Question

In the given figure, if $\angle AOC={120}^{\circ }$, then $\angle ABC$ = _______.

• A. ${50}^{\circ }$
• B. ${120}^{\circ }$
• C. ${60}^{\circ }$
• D. ${100}^{\circ }$

Answer 1

The correct option is B ${120}^{\circ }$

Given that $\angle AOC={120}^{\circ }.$

Consider the arc ABC. Take a point on the circumference of the circle outisde this arc. Join AP and PC.

Arc ABC subtends $\angle AOC$ at centre and $\angle APC$ on the remaining part of the circle at a point P.
Since the angle subtended by an arc at the centre is double the angle subtended by it on any remaining part of the circle,
$\angle APC=\frac{1}{2}\angle AOC=\frac{1}{2}\times {120}^{\circ }={60}^{\circ }$
Consider the cyclic quadrilateral APCB. Since opposite angles are supplementary,
$\angle ABC+\angle APC={180}^{\circ }.$
$⟹\angle ABC={180}^{\circ }-\angle APC={180}^{\circ }-{60}^{\circ }={120}^{\circ }$