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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.
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Solution
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
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