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

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 180^\circ.  In the figure above, to prove that vertical angles are congruent,  we have to show that \angle 1 and \angle 3 are congruent or \angle 2 and \angle 4 are congruent.

Theorem

Vertical angles are congruent.

Proof

We show that \angle 1 \cong \angle 3.

m \angle 1 + m \angle 2 = 180^\circ ** Linear pair of angles are supplementary.

m \angle 2 + m\angle 3 = 180^\circ **Linear pair of angles are supplementary.

m \angle 1 + m \angle 2 = m \angle 2 + m \angle 3 ** Substitution property of equality; that is 180^\circ = 180^\circ.

Substracting \angle 2 from both sides, we have

m \angle 1 = m \angle 3.

Therefore, vertical angles are congruent. \blacksquare

As an exercise, show that m \angle 2 = m \angle 4.

 

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