A pair of angles whose sides form two lines is called vertical angles. In the figure below, angles 1 and 3 are vertical angles since their sides form lines l and m. Similarly, angles 2 and 4 are vertical angles for the same reason.
Vertical angles are congruent and it is easy to prove. We just use the fact that a linear pair of angles are supplementary; that is their measures add up to $latex 180^\circ$. In the figure above, to prove that vertical angles are congruent, we have to show that $latex \angle 1$ and $latex \angle 3$ are congruent or $latex \angle 2$ and $latex \angle 4$ are congruent.
Vertical angles are congruent.
We show that $latex \angle 1 \cong \angle 3$.
$latex m \angle 1 + m \angle 2 = 180^\circ$ ** Linear pair of angles are supplementary.
$latex m \angle 2 + m\angle 3 = 180^\circ$ **Linear pair of angles are supplementary.
$latex m \angle 1 + m \angle 2 = m \angle 2 + m \angle 3$ ** Substitution property of equality; that is $latex 180^\circ = 180^\circ$.
Substracting $latex \angle 2$ from both sides, we have
$latex m \angle 1 = m \angle 3$.
Therefore, vertical angles are congruent. $latex \blacksquare$
As an exercise, show that $latex m \angle 2 = m \angle 4$.