Question

OP bisects AOC , OQ bisects COB and OP OQ . Show that A, O, B are collinear.

Answer 1

Answer:

Given,

Since OP is the bisector of ∠AOC 

∴∠AOP = ∠COP                                          [1]

Since OQ is the bisector of ∠BOC

∴∠BOQ = ∠COQ                                          [2]

Now consider ∠AOB

= ∠AOP + ∠COP + ∠COQ + ∠BOQ

= ∠COP + ∠COP + ∠COQ + ∠COQ

from[1] and [2]

= 2[∠COP + ∠COQ]

= 2∠POQ

= 2(90°) 

∴ OP ⊥ OQ=180°

Thus we can say that A,O and B lies on the same line   {linear pair}.

And since, A, O and B lie on the same line, they are collinear.

Hence, proved.