$

"1

#"

2P

rove

:A $

B$#

A$C$

Pro

of:

Sta

tem

ents

Rea

sons

1.D!

E!|| B!

C!1.

Giv

en

2."

1 "

"4

2.C

orr.

#ar

e "

."

2 "

"3

3."

1 "

"2

3.G

iven

4."

3 "

"4

4.C

ongr

uenc

e of

#is

tra

nsiti

ve.

5.A!

B!"

A!C!

5.If

2 #

of a

!ar

e "

,the

n th

e si

des

oppo

site

th

ose

#ar

e "

.

11.S

PORT

SA

pen

nant

for

the

spo

rts

team

s at

Lin

coln

Hig

h Sc

hool

is in

the

sha

pe o

f an

isos

cele

s tr

iang

le.I

f th

e m

easu

re

of t

he v

erte

x an

gle

is 1

8,fi

nd t

he m

easu

re o

f ea

ch b

ase

angl

e.81

,81

Linc

oln

Haw

ks

E DBC

A12

3 4

GMLK

J

H

UR

TV

S

Pra

ctic

e (A

vera

ge)

Isos

cele

s Tr

iang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csIs

osce

les

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

7G

lenc

oe G

eom

etry

Lesson 4-6

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n a

rt?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 4-

6 at

the

top

of

page

216

in y

our

text

book

.

•W

hy d

o yo

u th

ink

that

isos

cele

s an

d eq

uila

tera

l tri

angl

es a

re u

sed

mor

eof

ten

than

sca

lene

tri

angl

es in

art

?S

ampl

e an

swer

:The

irsy

mm

etry

is p

leas

ing

to t

he e

ye.

•W

hy m

ight

isos

cele

s ri

ght

tria

ngle

s be

use

d in

art

?S

ampl

e an

swer

:Tw

o co

ngru

ent

isos

cele

s ri

ght

tria

ngle

s ca

n be

pla

ced

toge

ther

to

form

a s

quar

e.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.W

hat

kind

of

tria

ngle

is !

QR

S?

isos

cele

sb.

Nam

e th

e le

gs o

f !Q

RS

.Q!

S!,R!

S!c.

Nam

e th

e ba

se o

f !Q

RS

.Q!

R!d.

Nam

e th

e ve

rtex

ang

le o

f !Q

RS

."

Se.

Nam

e th

e ba

se a

ngle

s of

!Q

RS

."

Q,"

R

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.If

a t

rian

gle

has

thre

e co

ngru

ent

side

s,th

en it

has

thr

ee c

ongr

uent

ang

les.

alw

ays

b.If

a t

rian

gle

is is

osce

les,

then

it is

equ

ilate

ral.

som

etim

esc.

If a

rig

ht t

rian

gle

is is

osce

les,

then

it is

equ

ilate

ral.

neve

rd.

The

larg

est

angl

e of

an

isos

cele

s tr

iang

le is

obt

use.

som

etim

ese.

If a

rig

ht t

rian

gle

has

a 45

°an

gle,

then

it is

isos

cele

s.al

way

sf.

If a

n is

osce

les

tria

ngle

has

thr

ee a

cute

ang

les,

then

it is

equ

ilate

ral.

som

etim

esg.

The

ver

tex

angl

e of

an

isos

cele

s tr

iang

le is

the

larg

est

angl

e of

the

tri

angl

e.so

met

imes

3.G

ive

the

mea

sure

s of

the

thr

ee a

ngle

s of

eac

h tr

iang

le.

a.an

equ

ilate

ral t

rian

gle

60,6

0,60

b.an

isos

cele

s ri

ght

tria

ngle

45,4

5,90

c.an

isos

cele

s tr

iang

le in

whi

ch t

he m

easu

re o

f th

e ve

rtex

ang

le is

70

70,5

5,55

d.an

isos

cele

s tr

iang

le in

whi

ch t

he m

easu

re o

f a

base

ang

le is

70

70,7

0,40

e.an

isos

cele

s tr

iang

le in

whi

ch t

he m

easu

re o

f th

e ve

rtex

ang

le is

tw

ice

the

mea

sure

of

one

of t

he b

ase

angl

es90

,45,

45

Hel

pin

g Y

ou

Rem

emb

er4.

