# Proof that Vertical Angles Are Congruent

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.

Theorem

Vertical angles are congruent.

Proof

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\$.

## 1 thought on “Proof that Vertical Angles Are Congruent”

1. Stephanie on said:

Nice!