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.

Question

P is a point on the bisector of an angle &#x2220; ABC . If the line through P parallel to AB meets BC at Q , prove that the triangle BPQ is isosceles.

Philoid · Accepted Answer

Given that P is the point on the bisector of an angle ∠ABC, and PQ ‖ AB

We have to prove that BPQ is isosceles

Since,

BP is the bisector of ∠ABC = ∠ABP = ∠PBC (i)

Now,

PQ ‖ AB

∠BPQ = ∠ABP (ii) [Alternate angles]

From (i) and (ii), we get

∠BPQ = ∠PBC

Or,

∠BPQ = ∠PBQ

Now, in

∠BPQ = ∠PBQ

is an isosceles triangle

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.