If a

the

orem

and

its

conv

erse

are

bot

h tr

ue,y

ou c

an o

ften

rem

embe

r th

em m

ost

easi

ly b

yco

mbi

ning

the

m in

to a

n “i

f-an

d-on

ly-if

”sta

tem

ent.

Wri

te s

uch

a st

atem

ent

for

the

Isos

cele

sT

rian

gle

The

orem

and

its

conv

erse

.S

ampl

e an

swer

:Tw

o si

des

of a

tri

angl

e ar

eco

ngru

ent

if an

d on

ly if

the

ang

les

oppo

site

tho

se s

ides

are

con

grue

nt.

R Q

S

©G

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8G

lenc

oe G

eom

etry

Tria

ngle

Cha

lleng

esSo

me

prob

lem

s in

clud

e di

agra

ms.

If y

ou a

re n

ot s

ure

how

to

solv

e th

epr

oble

m,b

egin

by

usin

g th

e gi

ven

info

rmat

ion.

Fin

d th

e m

easu

res

of a

s m

any

angl

es a

s yo

u ca

n,w

riti

ng e

ach

mea

sure

on

the

diag

ram

.Thi

s m

ay g

ive

you

mor

e cl

ues

to t

he s

olut

ion.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-6

4-6

1.G

iven

:B

E!

BF

, "B

FG

# "

BE

F #

"B

ED

, m"

BF

E !

82 a

ndA

BF

Gan

d B

CD

Eea

ch h

ave

oppo

site

sid

es p

aral

lel a

ndco

ngru

ent.

Fin

d m

"A

BC

.14

8

3.G

iven

:m

"U

ZY

!90

,m"

ZW

X!

45,

!Y

ZU

#!

VW

X,U

VX

Yis

asq

uare

(al

l sid

es c

ongr

uent

,all

angl

es r

ight

ang

les)

.F

ind

m"

WZ

Y.

45

2.G

iven

:A

C!

AD

,and

A $B $

#B $

D $,

m"

DA

C!

44 a

ndC $

E $bi

sect

s "

AC

D.

Fin

d m

"D

EC

.78

4.G

iven

:m

"N

!12

0,J $N $

#M $

N $,

!JN

M#

!K

LM

.F

ind

m"

JKM

.15

J

K

L

MN

A

DC

BE

UV

W

XY

Z

A

GD

FE

CB



Stu

dy

Gu

ide

and I

nte

rven

tion

Tria

ngle

s an

d C

oord

inat

e P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-7

4-7

©G

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9G

lenc

oe G

eom

etry

Lesson 4-7

Posi

tio

n a

nd

Lab

el T

rian

gle

sA

coo

rdin

ate

proo

f us

es p

oint

s,di

stan

ces,

and

slop

es t

opr

ove

geom

etri

c pr

oper

ties

.The

fir

st s

tep

in w

riti

ng a

coo

rdin

ate

proo

f is

to

plac

e a

figu

re o

nth

e co

ordi

nate

pla

ne a

nd la

bel t

he v

erti

ces.

Use

the

fol

low

ing

guid

elin

es.

1.U

se th

e or

igin

as

a ve

rtex

or

cent

er o

f the

figu

re.

2.P

lace

at l

east

one

sid

e of

the

poly

gon

on a

n ax

is.

3.K

eep

the

figur

e in

the

first

qua

dran

t if p

ossi

ble.

4.U

se c

oord

inat

es th

at m

ake

the

com

puta

tions

as

sim

ple

as p

ossi

ble.

