4.5 Proving Δs are  : ASA and AAS

Objectives: Use the ASA Postulate to prove triangles congruent Use the AAS Theorem to prove triangles congruent

Postulate 4.3 (ASA): Angle-Side-Angle Congruence Postulate 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 triangles are congruent.

Theorem 4.5 (AAS): Angle-Angle-Side 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 triangles are congruent.

Proof of the Angle-Angle-Side (AAS) Congruence Theorem Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF D A B F C Paragraph Proof You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. E

Example 1: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 1: In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Example 2: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Example 2: In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Example 3: Given: AD║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC. Use the fact that AD ║EC to identify a pair of congruent angles.

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given If || lines, then alt. int. s are  Vertical Angles Theorem ASA Congruence Postulate

Assignment Geometry: Pg. 210 #9 – 14, 25 – 28 Pre-AP Geometry: Pg. 211 #10 – 18 evens, #25 - 28