Prove Triangles Congruent by ASA &

AAS

Lesson 4.10Use two more methods to prove

congruences

Vocabulary A flow proof uses arrows to show the flow of a

logical argument. ASA Congruence Postulate: If two angles and

the included side of one triangle are congruent to two angles & included side of a second triangle, then the two triangles are congruent

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.

Triangle Congruence ReviewPostulates we can use CAN’T USE

Warm-Up Exercises

ANSWER Yes; HL Thm.

Lesson 4.5, For use with pages 249-255

Tell whether the pair of triangles is congruent or not and why.

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.

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

ASA Congruence Postulate

AAS Congruence Theorem

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 ? Explain. RST VUT

RST VUT AAS

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

EXAMPLE 3 Write a flow proof

In the diagram, CE BD and ∠CAB CAD.

Write a flow proof to show ABE ADE

GIVEN CE BD,∠CAB CAD

PROVE ABE ADE