Angle-Side-Angle Congruence Theorem
If two angles and the included side in a triangle are congruent with corresponding parts in another triangle, then the triangles are congruent.
This can be proven
Consider the triangles and where
If either of these can be mapped onto the other using , then they are congruent. As is congruent with there is a rigid motion that maps one of these onto the other. This can be performed for one of the triangles, which leads to the two congruent sides overlapping.
The triangle can now be in the line If the image of falls onto the triangles will completely overlap. As the angles and are congruent, the ray will be mapped onto the ray Similarly, will be mapped onto
Thus, the intersection of and which is will be mapped onto the intersection of and which is
There is a rigid motion that maps onto Consequently, and are indeed congruent.