Pos

itio

n a

n e

quil

ater

al t

rian

gle

on t

he

coor

din

ate

pla

ne

so t

hat

its

sid

es a

re a

un

its

lon

g an

d

one

sid

e is

on

th

e p

osit

ive

x-ax

is.

Star

t w

ith

R(0

,0).

If R

Tis

a,t

hen

anot

her

vert

ex is

T(a

,0).

For

vert

ex S

,the

x-c

oord

inat

e is

%a 2% .U

se b

for

the

y-co

ordi

nate

,

so t

he v

erte

x is

S%%a 2% ,

b &.

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

1.2.

3.

C(p

,q)

T(2

a,2a

)E

(%2g

,0);

F(0

,b)

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

4.is

osce

les

tria

ngle

5.

isos

cele

s ri

ght

!D

EF

6.eq

uila

tera

l tri

angl

e !

EQ

I!

RS

T w

ith

base

R $S$

wit

h le

gs e

unit

s lo

ngw

ith

vert

ex Q

(0,a

) an

d4a

unit

s lo

ngsi

des

2bun

its

long x

y

I( b, 0

)E

( –b,

0)

Q( 0

, a)

x

y

E( e

, 0)

F( e

, e)

D( 0

, 0)

x

yT

( 2a,

b)

R( 0

, 0)

S( 4

a, 0

)

Sam

ple

answ

ers

are

give

n.

x

y

G( 2

g, 0

)

F( ?

, b)

E( ?

, ?)

x

y

S( 2

a, 0

)

T( ?

, ?)

R( 0

, 0)

x

y

B( 2

p, 0

)

C( ?

, q)

A( 0

, 0)

x

y

T( a

, 0)

R( 0

, 0)S

#a – 2, b$

Exercis

esExercis

es

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill22

0G

lenc

oe G

eom

etry

Wri

te C

oo

rdin

ate

Pro

ofs

Coo

rdin

ate

proo

fs c

an b

e us

ed t

o pr

ove

theo

rem

s an

d to

veri

fy p

rope

rtie

s.M

any

coor

dina

te p

roof

s us

e th

e D

ista

nce

Form

ula,

Slop

e Fo

rmul

a,or

Mid

poin

t T

heor

em.

Pro

ve t

hat

a s

egm

ent

from

th

e ve

rtex

an

gle

of a

n i

sosc

eles

tri

angl

e to

th

e m

idp

oin

t of

th

e ba

se

is p

erp

end

icu

lar

to t

he

base

.F

irst

,pos

itio

n an

d la

bel a

n is

osce

les

tria

ngle

on

the

coor

dina

te

plan

e.O

ne w

ay is

to

use

T(a

,0),

R(#

a,0)

,and

S(0

,c).

The

n U

(0,0

) is

the

mid

poin

t of

R $T$

.

Giv

en:I

sosc

eles

!R

ST

;Uis

the

mid

poin

t of

bas

e R $

T$.

Pro

ve:S $

U$"

R$T$

Pro

of:

Uis

the

mid

poin

t of

R $T$

so t

he c

oord

inat

es o

f Uar

e %%#

a 2"a

%,%

0" 2

0%

&!(0

,0).

Thu

s S$U$

lies

on

the

y-ax

is,a

nd !

RS

Tw

as p

lace

d so

R$T$

lies

on t

he x

-axi

s.T

he a

xes

are

perp

endi

cula

r,so

S $U$

"R$

T$.

Pro

ve t

hat

th

e se

gmen

ts j

oin

ing

the

mid

poi

nts

of

the

sid

es o

f a

righ

t tr

ian

gle

form

a ri

ght

tria

ngl

e.

Sam

ple

answ

er:P

ositi

on a

nd la

bel r

ight

!A

BC

with

the

coo

rdin

ates

A

(0,0

),B

(0,2

b),a

nd C

(2a,

0).

The

mid

poin

t P

of B

Cis

#&0$ 2

2a &,&

2b2$

0&

$"(a

,b).

