# Proof That The Opposite Sides of a Parallelogram Are Congruent

A parallelogram is a quadrilateral whose opposite sides are parallel. In the figure below, $PQRS$ is a parallelogram. $PQ$ is parallel to $RS$ and $PS$ is parallel to $QR$. In this post, aside from being parallel, we will also prove that the opposite sides of a parallelogram are congruent.

Theorem

The opposite sides of a parallelogram are congruent.

Given

Parallelogram $PQRS$.

What To Show $\overline{PQ} \cong \overline{SR}$ and $\overline{PS} \cong \overline{QR}$.

Proof

Draw diagonal $\overline{QS}$. Since $\overline{QS}$ is a transversal to parallel lines $\overline{PQ}$ and $\overline {SR}$, $\angle PSQ \cong \angle SQR$ because they are alternate interior angles. (A)

Now, segment is congruent to itself (reflexive proeprty) so, $\overline{QS} \cong \overline{QS}$. (S)

Since $\overline{PS}$  is parallel to $\overline {QR}$ $\angle PQS \cong \angle QSR$ because they are alternate interior angles. (A)

By ASA congruence, $\triangle PQS \cong \triangle QSR$.

Since corresponding parts of congruent triangles are congruent, $\overline{PQ} \cong \overline{SR}$ and $\overline{PS} \cong \overline{QR}$.