EXAMPLE 1 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.

SOLUTION

The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem.

a.

EXAMPLE 1 Identify congruent triangles

b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent.

c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate.

EXAMPLE 2 Prove the AAS Congruence Theorem

Prove the Angle-Angle-Side Congruence Theorem.

Write a proof.

GIVEN BC EF A D, C F,

PROVE ABC DEF

GUIDED PRACTICE for Examples 1 and 2

SOLUTION

1.

GivenS U

The vertical angles are congruent

RTS UTV

GivenRS UV

STATEMENTS REASONS

In the diagram at the right, what postulate or theorem can you use to prove that RST VUT ? Explain.

GUIDED PRACTICE for Examples 1 and 2

Therefore are congruent because vertical angles are congruent so two pairs of angles and a pair of non included side are congruent. The triangle are congruent by AAS Congruence Theorem.

RTS UTV

ANSWER

GUIDED PRACTICE for Examples 1 and 2

2. Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof.

1. Draw BD parallel to AC . 1. Parallel Postulate

PROVE 3 = 180°1m 2m m+ +

2. Angle Addition Postulate and definition of straight angle

2. 4m 2m 5m+ + = 180°

3. Alternate Interior Angles Theorem

3. 1 4 , 3 5

5. Substitution Property of Equality

5. 1m 2m 3m+ + = 180°

4. Definition of congruent angles

4. 1m = 4m 3m = 5m,

STATEMENTS REASONS

GIVEN ABC