In the following figures, the sides AB and BC and the median AD of the triangle ABC are respectively equal to the sides PQ and QR and median PS of the triangle PQR. Prove that △ ABC and △ PQR are congruent.

Given,

BC = QR = x (let)

AD and PS are median of triangle ABC and PQR.

∴ BD = $\dfrac{1}{2}BC = \dfrac{x}{2}$

and

QS = $\dfrac{1}{2}QR = \dfrac{x}{2}$.

In △ ABD and △ PQS,

⇒ AB = PQ (Given)

⇒ AD = PS (Given)

⇒ BD = QS (Proved above)

∴ △ ABD ≅ △ PQS (By S.S.S. axiom).

We know that,

Corresponding parts of congruent triangle are equal.

∴ ∠B = ∠Q.

In △ ABC and △ PQR,

⇒ AB = PQ (Given)

⇒ BC = QR (Given)

⇒ ∠B = ∠Q (Proved above)

∴ △ ABC ≅ △ PQR (By S.A.S. axiom).

Hence, proved that △ ABC ≅ △ PQR.