Question

If two straight lines intersect each other prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

Answer 1

Given: AB and CD intersect each other at O. OE is the bisector of $\angle AOD$ and EO is produced to F.
To prove OF is the bisector of $\angle BOC$
Proof: $\because \angle AOD=\angle BOC$
(Vertically opposite angles)
$\because OE$ is the bisector of $\angle AOD$
$\therefore \angle 1=\angle 2$
$\because$ AB and EF intersect each other at O
$\therefore \angle 1=\angle 4$ (Vertically opposite angles)
Similarly, CD and EF intersect each other at O
$\therefore \angle 2=\angle 3$
But $\angle 1=\angle 2$
$\therefore \angle 3=\angle 4$
$\therefore$ OF is the bisector of $\angle BOC$