Proving Triangles Congruent – AAS and ASA Section 4.5 Proving Triangles Congruent – AAS and ASA

An included side is the side located between two consecutive angles of a polygon. In ∆ABC, is the included side between ÐA and ÐC.

4. Alternate Interior Angles Example 1: Write a two-column proof. Given: L is the midpoint of Prove: ∆WRL @ ∆EDL Statements Reasons 1. L is the midpoint of 1. Given 2. Given 3. Midpoint Theorem 4. ÐW @ ÐE 4. Alternate Interior Angles 5. ÐWLR @ ÐELD 5. Vertical Angles Theorem 6. ∆WRL @ ∆EDL 6. ASA

Example 2: Write a paragraph proof. Given: ÐNKL @ ÐNJM Statements Reasons 1. ÐNKL @ ÐNJM 1. Given 2. 2. Given 3. ÐN @ ÐN 3. Reflexive Property 4. ∆NKL @ ∆NJM 4. AAS 5. 5. CPCTC

Example 3: MANUFACTURING Barbara designs a paper template for a certain envelope. She designs the top and bottom flaps to be isosceles triangles that have congruent bases and base angles. If EV = 8 cm and the height of the isosceles triangle is 3 cm, find PO. In order to determine the length of PO, we must first prove that the two triangles are congruent. ___ ΔENV  ΔPOL by ASA. NV  EN by definition of isosceles triangle ___ EN  PO by CPCTC. ___ NV  PO by the Transitive Property of Congruence. ___ Since the height is 3 centimeters, we can use the Pythagorean theorem to calculate PO. The altitude of the triangle connects to the midpoint of the base, so each half is 4. Therefore, the measure of PO is 5 centimeters. PO = 5 cm

Concept