Question

In $\Delta ABC,$ if bisectors of $\angle ABC$ and $\angle ACB$ intersect at O at angle of ${120}^{\circ }$ , then find the measure of $\angle A.$

Answer 1

In $\Delta ABC,$ bisectors of $\angle B$ and $\angle C$ intersect at O and $\angle BOC={120}^{\circ }$

But $\angle BOC={90}^{\circ }+\frac{1}{2}\angle A$

${90}^{\circ }+\frac{1}{2}\angle A={120}^{\circ }$

$\Rightarrow \frac{1}{2}\angle A={120}^{\circ }-{90}^{\circ }={30}^{\circ }$

$\therefore \angle A=2\times {30}^{\circ }={60}^{\circ }$