Question

prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Answer 1

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.