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Name ——————————————————————— Date ————————————Name ——————————————————————— Date ————————————

Use two more methods to prove congruences.

VocabularyA flow proof uses arrows to show the flow of a logical argument.

Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate: 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.

Theorem 6 Angle-Angle-Side (AAS) Congruence Theorem: 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.

goal

Identify congruent triangles

Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

a. b. c.

Solution

a. The vertical angles are congruent, so three pairs of angles are congruent. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent.

b. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate.

c. Two pairs of angles and a non-included pair of sides are congruent. The triangles are congruent by the AAS Congruence Theorem.

example 1

Study guideFor use with the lesson “Prove Triangles Congruent by ASA and AAS“

GeometryChapter Resource Book4-82

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Name ——————————————————————— Date ————————————

Study guide continuedFor use with the lesson “Prove Triangles Congruent by ASA and AAS“

exercises for example 1

Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.

1. 2. 3.

Write a flow proof

In the diagram, ∠G > ∠B and }

CB }

GA . B C

G A

Write a flow proof to show nGCA > nBAC.

Solution

Given: ∠ G > ∠ B, }

CB }

GA

Prove: n GCA > n BAC

}

CB }

GA

∠BCA > ∠GAC

Given Alternate Interior ?

∠G > ∠B

nGCA > nBAC

Given AAS Congruence Theorem

}

AC > }

AC

Reflexive Property

exercises for example 2

Write a flow proof to show that the triangles are congruent.

4. Given: ∠ PQS > ∠ RQS 5. Given: ∠ OMN > ∠ ONM

∠ QSP > ∠ QSR ∠ LMO > ∠ JNO

Prove: nPQS > nRQS Prove: n MJN > n NLM

P R

S

N

J

O L

M

example 2 Less

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GeometryChapter Resource Book 4-83

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Angles Postulate; 5. Given; 6. ASA Congruence Postulate

18. It is given that ∠ B ù ∠ D. and that }

AC ù }

EC . By the Vertical Angles Theorem, ∠ BCA ù ∠ DCE. nABC ù nEDC by the AAS Congruence Theorem.

Practice Level C

1. } FE > } TR or } DE > }

QR 2. ∠ F > ∠ T

3. } DF > }

QT 4. No 5. Yes; ∠ KNL > ∠ MLN by Alternate Interior Angles Theorem, } LN > } LN by Reflexive Property of Congruence, n KLN ù n MNL by ASA Congruence Postulate

6. Yes; } TX > } VY by summation of congruent parts, } YX > } YX by Reflexive Property of Congruence, } XZ > } YW by summation of congruent parts, n TXZ ù n VYW by SAS Congruence Theorem 7. No, ∠ M and ∠ Y are not corresponding angles. 8. No, } JR and } YZ are not corresponding sides. 9. Yes, AAS Congruence Theorem 10. No, the congruent sides are not corresponding sides. 11. Two pairs of corresponding sides (

} AF ù } BF , } FD ù

} FC )

and the corresponding included angles (∠ AFD ù ∠ BFC, by Vertical Angles Theorem) are congruent. 12. Two pairs of corresponding angles (∠ ACE ù ∠ DBA, ∠ AEC ù ∠ DAB) and the a corresponding non-included side (

} AC ù } DB ,

by summation of congruent parts) are congruent.

13. ∠ ACD ù ∠ ABD is given. ∠ BDC ù ∠ ABD by Alternate Interior Angles Theorem. ∠ ACD ù ∠ BDC by Transitive Property of Congruence.

} DC ù } DC by Reflexive Property

of Congruence. ∠ ADF ù ∠ BCF because n ADF ù n BCF by SAS Congruence Theorem. Then ∠ ADC ù ∠ BCD by summation of congruent parts.

14.

Statements Reasons

1. } AB i } DC 1. Given2. ∠ ADB ù ∠ CBD 2. Given3. ∠ ABD ù ∠ CDB 3. Alternate Interior

Angles Theorem4. } DB > } DB 4. Reflexive Property

of Congruence5. n ABD ù n CDB 5. ASA Congruence

Postulate

15. 1. Given; 2. Given; 3. Reflexive Property of Congruence; 4. AAS Congruence Theorem; 5. Corresponding parts of congruent triangles are congruent; 6. Alternate Interior Angles Converse

16. 1. Given; 2. Given; 3. Vertical Angles Theorem; 4. ASA Congruence Postulate; 5. Corresponding parts of congruent triangles are congruent.

17.

