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