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

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

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

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

6. 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

7. Concept