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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