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 triangle BPQ is isosceles.  

 

Answer 1

Given that P is a point on the bisector of an angle ABC, and PQ|| AB.
We have to prove that ΔBPQis isosceles 
Since,  

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

Now, 

PQllAB 

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

[alternative angles] 

From (1) and (2), we get 

∠BPQ=∠PBC(or)∠BPQ=∠PBQ 

Now, 
In , ΔBPQ 

∠BPQ=∠PBQ 

⇒ΔBPQ  is an isosceles triangle.
∴Hence proved