# A Proof of The Trapezoid Angle Theorem

A trapezoid is a quadrilateral whose two sides are parallel. In the figure below, \$latex ABCD\$ is a trapezoid and \$latex AB\$ is parallel to \$latex 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 \$latex m \angle A = m \angle D = 180\$ degrees and \$latex 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 \$latex DA\$ at \$latex A\$ and locate point \$latex E\$ on the extended part as shown above. Label the angles angles as 1 and 2. \$latex m \angle 1 + m \angle 2 = 180\$ because they are a linear pair.

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

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

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