Question

$\text{P is a point on the bisector of}\angle ABC.\text{If the line through P, parallel to BA meets BC at Q, prove that}\Delta BPQ\text{is an isosceles triangle.}$

Answer 1

Given: BP is the bisector of ∠ ABC, and BA ∥ QP
To prove: Δ BPQ is an isosceles triangle
Proof:
∵∠1=∠2
Given, BP is the bisector of ∠ABC

And, ∠1=∠3
Alternate interior angles

∴∠2=∠3

So, PQ=BQ
In a triangle, sides opposite

But these are sides of ∆BPQ.
Hence, ∆BPQ is an isosceles triangle.