1. Prove Triangles Congruent by ASA & AAS
Lesson 4.10
Use two more methods to prove congruences

2. 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.

3. Triangle Congruence Review
Postulates we can use CAN’T USE

4. Lesson 4.5, For use with pages 249-255
Tell whether the pair of triangles is congruent or not and why.
ANSWER
Yes; HL Thm.

5. 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.

6. EXAMPLE 1
Identify congruent triangles
There is not enough information to prove the triangles are congruent, because no sides are known to be congruent.
Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate.

7. 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

8. ASA Congruence Postulate

9. AAS Congruence Theorem

12. AAS GUIDED PRACTICE for Examples 1 and 2
In the diagram at the right, what postulate or theorem can you use to prove that
RST VUT
? Explain.
SOLUTION
STATEMENTS
REASONS
Given
S U
Given
RS UV
The vertical angles are congruent
RTS UTV
RST VUT
AAS

13. GUIDED PRACTICE for Examples 1 and 2
Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof.
GIVEN
ABC
PROVE 3
= 180° 1 m 2 +
STATEMENTS
REASONS 1.
Draw BD parallel to AC .
Parallel Postulate 2.
Angle Addition Postulate and definition of straight angle 4 m 2 5 +
= 180° 3.
Alternate Interior Angles Theorem 1 4 , 3 5 4.
Definition of congruent angles 1 m = 4 3 5 , 5.
Substitution Property of Equality 1 m 2 3 +
= 180°

14. 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