Question

P is a point on the bisector of ∠ ABC. If the line through P, parallel to BA meets BC at Q, prove that △ BPQ is an isosceles triangle.

Answer 1

We know that AB || PQ and BP is a transversal

From the figure we know that ∠ ABP and ∠ BPQ are alternate angles

So we get

∠ ABP = ∠ BPQ ….. (1)

We also know that BP is the bisector of ∠ ABC

So we get

∠ ABP = ∠ PBC and ∠ ABP = ∠ PBQ …. (2)

By considering equation (1) and (2) we get

∠ BPQ = ∠ PBQ

We know that the sides opposite to equal angles are equal

PQ = BQ

Therefore, it is proved that △ BPQ is an isosceles triangle.