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. Which of the following is correct?

• A. $\angle MOC=\angle BAC$
  
• B. $\angle MOC=\angle ABC$
  
• C. $\angle MOC=\angle MBC$
  
• D. None of these

Answer 1

The correct option is B

$\angle MOC=\angle ABC$

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


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