Parallelogram Opposite Angles Theorem

In a parallelogram, the opposite angles are congruent.

Consider the parallelogram PQRS,PQRS, the following angles are then congruent: PQRRSP and SPQQRS. \angle PQR \cong \angle RSP\text{ and }\angle SPQ \cong \angle QRS.

This can be proven with 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 angles PQSQSR\angle PQS\cong\angle QSR and PSQRQS.\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 As the triangles are congruent, all their corresponding parts are congruent. Therefore, the angles SPQ\angle SPQ and QRS\angle QRS are congruent as well.

Since the angles PQR\angle PQR and QRS\angle QRS are the sum of the same two congruent angles, they are congruent.

The proof can be summarized in a flowchart proof.

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