5.3 Proving Triangles are Congruent: ASA and AAS

[Pages:4]5.3 Proving Triangles are Congruent:

ASA and AAS

Goal Show triangles are congruent using ASA and AAS.

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 Side A**C** c *D**F*, and Angle aC c aF,

then TABC c T DEF .

E

B

D

F

A

C

Follow-Up

Use TTGL shown. Complete the table.

Angles

Included Side

T

aT and aG

T&G*

aG and aL

G&L*

aT and aL

T&L

G

L

Draw any TABC in the space below. Complete the table.

Angles

Non-Included Sides

B

aA and aB

A&C* and B&C*

aB and aC

A&B* and A&C*

A

aA and aC

A&B* and B&C*

C

114 Geometry, Concepts and Skills Notetaking Guide ? Chapter 5

Example 1 Determine When to Use ASA

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

a.

B

D

b. T

Z

A

C

R

SX

Y

Solution

a. In TABC, the included side between the marked angles is B&C* . In TDCB, the included side is B&C* . These sides are congruent by the Reflexive Property. So, with aACB c a DBC , aABC c a DCB , and B&C* c B&C* , you can use the ASA Congruence Postulate to show that TABC c TDCB.

b. In TRST, the included side is R&S* . In TYXZ, the included side is X&Y* . Since these sides are not marked congruent, 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

E

If Angle aA c aD, and

Angle aC c aF, and

BD

F

Side *B*C** c *E*F**,

then TABC c T DEF .

A

C

Follow-Up

In Example 1, part (b), can you use the AAS Congruence Theorem to show that the triangles are congruent?

Yes

Lesson 5.3 ? Geometry, Concepts and Skills Notetaking Guide 115

Example 2 Determine What Information is Missing

What additional congruence is needed K

N

to show that TJKL c TNML by the

AAS Congruence Theorem?

L

Solution

J

M

You are given *K**L* c M&L*. You know aKLJ c aMLN because they

are vertical angles. The angles that make *K**L* and M&L* the non-included sides are a J and a N . So, to use the AAS

Congruence Theorem you need the congruence aJ c aN .

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

Q c.

U

Z 1

2

G

F

J

M

P

4

3

W

X

Solution a. Use the AAS Congruence Theorem.

Side *E*F** c J**H** is given. Angle aE c aJ is given. Angle aFGE c aHGJ since vertical angles are congruent.

b. You only know that M***P* c Q**N** and N**P** c N**P**. You cannot conclude that the triangles are congruent.

c. Use the ASA Congruence Postulate. Angle a1 c a3 since alternate interior angles are congruent. Angle a2 c a4 since alternate interior angles are congruent. Side W***Z** c W***Z** by the Reflexive Property of Congruence.

116 Geometry, Concepts and Skills Notetaking Guide ? Chapter 5

Example 4 Prove Triangles are Congruent

Use the information given in the

A

diagram to prove TABD c TEBC.

C

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

B

D

E

Statements

Reasons

1. B**D** c *B*C** 2. A**D** E**C** 3. aD c aC 4. aABD c aEBC 5. TABD c TEBC

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

Follow-Up

In Example 4, how can you use the AAS Congruence Theorem to prove that the triangles are congruent? Replace Statement 3 with aA c aE and Reason 5 with AAS Congruence Theorem.

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

1. A

1 4

B 2.

G

E

J 3. K

2

3

D

C

P

H

D

F

N L

M

Yes; ASA

No

Yes; AAS

Lesson 5.3 ? Geometry, Concepts and Skills Notetaking Guide 117

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