Question

In the given figure, $BOC$ is a diameter of a circle and $AB=AC.$ Then, $\angle ABC=?$

(a) ${30}^{\circ }$

(b) ${45}^{\circ }$

(c) ${60}^{\circ }$

(d) ${90}^{\circ }$

Answer 1

The correct option is (b): ${45}^{\circ }$

Given: $AB=AC$ and $BOC$ is the diameter of the circle.

$\Rightarrow \angle ABC=\angle ACB={45}^{\circ }\dots \dots \left(i\right)$

Since an angle in a semicircle is a right angle.

$\therefore \angle BAC={90}^{\circ }$

And, $\angle ABC+\angle ACB+\angle BAC={180}^{\circ }$ [Angle Sum property]

$\Rightarrow 2\angle ABC+{90}^{\circ }={180}^{\circ }$ [Using $eq.\left(i\right)$]

$\Rightarrow \angle ABC={45}^{\circ }$