5.3 Proving Triangles are Congruent: ASA and AAS

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5.3 Proving Triangles are

Congruent: ASA and AAS

Goal

Show triangles are congruent using ASA and AAS.

Key Words

? vertical angles p. 75 ? alternate interior angles

p. 121

Geo-Activity Creating Congruent Triangles

1 Draw a segment 3 inches long.

Label the endpoints A and B.

2 Draw an angle measuring 45

at point A.

A

B

70 0 110

80 100

90

100 80

710106102050130

140401530016200

170 10

180 0

3

4

5

6

A

B

3 Draw an angle measuring 30

at point B.

4 Label the point where the angle

rays intersect as point C.

A

0 10 20

180

170

160

30 150 14040

70

60 50130 120

110

80 100

90

100 11 80 70

1

2

3

B

C

A

B

5 Compare your triangle to the triangles drawn by your classmates.

Are the triangles congruent?

The Geo-Activity above suggests the following postulate.

Visualize It!

In this triangle, A&C* is the included side between aA and aC.

included side

A

C

B

POSTULATE 14

Angle-Side-Angle Congruence Postulate (ASA)

Words If two angles and the included side of one triangle are

congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

Symbols If Angle aA c aD, and

B

Side A&C* c D&F*, and

Angle aC c aF,

A

E C

then TABC c TDEF.

D

F

250 Chapter 5 Congruent Triangles

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Student Help

STUDY TIP You can use the ASA Congruence Postulate because B&C* and F&E* are included between the congruent angles.

EXAMPLE 1 Determine When To Use ASA Congruence

Based on the diagram, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning.

a.

BE

D

b. T

Z

A

CF

R

SX

Y

Solution a. You are given that aC c aE, aB c aF, and B&C*c F&E*.

You can use the ASA Congruence Postulate to show that T ABC c T DFE.

b. You are given that aR c aY and aS c aX. You know that R&T* c Y&Z*, but these sides are not included between the congruent angles, so you cannot use the ASA Congruence Postulate.

THEOREM 5.1

Angle-Angle-Side Congruence Theorem (AAS)

Words If two angles and a non-included side of one triangle are

congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Symbols If Angle aA c aD, and

Angle aC c aF, and

Side B&C* c E&F*,

A

then TABC c TDEF.

BD

E C

D

F

EXAMPLE 2 Determine What Information is Missing

What additional congruence is needed

K

to show that TJKL c TNML by the

AAS Congruence Theorem?

N L

Solution

J

M

You are given K&L* c M&L**.

Because aKLJ and aMLN are vertical angles, aKLJ c aMLN.

The angles that make K&L* and M&L** the non-included sides are aJ and aN, so you need to know that aJ c aN.

5.3 Proving Triangles are Congruent: ASA and AAS 251

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Student Help

STUDY TIP The sides of MNQP look parallel but this information is not marked, so you cannot conclude that any angles are congruent.

EXAMPLE 3 Decide Whether Triangles are Congruent

Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use.

a. E

H

b. N

P c. U

Z

G

F

J

M

P

W

X

Solution a. E&F* c J&H* aE c aJ aFGE c aHGJ

Given Given Vertical Angles Theorem

Use the AAS Congruence Theorem to conclude that T EFG c T JHG.

b. Based on the diagram, you know only that M&P** c Q&N** and N&P* c N&P*. You cannot conclude that the triangles are congruent.

c. aUZW c aXWZ W&Z** c W&Z** aUWZ c aXZW

Alternate Interior Angles Theorem Reflexive Prop. of Congruence Alternate Interior Angles Theorem

Use the ASA Congruence Postulate to conclude that TWUZ c T ZXW.

Visualize It!

Because aA and aE are also alternate interior angles, you can show that TABD c TEBC by the AAS Congruence Theorem.

A

C

B

D

E

EXAMPLE 4 Prove Triangles are Congruent

A step in the Cat's Cradle string

game creates the triangles shown. Prove that T ABD c T EBC.

A

C

Solution

Given B&D* c B&C*, A&D* E&C* Prove TABD c TEBC

B

D

E

Statements

1. B&D* c B&C* 2. A&D* E&C* 3. aD c aC 4. aABD c aEBC 5. T ABD c T EBC

Reasons

1. Given 2. Given 3. Alternate Interior Angles Theorem 4. Vertical Angles Theorem 5. ASA Congruence Postulate

252 Chapter 5 Congruent Triangles

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Decide Whether Triangles are Congruent

1. Complete the statement: You can use the ASA Congruence Postulate when the congruent sides are __?__ between the corresponding congruent angles.

Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use.

2. A

B

3. E

G

J 4.

P

K

L

D

C

D

F

H

M

N

5.3 Exercises

Guided Practice

Vocabulary Check

Tell whether the side is included or not included between the given angles.

1. F&G* is __?__ between aF and aG.

H

2. G&H* is __?__ between aF and aG.

3. F&H** is __?__ between aH and aG.

F

G

4. H&G** is __?__ between aH and aG.

Skill Check

What congruence do you need to know in order to use the indicated postulate or theorem to conclude that the triangles are congruent?

