5 congruent chapter triangles - mrs....

ChapterChapter 55CongruentTrianglesCongruentTriangles

230

Where should a goalie

stand?

A soccer goalie instinctively imagines a triangle formed by the goal posts andthe ball. The best position to standallows the goalie to reach each side ofthe triangle in the same amount of time.

As the ball moves and the shape of the triangle changes, the goalie’s bestposition also changes.

Learn More About ItYou will learn more about soccer inExercises 21 and 22 on p. 278.

G

cgpe-0500 8/16/01 9:45 AM Page 230

Who usesCongruent Triangles?

How will you use these ideas?

• See how triangles are used in quilting. (p. 247)• Investigate origami patterns. (p. 255)• Identify triangles in skateboard ramps. (p. 257)• Analyze lighting for a stage production. (p. 262)• Look for patterns in string designs. (p. 269)• Find out how kaleidoscope patterns are created. (p. 285)

TYPE DESIGNERType designers design fonts that appear in books,magazines, newspapers, and other materials. Jonathan Hoefler hasdesigned many fonts that are widely used today. (p. 288)

FACILITIES PLANNERFacilities planners helpbusinesses determine the best locations for new buildings. They canhelp companies save money and run moreefficiently. (p. 275)

231

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232 Chapter 5 Congruent Triangles

Chapter Study GuideStudy Guide55

What’s the chapter about?

• Identifying corresponding parts of congruent triangles• Showing triangles are congruent• Using angle bisectors and perpendicular bisectors• Using reflections and line symmetry

Key Words

Take this quick quiz. If you are unsure of an answer, look at the

reference pages for help.

Vocabulary Check (refer to p. 192)

1. Which segment in the figure is the hypotenuse?

�A KL&* �B LN&* �C NM&& �D KM&**

Skill Check (refer to pp. 61, 128)

2. If maFGH � 100° and GJ&*(bisects aFGH, what is maFGJ ?

�F 25° �G 50° �H 90° �J 100°

3. Use the diagram at the right to determine which ofthe following statements is true.

�A a7 c a2 �B a5 c a2

�C a1 c a8 �D a3 c a7

PREVIEW

L

K

M

N

7865

3421

PREVIEW

Chapter Readiness QuizPREPAREPREPARE

Studying TrianglesVISUAL STRATEGYVISUAL STRATEGY

Visualize It!

• corresponding parts, p. 233• congruent figures, p. 233• proof, p. 243• distance from a point to a line,

p. 273

• equidistant, p. 273• perpendicular bisector, p. 274• reflection, p. 282• line of symmetry, p. 284

When working with overlappingtriangles, you may find it helpful to draw the triangles separately.

cgpe-05sg 07/24/2001 7:48 PM Page 232

The houses below are located in San Francisco, California. The twotriangles outlined on the houses have the same size and shape.

Suppose TABC slides over to fit on TDEF. When the vertices are

matched this way, aA and aD are corresponding angles and AC&*

and DF&** are corresponding sides.

Corresponding angles and corresponding sides are examples of.

Figures are if all pairs of corresponding angles arecongruent and all pairs of corresponding sides are congruent.

Congruent Not congruent

congruent

corresponding parts

5.1 Congruence and Triangles 233

Goal

Identify congruenttriangles andcorresponding parts.

Key Words

• corresponding parts

• congruent figures

5.15.1 Congruence and Triangles

Given that TJKLcTRST, list allcorresponding congruent parts.

Solution

The order of the letters in the names of the triangles shows which parts correspond.

Corresponding Angles Corresponding Sides

TJKLcTRST, so aJc aR. TJKLcTRST, so JK&* c RS&*.

TJKLcTRST, so aKc aS. TJKLcTRST, so KL&* c ST&*.

TJKLcTRST, so aLc aT. TJKLcTRST, so JL&* c RT&*.

J K

L

S R

T

EXAMPLE 111 List Corresponding Parts

LOOK BACK

To review vertices of atriangle, see p. 175.

Student Help

Same size and shape Different sizes or shapes

B

A C

E

D FF

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1. Given TSTUcTYXZ, list all 2. Given TABCcTDEF, findcorresponding congruent parts. the length of DF&* and maB.

