Question

In the adjoining figure, $ABC$ is a triangle in which $AD$ is the bisector of $\angle A$. If $AD\perp BC$, show that $\Delta ABC$ is isosceles.

Answer 1

Given : $AD$ bisects $\angle A$ and is perpendicular to $BC$.

To prove : $\Delta ABC$ is isoceles

Proof :

In $\Delta ABD$ & $\Delta ADC$

$\angle BAD=\angle DAC$ ($\because$ $D$ bisects $\angle A$)

$\angle ADB=\angle ADC$

$\Rightarrow \angle ABD=\angle ACD$

($\because$ if two angles in a triangle are equal to two angles in another triangle, then the third angle of both the triangles must also be equal)

$\Rightarrow \angle B=\angle C$

$\therefore$ $AB=AC$ (if two angles are equal then the side opposite to the angles will also be equal)

Hence proved.