Proof

Parallelogram Opposite Sides Theorem

The opposite sides in a parallelogram are congruent.

That is, if $PQRS$ is a parallelogram, then $\overline{PQ}\cong\overline{SR}\quad\text{and}\quad\overline{QR}\cong\overline{PS}.$

This can be proven using the ASA Congruence Theorem.

In the parallelogram $PQRS,$ the diagonal $\overline{QS}$ is drawn.

According to the definition of a parallelogram, the opposite sides are parallel, $PQ\parallel SR$ and $QR\parallel PS.$

Then, by the Alternate Interior Angles Theorem, the following angles are congruent: $\angle PQS\cong\angle QSR \text{ and }\angle PSQ\cong\angle RQS.$

Since the triangles have two congruent angles and share one side, $\overline{QS},$ the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. $\triangle PQS \cong \triangle RSQ$ In congruent triangles, all corresponding parts are congruent. Therefore, the opposite sides in the parallelogram are congruent as well.

This can be summarized by a two-column proof.

Statement	Reason
$PQRS$ is a parallelogram.	Given
Draw the diagonal $\overline{QS}.$	Diagonal to create triangles.
$PQ\parallel SR,$ $QR\parallel PS$	Parallel lines
$\angle PQS\cong\angle QSR,$ $\angle PSQ\cong\angle RQS$	Alternative Angle Theorem
$\triangle PQS \cong \triangle RSQ$	ASA Congruence Theorem
$PQ\cong SR,$ $QR\cong PS$	Parallel lines in a parallelogram are congruent.