Question

If the bisector of the base angles of a triangle enclosed an angle of ${135}^o$, prove that the triangle is a right triangle.

Answer 1

In $△ABC,BO$ and $CO$ are bisectors of angle $B$ and $C$ respectively.

Now, in $△ABC$,

$\angle A+\angle B+\angle C=180°$

$\angle B+\angle C=180°-\angle A.....\left(1\right)$

Now, in $△BOC$,

$\angle BOC+\frac{1}{2}\angle B+\frac{1}{2}\angle C=180°$

$\frac{1}{2}(\angle B+\angle C)=180°-135°(\because \angle BOC=135°)$

$\frac{1}{2}(180°-\angle A)=45°$

$\Rightarrow 180°-\angle A=90°$

$\Rightarrow \angle A=90°$

$\Rightarrow △ABC$ is a right triangle.

Hence proved.