The

mid

poin

t Q

of A

Cis

#&0$ 2

2a &,&

0$ 2

0&

$"(a

,0).

The

mid

poin

t R

of A

Bis

#&0$ 2

0&

,&0

$ 22b &

$"(0

,b).

The

slop

e of

R!P!

is &b a

% %b 0

&"

&0 a&"

0,so

the

seg

men

t is

hor

izon

tal.

The

slop

e of

P!Q!

is &b a

%%a0

&"

&b 0& ,w

hich

is u

ndef

ined

,so

the

segm

ent

is v

ertic

al.

"R

PQ

is a

rig

ht a

ngle

bec

ause

any

hor

izon

tal l

ine

is p

erpe

ndic

ular

to

any

vert

ical

line

.!P

RQ

has

a ri

ght

angl

e,so

!P

RQ

is a

rig

ht t

rian

gle.

x

y

C( 2

a, 0

)

B( 0

, 2b)

P Q

R

A( 0

, 0)

x

y

T( a

, 0)

U( 0

, 0)

R( –

a, 0

)

S( 0

, c)

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Tria

ngle

s an

d C

oord

inat

e P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-7

4-7

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Tria

ngle

s an

d C

oord

inat

e P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill22

1G

lenc

oe G

eom

etry

Lesson 4-7

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

1.ri

ght

!F

GH

wit

h le

gs

2.is

osce

les

!K

LP

wit

h 3.

isos

cele

s !

AN

Dw

ith

aun

its

and

bun

its

base

K$P$

6bun

its

long

base

A$D$

5alo

ng

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

4.5.

6.

A(0

,2a)

Z(b

,c)

M(0

,c)

7.8.

9.

Q(4

a,0)

R#&7 2& b

,c$

T(0

,b)

10.W

rite

a c

oord

inat

e pr

oof

to p

rove

tha

t in

an

isos

cele

s ri

ght

tria

ngle

,the

seg

men

t fr

omth

e ve

rtex

of

the

righ

t an

gle

to t

he m

idpo

int

of t

he h

ypot

enus

e is

per

pend

icul

ar t

o th

ehy

pote

nuse

.

Giv

en:

isos

cele

s ri

ght

!A

BC

wit

h "

AB

Cth

e ri

ght

angl

e an

d M

the

mid

poin

t of

A $C$

Pro

ve:

B$M$

"A$

C$

Pro

of:

The

Mid

poin

t Fo

rmul

a sh

ows

that

the

coo

rdin

ates

of

Mar

e #&0

$ 22a &

,&2a

2$0

&$o

r (a

,a).

The

slop

e of

A!C!

is

&2 0a %%

20 a&

"%

1.Th

e sl

ope

of B!

M!is

&a a% %

0 0&

"1.

The

prod

uct

of t

he s

lope

s is

%1,

so B!

M!!

A!C!

.

x

y

C( 2

a, 0

)

A( 0

, 2a)

M

B( 0

, 0)

x

y

U( a

, 0)

T( ?

, ?)

S( –

a, 0

)x

y

P( 7

b, 0

)

R( ?

, ?)

N( 0

, 0)

x

y

Q( ?

, ?)

R( 2

a, b

)

P( 0

, 0)

x

y

N( 3

b, 0

)

M( ?

, ?)

O( 0

, 0)

x

y

Y( 2

b, 0

)

Z( ?

, ?)

X( 0

, 0)

x

y

B( 2

a, 0

)

A( 0

, ?)

C( 0

, 0)

x

yN

#5 – 2a, b

$

A( 0

, 0)

D( 5

a, 0

)x

yL(

3b, c

)

K( 0

, 0)

P( 6

b, 0

)x

y F( 0

, a)

G( 0

, 0)

H( b

, 0)

Sam

ple

answ

ers

are

give

n.

