Question

If $m\angle \text{AOC}={135}^{\circ }$, then $x=?$

• A. ${22.5}^{\circ }$
• B. ${45}^{\circ }$
• C. ${35}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is A ${22.5}^{\circ }$

Given,
$m\angle \text{AOC}={135}^{\circ }$

Note: Angle subtended by the arc $\text{BC}$ at the center i.e. $\angle \text{COB}$ is double of the angle subtended by the arc $\text{BC}$ at the point D
$\Rightarrow m\angle \text{COB}=2x$

Where, $m\angle \text{COB}+m\angle \text{AOC}={180}^{\circ }$
$m\angle \text{COB}={180}^{\circ }-{135}^{\circ }$
$\because m\angle \text{AOB is the the angle subtended by the diameter AB of a circle is}{180}^{\circ }$

$m\angle \text{COB}={45}^{\circ }$
Hence,
$2x=m\angle \text{COB}$
$2x={45}^{\circ }$
$x={22.5}^{\circ }$