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.

Given:

Two straight lines intersect each other.

To do:

We have to prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

Solution:

Let $AB$ and $CD$ intersect each other at $O$.

$OE$ is the bisector of $\angle AOD$ and $EO$ is produced to $F$.

$\angle AOD = \angle BOC$              (Vertically opposite angles)

$OE$ is the bisector of $\angle AOD$

Therefore,

$\angle 1 = \angle 2$

$AB$ and $EF$ intersect each other at $O$

$\angle 1 = \angle 4$                  (Vertically opposite angles)

$CD$ and $EF$ intersect each other at $O$.

Therefore,

$\angle 2 = \angle 3$

This implies,

$\angle 3 = \angle 4$

$OF$ is the bisector of $\angle BOC$.

Therefore, the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

Hence proved.

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Updated on: 10-Oct-2022

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