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

To do:

We have to prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

Solution:

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

$\angle AOC = \angle BOD$             (Vertically opposite angles)

$OE$ and $OF$ are the bisectors of $\angle AOC$ and $\angle BOD$ respectively.

$\angle 1 = \angle 2$ and $\angle 3 = \angle 4$

$\angle 1 = \angle 2 = \frac{1}{2} \angle AOC$

$\angle 3 = \angle 4 = \frac{1}{2} \angle BOD$

This implies,

$\angle 1 = \angle 2 = \angle 3 = \angle 4$

$AOB$ is a straight line.

This implies,

$\angle BOD + \angle AOD = 180^o$            (Linear pair)

$\angle 3 + \angle 4 + \angle AOD = 180^o$

$\angle 3 + \angle 1 + \angle AOD = 180^o$            ($\angle 1 = \angle 4$)

Therefore,

$EOF$ is a straight line.

Hence proved.

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Updated on: 10-Oct-2022

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