Vertical Angles Theorem

To prove that vertical angles are congruent, it will be shown that $\angle 1$ and $\angle 3,$ in the figure below, have the same measure.

Notice that $\angle 1$ and $\angle 2$ are supplementary because they form a straight angle. Thus, since $m\angle 1+m\angle 2=180^{\circ },$ $m\angle 2$ can be expressed as follows. In the same way, since $\angle 2$ and $\angle 3$ are supplementary, $m\angle 2$ can be expressed in another way. By transitivity, the equations for $m\angle 2$ can be set equal to each other. By simplifying, the following equality yields. Thus, $\angle 1\cong \angle 3.$ Therefore, vertical angles are congruent.
This reasoning can be summarized in a flowchart proof.