In a $\triangle ABC$, it is given that $AB = AC$ and the bisectors of $\angle B$ and $\angle C$ intersect at $O$. If $M$ is a point on $BO$ produced, prove that $\angle MOC = \angle ABC$.
Given:
In a $\triangle ABC$, it is given that $AB = AC$ and the bisectors of $\angle B$ and $\angle C$ intersect at $O$.
$M$ is a point on $BO$ produced.
To do:
We have to prove that $\angle MOC = \angle ABC$.
Solution:
In $\triangle ABC, AB = BC$
This implies,
$\angle C = \angle B$
$OB$ and $OC$ are the bisectors of $\angle B$ and $\angle C$
This implies,
$\angle 1 = \angle 2 = \frac{1}{2}\angle B$
In $\triangle OBC$,
$\angle MOC = \angle 1 + \angle 2$
$ = \angle 1 + \angle 1$
$= 2\angle 1$
$= \angle B$
Hence, $\angle MOC = \angle ABC$.
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