Parallelogram Opposite Sides Theorem

The opposite sides in a parallelogram are congruent.

That is, if PQRSPQRS is a parallelogram, then PQSRandQRPS. \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,PQRS, the diagonal QS\overline{QS} is drawn.

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

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

Since the triangles have two congruent angles and share one side, QS,\overline{QS}, the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. PQSRSQ \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
PQRSPQRS is a parallelogram. Given
Draw the diagonal QS.\overline{QS}. Diagonal to create triangles.
PQSR,PQ\parallel SR, QRPSQR\parallel PS Parallel lines
PQSQSR,\angle PQS\cong\angle QSR, PSQRQS\angle PSQ\cong\angle RQS Alternative Angle Theorem
PQSRSQ\triangle PQS \cong \triangle RSQ ASA Congruence Theorem
PQSR,PQ\cong SR, QRPSQR\cong PS Parallel lines in a parallelogram are congruent.

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