Lines AB and CD intersect at point O, such that
∠AOC = ∠BOD (vertically angles) …(1)
Also OP is the bisector of AOC and OQ is the bisector of BOD
To Prove: POQ is a straight line.
OP is the bisector of ∠AOC:
∠AOP = ∠COP …(2)
OQ is the bisector of ∠BOD:
∠BOQ = ∠QOD …(3)
Now,
Sum of the angles around a point is 360°.
∠AOC + ∠BOD + ∠AOP + ∠COP + ∠BOQ + ∠QOD = 360°
∠BOQ + ∠QOD + ∠DOA + ∠AOP + ∠POC + ∠COB = 360°
2∠QOD + 2∠DOA + 2∠AOP = 360° (Using (1), (2) and (3))
∠QOD + ∠DOA + ∠AOP = 180°
POQ = 180°
Which shows that, the bisectors of pair of vertically opposite angles are on the same straight line.
Hence Proved.
