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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.