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{{ printedBook.courseTrack.name }} {{ printedBook.name }} In a parallelogram, the opposite angles are congruent.

For the parallelogram $PQRS,$ the following statement holds true.

$∠Q≅∠Sand∠P≅∠R$

This theorem can be proved by using congruent triangles. Consider the parallelogram $PQRS$ and its diagonal $PR.$

Opposite sides of a parallelogram are parallel. Therefore, by the Alternate Interior Angles Theorem it can be stated that $∠QPR≅∠SRP$ and $∠QRP≅∠SPR.$ Furthermore, by the Reflexive Property of Congruence, $PR$ is congruent to itself.

Two angles of $△PQR$ and their included side are congruent to two angles of $△RSP$ and their included side. By the Angle-Side-Angle Congruence Theorem, $△PQR$ and $△RSP$ are congruent triangles. $△PQR≅△RSP $ Since corresponding parts of congruent figures are congruent, $∠Q$ and $∠S$ are congruent angles.

By drawing the diagonal $QS $ and using a similar procedure, it can be shown that $∠P$ and $∠R$ are also congruent angles.

$∠Q≅∠Sand∠P≅∠R$

In the parallelogram $PQRS,$ the diagonal $QS $ is drawn.

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

Then, by the Alternate Interior Angles Theorem, the angles $∠PQS≅∠QSR$ and $∠PSQ≅∠RQS.$

Since the triangles have two congruent angles and share one side, $QS ,$ the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. $△PQS≅△RSQ$ As the triangles are congruent, all their corresponding parts are congruent. Therefore, the angles $∠SPQ$ and $∠QRS$ are congruent as well.

Since the angles $∠PQR$ and $∠QRS$ are the sum of the same two congruent angles, they are congruent.

The proof can be summarized in a flowchart proof.