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Proof

# Parallelogram Opposite Angles Theorem

In a parallelogram, the opposite angles are congruent.

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

This can be proven with the ASA Congruence Theorem.

In the parallelogram $PQRS,$ 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 angles $\angle PQS\cong\angle QSR$ 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. $\triangle PQS \cong \triangle RSQ$ As the triangles are congruent, all their corresponding parts are congruent. Therefore, the angles $\angle SPQ$ and $\angle QRS$ are congruent as well.

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

The proof can be summarized in a flowchart proof.