Question

relating congruence theorems
explain how the angle - angle - side congruence theorem is an extension of the angle - side - angle congruence theorem. be sure to discuss the information you would need for each theorem.

Explanation:

The Angle - Side - Angle (ASA) theorem requires two angles and the included side of one triangle to be congruent to the corresponding parts of another triangle. The Angle - Angle - Side (AAS) theorem has two angles and a non - included side. But since the sum of angles in a triangle is \(180^{\circ}\), if two angles are known (in AAS), the third angle is determined. So, AAS effectively uses the ASA concept (by getting the third angle) with a non - included side instead of the included side.

Answer:

The Angle - Angle - Side (AAS) theorem is an extension of the Angle - Side - Angle (ASA) theorem. In ASA, we have two angles and the included side. In AAS, we have two angles and a non - included side. But because the sum of angles in a triangle is \(180^{\circ}\), when two angles are given (in AAS), the third angle is known. So, AAS can be thought of as using the ASA principle (since we can find the third angle) but with a non - included side. For ASA, we need two angles and the side between them. For AAS, we need two angles and a side not between them (but we can find the third angle using the angle sum property of a triangle).