Question

In the given figure, $O$ is the centre of the circle, $\angle ACB={54}^{\circ }$ and $BCE$ is a straight line. Find $x.$

• A. ${126}^{\circ }$
• B. ${54}^{\circ }$
• C. ${108}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

Answer: (C)

${108}^{\circ }$

$O$ is the center of the circle. $\angle ACB={54}^{\circ }$
To find, $\angle AOE$
$\angle ACE={180}^{\circ }-{54}^{\circ }={126}^{\circ }$ ($BCE$ is a straight line)
$\Rightarrow$ $\angle ADE=\angle ACE={126}^{\circ }$ (Angles in the same segement are equal)
Reflex $\angle AOE=2\times \angle ADE=2\times {126}^{\circ }={252}^{\circ }$ (Angle at the centre$=2\times$Angle at any other pt. on remaining part of the circle)
$\therefore$ $\angle AOE=360-$ reflex $\angle AOE$
$x={360}^{\circ }-{252}^{\circ }={108}^{\circ }$