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 . In the figure above, to prove that vertical angles are congruent, we have to show that and are congruent or and are congruent.
Vertical angles are congruent.
We show that .
** Linear pair of angles are supplementary.
**Linear pair of angles are supplementary.
** Substitution property of equality; that is .
Substracting from both sides, we have
Therefore, vertical angles are congruent.
As an exercise, show that .