©G

lenc

oe/M

cGra

w-H

ill22

2G

lenc

oe G

eom

etry

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

1.eq

uila

tera

l !S

WY

wit

h 2.

isos

cele

s !

BL

Pw

ith

3.is

osce

les

righ

t !

DG

Jsi

des

%1 4% alo

ngba

se B$

L$3b

uni

ts lo

ngw

ith

hypo

tenu

se D$

J$an

dle

gs 2

aun

its

long

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

4.5.

6.

S#&1 6& b

,c$

C(3

a,0)

,E(0

,c)

M(0

,c),

N(%

2b,0

)

NEI

GH

BO

RH

OO

DS

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.K

arin

a liv

es 6

mile

s ea

st a

nd 4

mile

s no

rth

of h

er h

igh

scho

ol.A

fter

sch

ool s

he w

orks

par

tti

me

at t

he m

all i

n a

mus

ic s

tore

.The

mal

l is

2 m

iles

wes

t an

d 3

mile

s no

rth

of t

he s

choo

l.

7.W

rite

a c

oord

inat

e pr

oof

to p

rove

tha

t K

arin

a’s

high

sch

ool,

her

hom

e,an

d th

e m

all a

reat

the

ver

tice

s of

a r

ight

tri

angl

e.

Giv

en:

!S

KM

Pro

ve:

!S

KM

is a

rig

ht t

rian

gle.

Pro

of:

Slo

pe o

f S

K"

&4 6% %

0 0&

or &2 3&

Slo

pe o

f S

M"

& %3 2% %0 0

&or

%&3 2&

Sin

ce t

he s

lope

of

S!M!is

the

neg

ativ

e re

cipr

ocal

of

the

slop

e of

S!K!

,S!M!

!S!K!

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4-7

4-7



Rea

din

g t

o L

earn

Math

emati

csTr

iang

les

and

Coo

rdin

ate

Pro

of

NA

ME

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4-7

4-7

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Lesson 4-7

Pre-

Act

ivit

yH

ow c

an t

he

coor

din

ate

pla

ne

be u

sefu

l in

pro

ofs?

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d th

e in

trod

ucti

on t

o L

esso

n 4-

7 at

the

top

of

page

222

in y

our

text

book

.

Fro

m t

he c

oord

inat

es o

f A,B

,and

Cin

the

dra

win

g in

you

r te

xtbo

ok,w

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do y

ou k

now

abo

ut !

AB

C?

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ple

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Cis

isos

cele

sw

ith "

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the

ver

tex

angl

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din

g t

he

Less

on

1.F

ind

the

mis

sing

coo

rdin

ates

of

each

tri

angl

e.a.

b.

R(0

,b),

S(0

,0),

T#a,

&b 2& $D

(0,0

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(0,a

),F

(a,a

)

2.R

efer

to

the

figu

re.

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ind

the

slop

e of

S$R$

and

the

slop

e of

S$T$

.1;

%1

b.F

ind

the

prod

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of t

he s

lope

s of

S$R$

and

S$T$.W

hat

does

thi

s te

ll yo

u ab

out

S $R$an

d S$T$

?%

1;S!R!

!S!T!

c.W

hat

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you

r an

swer

fro

m p

art

b te

ll yo

u ab

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ST

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ampl

e an

swer

:!R

ST

is a

rig

ht t

rian

gle

with

"S

as t

he r

ight

ang

le.

d.F

ind

SR

and

ST

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t do

es t

his

tell

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abou

t S$R$

and

S$T$?

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2a2

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a%

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or a

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our

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par

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abou

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ple

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isos

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ver

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to

desc

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emb

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3.M

any

stud

ents

fin

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eas

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emem

ber

mat

hem

atic

al f

orm

ulas

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can

put

them

into

wor

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a c

ompa

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ay.H

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se t

his

appr

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to

rem

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fig

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2.40

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oin

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n e

ach

cir

cle?

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ist

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ices

of

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En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

4-7

4-7


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