explain how the angle - angle - side congruence theorem is an extension of the angle - side - angle…

Brief Explanations:

The angle - side - angle (ASA) congruence theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. The angle - angle - side (AAS) congruence theorem says that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent. Since the sum of the interior angles of a triangle is 180 degrees, if two angles of one triangle are congruent to two angles of another triangle, the third angles must also be congruent. So, in AAS, once we have two pairs of congruent angles, the third pair is also congruent. Then, with a non - included side congruent, we can use ASA (by considering the third congruent angle and the given side) to prove triangle congruence. Thus, AAS is an extension of ASA. For ASA, we need two angles and the side between them. For AAS, we need two angles and a non - included side.

Answer:

The angle - angle - side (AAS) congruence theorem is an extension of the angle - side - angle (ASA) congruence theorem because when we have two pairs of congruent angles in AAS, the third pair of angles is also congruent due to the angle - sum property of triangles. Then, with a non - included side congruent, we can use ASA to prove triangle congruence. For ASA, we need two angles and the included side. For AAS, we need two angles and a non - included side.