Question

Prove that if one pair of opposite sides of a quadrilateral are equal and parallel, then it is a parallelogram.

Answer 1

Consider parallelogram quadrilateral PQRS with PQ = SR and PQ || SR.

In ΔPQS and ΔRSQ:

PQ = SR (Given)

∠PQS = ∠RSQ (As PQ || SR)

SQ = SQ (Common side)

As the two sides and the included angle are equal to the two sides and the included angle of the other triangle, ΔPQS ≅ ΔRSQ.

Corresponding parts of congruent triangles are congruent.

∴ PS = QR

∠PSQ = ∠RQS

Using converse of alternate angle axiom, we have:

PS || RQ

Thus, both the pairs of quadrilateral PQRS are equal and parallel.

Hence, PQRS is a parallelogram.