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.

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

trapezoid angle theorem 3

 

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.

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