Question

prove that a pair of vertically opposite angles are equal.

Answer 1

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°

∠1+∠2+∠3=360°÷2=180°

⇒ ∠DOY + ∠AOD + ∠AOX = 180°
⇒ ∠XOY = 180°
∴ The bisectors of pair of vertically opposite angles are on the same straight line.