The given figure shows a △ABC in which AB = AC and BP = CQ.

Prove that :

(i) △ABQ ≡ △ACP.

(ii) △APQ is isosceles.

(i) Given: BP = CQ and AB = AC

To Prove: △ABQ ≡ △ACP.

Proof: BP = CQ

⇒ BP + PQ = CQ + PQ

⇒ BQ = CP

In △ABQ and △ACP,

AB = AC (Given)

BQ = CP (Proved above)

∠ABQ = ∠ACP (Isosceles triangle property)

By SAS congruency criterion,

Hence, △ABQ ≅ △ACP.

(ii) To prove: △APQ is isosceles.

Proof: From (i), △ABQ ≅ △ACP

By corresponding parts of congruent triangles,

AP = AQ

Thus, △APQ has two equal sides AP = AQ, making it an isosceles triangle.

Hence, △APQ is an isosceles triangle.