Two sides AB and BC and median AD of triangle ABC are respectively equal to sides PQ and QR and median PN of △PQR. Show that :

(i) △ABD ≡ △PQN.

(ii) △ABC ≡ △PQR.

(i) Given: △ ABC and △ PQR in which AB = PQ, BC = QR and AD = PN.

To prove: △ABD ≡ △PQN.

Proof: Since, AD and PN are median of triangles ABC and PQR respectively,

⇒ $\dfrac{1}{2}$ BC = $\dfrac{1}{2}$ QR (Median divides opposite sides in two equal parts)

So, BD = QN ……………..(1)

Now, in △ ABD and △ PQN,

AB = PQ (Given)

BD = QN (From equation (1))

AD = PN (Given)

By SSS congruency criterion,

Hence, △ABD ≅ △PQN.

(ii) To prove: △ABC ≡ △PQR.

Poof: From △ABD ≅ △PQN,

By corresponding parts of congruent triangles,

∠ABC = ∠PQR

Now, in △ ABC and △ PQR,

AB = PQ (Given)

∠ABC = ∠PQR

BC = QR (Given)

By SAS congruency criterion,

Hence, △ABC ≅ △PQR.