5.6 Vocabulary Included Side ASA Triangle Congruence AAS Triangle Congruence

An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

In Lesson 5-2, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you shortcuts for determining two triangles congruent. The first shortcuts were SAS and SSS (and we proved HL). A third “rigidity” shortcut is ASA which states that if two angles and an included side are given, the triangle can have only one shape.

Thrm 5-10

Example 1: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent as marked. Explain. Two congruent angle pairs are given, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent. What if you added additional information that you “know” to be true? Third angles, reflexive side?

Check It Out! Example 2 Mark what you know to be true. Determine if you can use ASA to prove NKL  LMN. Can you use SAS? Explain. By the Alternate Interior Angles Theorem. KLN  MNL. NL  NL by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied. However, SAS works!

You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). Thrm 5-11

Example 3: Using AAS to Prove Triangles Congruent Use AAS to prove the triangles congruent. Given: X  V, YZW  YWZ, XY  VY Prove:  XYZ  VYW

Check It Out! Example 4 Use AAS to prove the triangles congruent. Given: JL bisects KLM, K  M Prove: JKL  JML

Example 5: Given: FAB  GED, ABC   EDC, AC  EC Prove: ABC  EDC