Statements Reasons

1. ∠ MJL ù ∠ KJL 1. Given 2. ∠ MLJ ù ∠ KLJ 2. Given 3. } JL ù } JL 3. Reflexive Property of

Congruence 4. n MJL ù n KJL 4. ASA Congruence

Postulate 5. } LN ù } LN 5. Reflexive Property of

Congruence 6. } ML ù } KL 6. Corresponding parts

of congruent triangles are congruent.

7. m∠ MLJ 1 7. Linear Pair Postulate m∠ MLN 5 1808

8. ∠ MLJ and ∠ MLN 8. Definition of are supplementary. supplementary angles

9. m∠ KLJ 1 9. Linear Pair Postulate m∠ KLN 5 1808

10. ∠ KLJ and ∠ KLN 10. Definition of are supplementary. supplementary angles

11. ∠ MLN ù ∠ KLN 11. Congruent Supple-ments Theorem

12. n MLN ù n KLN 12. SAS Congruence Theorem

Study Guide

1. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate. 2. Two pairs of angles and a non-included pair of sides are congruent. The triangles are congruent by the AAS Congruence Theorem. 3. Two pairs of sides and a pair of angles are congruent. This is not enough information to prove that the triangles are congruent.

4. aPQS c aRQS

aQSP c aQSR TPQS c TRQS

QS c QS

Given

ASA CongruencePostulate

Given

ReflexiveProperty

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GeometryChapter Resource Book A55

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5. aOMN c aONM

TMJN c TNLM

MN c NM

Given

aLMN c aJNM

Given

AAS CongruenceTheorem

aLMO c aJNO

Angle AdditionPostulate;

Definition ofCongruence Reflexive

Property

Interdisciplinary Application

1.

Statements Reasons

1. ∠ ABS > ∠ CBT 1. Given2. ∠ SAB > ∠ TCB 2. Def. of a rectangle3. } SA >

} TC 3. Def. of a rectangle

4. nSAB > nTCB 4. AAS Congruence Thm.

2. Yes, since nSAB >nTCB, }

AB > }

CB . So, B is the midpoint of

} AC .

3.

A B

M

C

S T

4. Sample answer:

Statements Reasons

1. } SM > } TM 1. Given (from Ex. 3)

2. } BM > } BM 2. Reflexive Prop.3. ∠ ASB > ∠ CTB 3. Def. of

> triangles4. m∠ ASB > m∠ CTB 4. Def. of

congruence5. m∠ TSB 5 90° 2 m∠ ASB 5. Def. of a

m∠ STB 5 90° 2 m∠ CTB rectangle and complementary angles

6. m∠ STB 5 90° 2 m∠ ASB 6. By substitution m∠ STB 5 m∠ TSB

7. ∠ STB > ∠ TSB 7. Def. of congruence

8. nSMB > nTMB 8. SAS Congruence Postulate

5. a. m∠ ABS 5 53° b. m∠ CBT 5 53° c. m∠ SAB 5 90° d. m∠ TSB 5 53°

e. m∠ TCB 5 90° f. m∠ STB 5 53° g. m∠ ASB 5 37° h. m∠ CTB 5 37°

Challenge Practice

1.

x

y

1

21

(0, 5)

(1, 1)

(4, 3)

(3, 7)(2, 4)

2. (1, 6)

(1, 0)

(5, 2)

(5, 4)

(4, 3)

x

y

1

4

Proofs will vary. Proofs will vary.

3. a.

12 cm15 cm

9 cm

b.

12 cm

13 cm

15 cm

4 cm5 cm

c.

12 cm15 cm15 cm

4 cm5 cm9 cm

13 cm d.

15 cm15 cm

4 cm14 cm

13 cm

The two triangles in the figure have two sets of congruent sides measuring 15 centimeters and 13 centimeters. They also each have a non-included angle congruent. However, from the diagram you can see that the larger triangle has a base of 14 centimeters and the smaller triangle has a base of 4 centimeters. Because these measures are not congruent, the triangles are not congruent.

4. The SSA case is valid when the non-included angle is a right angle because then the hypotenuse and leg of the first triangle would be congruent to the hypotenuse and leg of the second triangle. Using the HL Congruence Theorem, you can conclude that the two triangles are congruent.

5. Statements Reasons

1. } AE i } BF , }

CE i }

DF 1. Given 2. ∠ EAC > ∠ FBD 2. Corresponding

Angles Postulate 3. } AB >

} CD 3. Given

4. AB 5 CD 4. Definition of congruent segments

5. AC 5 AB 1 BC 5. Segment Addition BD 5 BC 1 CD Postulate

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