Prove that the bisectors of pair of Vertically opposite angles are on the same straight line..

(please i want the answer by the expert..)

Hi!
Here is the answer to your question.
 
 
AB and CD are straight lines intersecting at O. OX the bisector of angles ∠AOC and OY is the OY is the bisector of ∠BOD.
 
OY is the bisector of ∠BOD.
∴ ∠1 = ∠6  … (1)
OX is the bisector of ∠AOC.
∴ ∠3 = ∠4  … (2)
∠2 = ∠5  … (3)  (Vertically opposite angles)
We know that, the sum of the angles formed at a point is 360°.
∴ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°
⇒ ∠1 + ∠2 + ∠3 + ∠3 + ∠2 + ∠1 = 360°  (Using (1), (2) and (3))
⇒ 2∠1 + 2∠2 + 2∠3 = 360°
⇒ 2(∠1 + ∠2 + ∠3) = 360°
⇒ ∠DOY + ∠AOD + ∠AOX = 180°
⇒ ∠XOY = 180°
∴ The bisectors of pair of vertically opposite angles are on the same straight line.
 
Cheers!

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