5. ASA Congruence Postulate

6. AAS Congruence Theorem

C

F

A

B

F

A

BD

E

C

E

D

Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use.

7. TRST and TTQR

8. TJKL and TNML

9. T ABC and TDEF

S

K

M

A

R P

T

L

B

F

J

N

C E

D

5.3 Proving Triangles are Congruent: ASA and AAS 253

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Practice and Applications

Extra Practice

See p. 683.

Recognizing Included Sides Tell whether the side is included or not included between the given angles.

10. A&B* is __?__ between aB and aBCA.

A

D

11. A&C* is __?__ between aBAC and aBCA.

12. A&C* is __?__ between aDAC and aD.

B

C

13. B&C* is __?__ between aCAB and aB.

Choosing ASA or AAS Tell which postulate or theorem you would use to show that the triangles are congruent. Explain your reasoning.

14.

A

15. J

P 16. R

B

C

D

L MN

E

F

U

S

T

Showing Triangles are Congruent Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

17. M P

S 18.

B

19. R

S

V

U

T

T

P

R

A

C

D

20. E

F

G

21.

X

W H

J

Y

22. N

M

Z

K

L

Homework Help

Example 1: Exs. 14?22 Example 2: Exs. 24?30 Example 3: Exs. 17?22 Example 4: Exs. 34, 35

23. Sea Plane In the sea plane at the right, aL and aR are

right angles, K&L* c Q&R*, and aJ c aP.

What postulate or theorem

allows you to conclude that T JKL c T PQR ?

254 Chapter 5 Congruent Triangles

L

R

J

P

K

OE

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Origami

ORIGAMI is the Japanese art of folding paper into objects. The seal shown in the photograph was folded from a single piece of square paper.

Application Links



Missing Information Determine what congruence is needed in order to use the indicated postulate or theorem to show that the triangles are congruent.

24. ASA Congruence Postulate

25. AAS Congruence Theorem

26. ASA Congruence Postulate

A

D

E

B

CF

J

W

K P

X

Z

L

P

R

Y

Origami The triangles below show a step in folding an origami seal.

State the third congruence needed to prove that TABC c TABD

using the indicated postulate or theorem.

27. Given aCBA c aDBA, B&A* c B&A*

B

Use the AAS Congruence Theorem.

28. Given aC c aD, A&C* c A&D*

C

D

Use the ASA Congruence Postulate.

29. Given aC c aD, aCAB c aDAB

Use the ASA Congruence Postulate.

30. Given aC c aD, A&C* c A&D*

Use the SAS Congruence Postulate.

A

Visualize It! Use the given information to make a sketch of TPQR

and TSTU. Mark the triangles with the given information.

31. aQ c aT aR c aU

Q&R* c T&U*

32. aP c aS aR c aU

P&Q* c &ST*

33. aQ c aT aS c aP

Q&R* c T&U*

34. Logical Reasoning Fill in the missing statements and reasons.

Given G&F* c G&L*

F&H* L&K*

Prove TFGH c TLGK

H F

L G

K

Statements

1. G&F* c G&L* 2. F&H* L&K* 3. aF c aL 4. _________?_________ 5. TFGH c TLGK

Reasons

1. Given 2. _________?_________ 3. _________?_________ 4. Vertical Angles Theorem 5. _________?_________

5.3 Proving Triangles are Congruent: ASA and AAS 255

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35. Logical Reasoning Fill in the missing statements and reasons.

Given B&C* c E&C*

E

A&B* A&D*

D&E* A&D*

A

Prove TABC c TDEC

C

D

B

Statements

Reasons

1. _________?_________ 2. A&B* A&D* 3. D&E* A&D* 4. aA and aD are right A.

5. _________?_________ 6. aACB c aDCE 7. T ABC c TDEC

1. Given 2. _________?_________ 3. _________?_________ 4. _________?_________ 5. Right angles are congruent. 6. _________?_________ 7. _________?_________

Standardized Test Practice

Mixed Review

36. Multiple Choice Which theorem or postulate cannot be used to show that TPQR c TSQU?

A ASA

P

B AAS

U P

C SAS

D SSS

R

S

Finding a Missing Length Find the unknown side length. Round your answer to the nearest tenth. (Lesson 4.4)

37. 18

b

38.

6

10 11 c

39.

a

15

26

Identifying Congruent Parts In the diagram, TABC c TDEF.

Complete the statement. (Lesson 5.1)

40. aC c __?__

41. B&A* c __?__

B

CD

42. aD c __?__

43. E&F* c __?__

44. aB c __?__

45. C&B* c __?__

AF

E

Algebra Skills

Fraction Operations Divide. Write your answer in simplest form. (Skills Review, p. 659)

46. 12 6

47. 45 4

48. 16 23

49. 58 156

50. 79 27

51. 130 1225

52. 8 34

53. 141 12

256 Chapter 5 Congruent Triangles

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