Recall that if the measure of two angles add to 180, then these angles are called supplementary. Let's focus on two angles like this. We know that AB is a transversal to AD and BC, and angles ∠ A and ∠ B are consecutive interior angles.
Since angles ∠ A and ∠ B are supplementary, the Consecutive Interior Angles Converse Theorem guarantees that AD and BC are parallel. We can also look at a different pair of angles.
Angles ∠ A and ∠ D are also supplementary consecutive interior angles, so segments AB and DC are also parallel. Since the opposite sides of ABCD are parallel, by definition, it is a parallelogram. We can summarize the argument above in a paragraph proof.
Completed Proof
2
&Given:&&Both pairs of opposite angles
& &&of a quadrilateral are congruent.
&Prove:&&The quadrilateral is a parallelogram.
Proof.
Since opposite angles have the same measure and the sum of the angle measures in a quadrilateral is 360, the sum of the measures of any two consecutive angles is 180. Applying the Consecutive Interior Angles Converse Theorem for the supplementary consecutive interior angles ∠ A and ∠ B, we get that segments AD and BC are parallel. Using supplementary consecutive interior angles ∠ A and ∠ D, we also get that segments AB and DC are parallel. Since opposite sides are parallel, by definition, ABCD is a parallelogram.
