Question

O is the centre of the circle. $\angle$OAB = 20°, $\angle$OCB = 55°. Find $\angle$BOC and $\angle$AOC.

Answer 1

Given : $\angle$OAB = 20°, $\angle$OCB = $\angle$55°
To find : $\angle$BOC = ? and $\angle$AOC = ?
Proof :
Join AC, OA = OB [radius of circle]
$\Rightarrow$ $\angle$OAB = $\angle$OBA = 20º
Now, As OC = OB [radii of circle]
$\therefore \angle$OCB = $\angle$OBC [Angle opposite to equal sides are equal]
$\angle$OBC = $\angle$OBA + $\angle$ABC
$\angle$ABC = $\angle$OBC – $\angle$OBA = 55º – 20º = 35º
and $\angle$AOC = 2$\angle$ABC = 2(35º) = 70º [Angle subtended by an arc at centre is twice the angle subtended by it on any other part on the circle]
Now, In $△$AOB
$\angle$AOB + $\angle$OAB + $\angle$OBA = 180º
$\angle$AOC + $\angle$BOC + $\angle$OAB + $\angle$OBA = 180º
70º + $\angle$BOC + 20º + 20º = 180º
$\Rightarrow \angle$BOC = 70º.