Question

In a $\Delta ABC,\text{it is given that}AB=AC\text{and the bisectors of}\angle Band\angle C\text{intersect at O. If M is a point on BO produced, prove that}\angle MOC=\angle ABC.$

Answer 1

Given : In $\Delta ABC,AB=AC\text{the bisectors of}\angle Band\angle C\text{intersect at O.M is any point on BO produced.}$

To prove : $\angle MOC=\angle ABC$

Proof : In $\Delta ABC,AB=BC$

$\angle C=\angle B$

OB and OC are the bisectors of $\angle Band\angle C$

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

Now in $\angle OBC,$

Ext. $\angle MOC=\text{Interior opposite angles}$

$\angle 1+\angle 2$

=$\angle 1+\angle 1=2\angle 1=\angle B$

Hence $\angle MOC=\angle ABC$