Converse Parallelogram Opposite Sides Theorem

This theorem can be proven by using congruent triangles. Consider the quadrilateral PQRS, whose opposite sides are congruent, and its diagonal PR. By the Reflexive Property of Congruence, this diagonal is congruent to itself.

Therefore, by the Side-Side-Side Congruence Theorem, △ PQR and △ RSP are congruent triangles. PQ≅ RS QR≅ SP PR≅ PR ⇒ △ PQR ≅ △ RSP Since corresponding parts of congruent figures are congruent, corresponding angles of △ PQR and △ RSP are congruent.

Finally, by the Converse of the Alternate Interior Angles Theorem, PQ is parallel to RS and QR is parallel to SP. Therefore, by the definition of a parallelogram, PQRS is a parallelogram.

This proves the theorem.