The opposite sides in a parallelogram are congruent.

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

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. $$△PQS\cong △RSQ$$ 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.