# A Proof of The Trapezoid Angle Theorem

A trapezoid is a quadrilateral whose two sides are parallel. In the figure below, $ABCD$ is a trapezoid and $AB$ is parallel to $CD$. In this post, we are going to prove that in a trapezoid, the consecutive angles between a pair of parallel sides are complementary. That is, we are going to prove in the figure above that $m \angle A = m \angle D = 180$ degrees and $m \angle B + m \angle C = 180$.

The Trapezoid Angle Theorem.

In a trapezoid, the consecutive angles between a pair of parallel sides are complementary.

Proof

Extend line segment $DA$ at $A$ and locate point $E$ on the extended part as shown above. Label the angles angles as 1 and 2.

$m \angle 1 + m \angle 2 = 180$ because they are a linear pair.

$m \angle 2 = m \angle D$ because they are corresponding angles of the parallel lines containing $AB$ and $CD$.

$m \angle 1 + m \angle D = 180$ by substitution.

$\angle 1$ and $\angle D$ are supplementary by definition of supplementary angles.