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 $latex m \angle B + m \angle C = 180$.

trapezoid angle theorem

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.

trapezoid angle theorem 3

 

$latex m \angle 1 + m \angle 2 = 180$ 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.

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