5.1 Congruence and Triangles - Murrieta Valley Unified School District

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5.1 Congruence and Triangles

Goal

Identify congruent triangles and corresponding parts.

Key Words

? corresponding parts ? congruent figures

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

E B

D

F

A

C

Suppose T ABC slides over to fit on T DEF. When the vertices are matched this way, a A and aD are corresponding angles and A&C* and D&F** are corresponding sides.

Corresponding angles and corresponding sides are examples of corresponding parts .

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

Congruent

Not congruent

Student Help

LOOK BACK To review vertices of a triangle, see p. 175.

Same size and shape

Different sizes or shapes

EXAMPLE 1 List Corresponding Parts

Given that TJKL c TRST, list all corresponding congruent parts.

LT

Solution

J

KS

R

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

Corresponding Angles

Corresponding Sides

TJKL c TRST, so aJ c aR.

TJKL c TRST, so J&K* c R&S*.

TJKL c TRST, so aK c aS.

TJKL c TRST, so K&L* c S&T*.

TJKL c TRST, so aL c aT.

TJKL c TRST, so J&L* c R&T*.

5.1 Congruence and Triangles 233

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Student Help

STUDY TIP To name the triangles in Example 2, you can start at any vertex. TACB c TFED

TBCA c TDEF

TCAB c TEFD

Be sure the letters are listed in corresponding order.

EXAMPLE 2 Write a Congruence Statement

The two triangles are congruent. a. Identify all corresponding congruent parts. b. Write a congruence statement.

A

F

B

CD

E

Solution

a. Corresponding Angles aA c aF aB c aD aC c aE

Corresponding Sides A&B* c F&D* B&C* c D&E* A&C* c F&E*

b. List the letters in the triangle names so that the corresponding

angles match. One possible congruence statement is TABC c TFDE.

EXAMPLE 3 Use Properties of Congruent Triangles

In the diagram, TPQR c TXYZ. P a. Find the length of X&Z*.

P

Y

X

95

b. Find maQ.

10

RZ

Solution a. Because X&Z* c P&R*, you know that XZ PR 10. b. Because aQ c aY, you know that maQ maY 95.

Name Corresponding Parts and Congruent Triangles

1. Given TSTU c TYXZ, list all

2. Given T ABC c TDEF, find

corresponding congruent parts.

the length of D&F* and maB.

U X

S

TY

C 3

A

B

D

E

28

Z

F

3. Which congruence statement is correct? Why?

A. T JKL c TMNP

K

B. T JKL c TNMP

C. T JKL c TNPM

J

LN

M P

234 Chapter 5 Congruent Triangles

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Student Help

STUDY TIP Since E&F is a shared side of both triangles, you know that E&F c E&F by the Reflexive Property of Congruence.

EXAMPLE 4 Determine Whether Triangles are Congruent

Use the two triangles at the right.

a. Identify all corresponding

congruent parts.

E

b. Determine whether the triangles

are congruent. If they are congruent, write a congruence statement.

D

G

Solution

a. Corresponding Angles Corresponding Sides

aD c aG

D&E** c G&E*

aDEF c aGEF

D&F** c G&F*

aDFE c aGFE

E&F* c E&F*

F

b. All three sets of corresponding angles are congruent and all three

sets of corresponding sides are congruent, so the two triangles are congruent. A congruence statement is TDEF c TGEF.

Visualize It!

Remember that when parallel lines are cut by a transversal, the alternate interior angles are congruent.

L

K

J

H

G

EXAMPLE 5 Determine Whether Triangles are Congruent

In the figure, H&G** L&K*. Determine

L

whether the triangles are congruent.

If so, write a congruence statement.

K J

H

G

Solution

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

aHJG c aKJL aH c aK aG c aL

Vertical angles are congruent. Alternate Interior Angles Theorem Alternate Interior Angles Theorem

The congruent sides are marked on the diagram, so H&J* c K&J*, H&G** c K&L*, and J&G* c &JL*. Since all corresponding parts are congruent, THJG c TKJL.

Determine Whether Triangles are Congruent

4. In the figure, X&Y* Z&W**. Determine

X

whether the two triangles are

congruent. If they are, write a

congruence statement.

Y

W V

Z

5.1 Congruence and Triangles 235

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5.1 Exercises

Guided Practice

Vocabulary Check

Determine whether the angles or sides are corresponding angles, corresponding sides, or neither.

1. aC and aL

2. A&C* and J&K*

B

K

3. B&C* and K&L*

4. aB and aL

A

CJ

L

Skill Check

Given that TXYZ c TEFD, determine the congruent side or angle

that corresponds to the side or angle.

5. aY

6. aD

7. X&Z*

8. F&D*

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

given that TLMN c TPQR.

N

9. maP __?__

10. maM __?__

P 45 P

11.6

8.5

11. QR __?__

12. LN __?__

105

L6M

R

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

E

K

F

GJ

L

Practice and Applications

Extra Practice

See p. 683.

Homework Help

Example 1: Exs. 14?19, 21?24

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

Corresponding Parts Determine whether the angles or sides are corresponding angles, corresponding sides, or neither.

14. aM and aS

15. aL and aR

M

S

16. M&N*** and S&T*

17. L&M** and T&S*

18. R&T* and L&N**

19. aN and aR

L

NR

T

20. Congruent Triangles Determine which of the following is a correct congruence statement for the triangles in Exercises 14?19. There is only one correct answer.

A. TLNM c TRTS B. TNML c TRST

C. TMNL c TSTR

236 Chapter 5 Congruent Triangles

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Identifying Congruent Parts In the diagram, TLMN c TPQR. Complete the statement with the corresponding congruent part.

21. aR c __?__

22. L&N* c __?__

L

P

23. aRPQ c __?__

24. Q&P* c __?__

25. TMNL c __?__ 26. TPRQ c __?__

M

P

NR

Naming Congruent Triangles Determine whether the congruence statement correctly describes the congruent triangles shown in the necklace.

27. TFGH c TNML

28. TGFH c TNLM

29. TGHF c TMLN 30. THFG c TLNM 31. THGF c TLNM

H

GM

L

32. TFHG c TLNM

F N

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.

P

S

34. A

B

F

C

P

RT

U

D

E

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.

CF

36.

B

A

C

E

A

BE

D

F

D

5.1 Congruence and Triangles 237

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