Let's say I have $\triangle$ ABC and $\triangle$ OPQ. Let's say that $\angle$ A $\cong$ $\angle$ O. Let's also say that segment AC $\cong$ segment OQ. I want to prove that these two triangles are congruent. Is it valid to reposition these triangles such that:
- Point A = point O. This means that $\triangle$ OPQ effectively becomes $\triangle$ APQ. This also effectively means that B, A, and P are colinear on segment BP and C, A, and Q are colinear on segment CQ.
- $\angle$ A is a vertical angle with $\angle$ O.
I can then prove the congruence of these two triangles using properties of parallel lines and their angles.