Question

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

Answer 1

Hence it is proved that the triangle BPQ is isosceles.

Given,

P is a point on the bisector of an angle ∠ABC

line through P parallel to AB meets BC at Q

we have,

∠ ABP = ∠ PBC ..........(1) (BP is a bisector of ∠ ABC)

PQ ║ AB  (given)

⇒ ∠ BPQ = ∠ ABP ............(2) (alternate angles)

from (1) and (2), we have,

∠ PBC = ∠ BPQ

⇒ ∠ PBQ = ∠ BPQ

In Δ BPQ,

∠ PBQ = ∠ BPQ

⇒ Δ BPQ is an isosceles triangle.

Hence it is proved that the triangle BPQ is isosceles.