Parallelogram Opposite Angles Theorem

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.