In a parallelogram, the opposite angles are congruent.

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

$\angle Q\cong \angle S\text{and}\angle P\cong \angle R$

Proof

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

Opposite sides of a parallelogram are parallel. Therefore, by the Alternate Interior Angles Theorem it can be stated that $\angle QPR\cong \angle SRP$ and $\angle QRP\cong \angle SPR.$ Furthermore, by the Reflexive Property of Congruence, $\overline{PR}$ 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. $$\begin{array}{c}△PQR\cong △RSP\end{array}$$ Since corresponding parts of congruent figures are congruent, $\angle Q$ and $\angle S$ are congruent angles.

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

$\angle Q\cong \angle S\text{and}\angle P\cong \angle R$

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. $$△PQS\cong △RSQ$$ 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.