Question

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

Answer 1

Given: Lines AB and CD intersect each other at O. OE and OF are the bisectors of $\angle AOC$ and $\angle BOD$ respectively
To prove OE and OF are in the same line
Proof: $\because \angle AOC=\angle BOD$
(Vertically opposite angles)
$\because$ OE and OF are the bisectors of $\angle AOC$ and $\angle BOD$
$\therefore \angle 1=\angle 2and\angle 3=\angle 4\Rightarrow \angle 1=\angle 2=\frac{1}{2}\angle AOC$ and
$\angle 3=\angle 4=\frac{1}{2}\angle BOD\therefore \angle 1=\angle 2=\angle 3=\angle 4\because AOBisaline$
$\Rightarrow \angle 3+\angle 4+\angle AOD={180}^0$ (Linear pair)
$\Rightarrow \angle 3+\angle 4+\angle AOD={180}^0$
$\Rightarrow \angle 3+\angle 1+\angle AOD={180}^0$ $(\because \angle 1=\angle 4)$
$\therefore$ EOF is a straight line