3. Which congruence statement is correct? Why?

A.TJKLcTMNP

B. TJKLcTNMP

C.TJKLcTNPM LJ

K

PN

M

BA

ED

C

F

28�

3

TS

U

Y

X

Z


The two triangles are congruent.

a. Identify all corresponding congruent parts.

b. Write a congruence statement.

Solution

a. Corresponding Angles Corresponding Sides

aAc aF AB&* c FD&*

aBc aD BC&* c DE&*

aC c aE AC&* c FE&*

b. List the letters in the triangle names so that the corresponding angles match. One possible congruence statement is TABCcTFDE.

EXAMPLE 222 Write a Congruence Statement

In the diagram, TPQRcTXYZ.

a. Find the length of XZ&*.

b. Find maQ.

Solution

a. Because XZ&* c PR&*, you know that XZ � PR � 10.

b. Because aQc aY, you know that maQ � maY � 95�.

R Z

P Y XP 95�

10

EXAMPLE 333 Use Properties of Congruent Triangles

Name Corresponding Parts and Congruent Triangles

STUDY TIP

To name the trianglesin Example 2, you canstart at any vertex.

TACBcTFED

TBCAcTDEF

TCABcTEFD

Be sure the letters arelisted in correspondingorder.

Student Help B C

A

D E

F

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Use the two triangles at the right.

a. Identify all corresponding congruent parts.

b. Determine whether the triangles are congruent. If they are congruent,write a congruence statement.

Solution

a. Corresponding Angles Corresponding Sides

aDc aG DE&** c GE&*

aDEFc aGEF DF&** c GF&*

aDFEc aGFE EF&* c EF&*

b. All three sets of corresponding angles are congruent and all three sets of corresponding sides are congruent, so the two triangles arecongruent. A congruence statement is TDEFcTGEF.

EXAMPLE 4 Determine Whether Triangles are Congruent

In the figure, HG&** � LK&*. Determine whether the triangles are congruent. If so, write a congruence statement.

Solution

Start by labeling any information you can conclude from the figure. You can list the following angles congruent.

aHJGc aKJL Vertical angles are congruent.

aHc aK Alternate Interior Angles Theorem

aG c aL Alternate Interior Angles Theorem

The congruent sides are marked on the diagram, so HJ&* c KJ&*, HG&** c KL&*,

and JG&*c JL&*. Since all corresponding parts are congruent, THJGcTKJL.

GH

J

L K

EXAMPLE 5 Determine Whether Triangles are Congruent

Determine Whether Triangles are Congruent

Remember that whenparallel lines are cut by a transversal, thealternate interior anglesare congruent.

J

GH

L K

Visualize It!

STUDY TIP

Since EF& is a sharedside of both triangles, you know that EF& c EF&by the Reflexive Propertyof Congruence.

Student Help

4. In the figure, XY&* � ZW&**. Determine whether the two triangles are congruent. If they are, write a congruence statement.

ZY

V

X W

D G

E

F

LAHGE_10_CSse_cp_233-239 4/23/09 2:31 AM Page 235


Determine whether the angles or sides are corresponding angles,

corresponding sides, or neither.

1. aC and aL 2. AC&* and JK&*

3. BC&* and KL&* 4. aB and aL

Given that TXYZ cTEFD, determine the congruent side or angle

that corresponds to the side or angle.

5. aY 6. aD 7. XZ&* 8. FD&*

In Exercises 9–12, find the missing length or angle measure,

given that TLMN c TPQR.

9. maP � __?__ 10. maM � __?__

11. QR � __?__ 12. LN � __?__

13. Determine whether the triangles are congruent. If they are, write a congruence statement.

Corresponding Parts Determine whether the angles or sides are

corresponding angles, corresponding sides, or neither.

14. aM and aS 15. aL and aR

16. MN&*** and ST&* 17. LM&** and TS&*

18. RT&* and LN&** 19. aN and aR

20. Congruent Triangles Determine which of the following is a correctcongruence statement for the triangles in Exercises 14–19. There isonly one correct answer.

A.TLNMcTRTS B. TNMLcTRST C. TMNLcTSTR

L

M S

N TR

Practice and Applications

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K

JF

E

G

Skill Check

Vocabulary Check

Guided Practice

Exercises5.15.1

Extra Practice

See p. 683.

