Prove Triangles Congruent by ASA & AAS Lesson 4.10 Use 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 Review Postulates we can use CAN’T USE

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.

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.

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.

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

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

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°

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