Question

If two straight lines intersect each other, then prove that the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.

Answer 1

Let AB and CD be the two lines intersecting at a point O and let ray OE bisect $\angle AOC$. Now, draw a ray OF in the opposite direction of OE, such that EOF is a straight line.
Let $\angle COE=1,\angle AOE=2,\angle BOF=3\text{and}\angle DOF=4$.
We know that vertically-opposite angles are equal.
$\therefore \angle 1=\angle 4\text{and}\angle 2=\angle 3$
But, $\angle 1=\angle 2$ [Since OE bisects $\angle AOC$ ]
$\therefore \angle 4=\angle 3$
Hence, the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.