Example 1: Exs. 14–19,21–24

Example 2: Exs. 21–36Example 3: Exs. 37–40Example 4: Exs. 42–47Example 5: Exs. 42–47

Homework Help

M RL

N PP

105�

45�

8.5

6

11.6

LJ

K

CA

B

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Identifying Congruent Parts In the diagram, TLMN c TPQR.

Complete the statement with the corresponding congruent part.

21. aRc __?__

22. LN&* c __?__

23. aRPQc __?__

24. QP&* c __?__

25. TMNLc __?__

26. TPRQc __?__

Naming Congruent Triangles Determine whether the congruence

statement correctly describes the congruent triangles shown

in the wooden floor.

27. TFGHcTNML

28. TGFHcTNLM

29. TGHFcTMLN

30. THFGcTLNM

31. THGFcTLNM

32. TFHGcTLNM

Writing Congruent Statements In Exercises 33 and 34, the

triangles are congruent. Identify all pairs of corresponding

congruent parts.Then write a congruence statement.

33. 34.

Marking Triangles In Exercises 35 and 36, TABC c TDEF. Draw

the triangles on your paper.Then mark every angle and side of

the triangles to show the corresponding congruent parts.

35. 36.

F

B

E

D

A C

E

F

DA

C

B

A

D

C

E

F

BP

RP

S

UT


M

L

N

P

R

P

NN

MM

LL

GGFF

HH

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Using Congruent Triangles In Exercises 37–40, TABC c TDEF.

37. Find the length of DF&* 38. Find the length of AB&*and maB. and maF.

39. Find the length of AB&* 40. Find the length of BC&*and maF. and maD.

41. Writing Congruence Statements Given that TFGH cTTSR,determine which congruence statement does not describe the triangles.

A.THGFcTRST B. TFHGcTTRS C. TGFHcTSRT

Congruent Triangles Determine whether the triangles are congruent.

If so, write a congruence statement.

42. 43.

44. 45.

46. 47. Z

W YXM

J K

L

DA

C

BER

T

SP

K

L M

J P N60�

30� 30�

60�

5 5

A

C

B

D

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F

C

BD F

A E86�

27� 115

B

A F E

C D

30�

50�

4

6

B

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C D39�

14

B

A D

C F E100�

5


HOMEWORK HELP

Extra help with problemsolving in Exs. 42–47 is at classzone.com

IStudent HelpI C L A S S Z O N E . C O M

cgpe-0501 07/24/2001 7:46 PM Page 238

Sculpture The diagram below represents a view of the base of a stainless

steel sculpture involving four triangles.

48. Explain how you know that aGJFc aHJF.

49. Explain how you know that aJGFc aJHF.

50. Do you have enough information to show that TGFJcTHFJ? Explain your reasoning.

Error Analysis In Exercises 51 and 52, refer to Jillian’s work and the

diagram below.

51. How does Jillian know that aBACc aDAE?

52. Is TABCcTADE? Explain your reasoning.

53. Multiple Choice Which congruence statementdoes not describe the triangles at the right?

�A TTSRcTXYZ �B TRSTcTXYZ

�C TSTRcTYXZ �D TTRScTXZY

Naming Properties Name the property that the statement

illustrates. (Lesson 2.6)

54. maJ � maJ 55. PQ&* c QP&*

56. If aPQRc aFGH, 57. If AB � CD and CD � FG, then aFGHc aPQR. then AB � FG.

Classifying Triangles Classify the triangle with the given side lengths

as acute, right, or obtuse. (Lesson 4.5)

58. 5, 2, 6 59. 5, 9, 10 60. 14, 5, 12

Evaluating Decimals Evaluate. (Skills Review, p. 655)

61. 4.15 � 7.8 62. 12.64 � 8.92 63. 14 � 5.33

64. 0.91 � 0.17 65. 1.476 � 6.2 � 9.4 66. 10.872 � 7.65

Algebra Skills

Mixed Review

Standardized Test

Practice


aABC c aADEaACB c aAEDaBAC c aDAE

The corresponding angles are congruent so TABC c TADE.

GJ

H

F

Z

XT Y

R

S

B

E

D

AC

Jillian

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5 congruent chapter triangles - mrs....

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