Question

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Then which of the following options is correct?

• A. $\angle MOC=\angle BAC$
• B. $\angle MOC=\angle ABC$
• C. $\angle MOC=\angle MBC$
• D. $\angle MOC=\angle OCB$

Answer 1

The correct option is B $\angle MOC=\angle ABC$

Bisectors of the angles B and C of an isosceles triangle $\Delta ABC$ with AB = AC intersect each other at O. BO is produced to a point M.


In $\Delta ABC$, we have
$AB=AC$ (given)
$\therefore \angle ABC=\angle ACB$ [ $\because$ Angles opposite to equal sides of a triangle are equal.]
$\Rightarrow \frac{1}{2}\angle ABC=\frac{1}{2}\angle ACB$
i.e., $\angle 1=\angle 2$
$\angle MOC=\angle 1+\angle 2$ [ $\because$ Exterior angle of a triangle is equal to the sum of interior opposite angles]
$\therefore \angle MOC=2\angle 1$ $[\because \angle 1=\angle 2]$
Hence, $\angle MOC=\angle ABC$.
[ $\because$ $\angle 1=\frac{1}{2}\angle ABC$ ]