Question

On a semi circle with $AB$ as diameter, a point $C$ is taken so that $\angle CAB={30}^{\circ }$. Find $\angle ACB\text{and}\angle ABC.$

Answer 1

A semicircle with $AB$ as diameter.

$\angle CAB={30}^{\circ }$

$\angle ACB={90}^{\circ }$ (Angle in a semi circle)

But $\angle CAB+\angle ACB+\angle ABC={180}^{\circ }$

$\Rightarrow {30}^{\circ }+{90}^{\circ }+\angle ABC={180}^{\circ }$

$\Rightarrow {120}^{\circ }+\angle ABC={180}^{\circ }$

$\therefore \angle ABC={180}^{\circ }-{120}^{\circ }={60}^{\circ }$

Hence, $\angle ACB={90}^{\circ }$ and $\angle ABC={60}^{\circ }$