Question

In the given figure, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°; find:

1. ∠ACB, 
2. ∠OBC, 
3. ∠OAB, 
4. ∠CBA.

Answer 1

Here, ${\displaystyle \angle ACB=\frac{1}{2} \text{Reflex} (\angle AOB)}$

= ${\displaystyle \frac{1}{2}({360}^{\circ }-{140}^{\circ })}$

= 110°

(Angle at the centre is double the angle at the circumference subtended by the same chord)

Now, OA = OB (Radii of same circle)

∴ ∠OBA = ∠OAB

= ${\displaystyle \frac{{180}^{\circ }-{140}^{\circ }}{2}}$

= 20°

∴  ∠CAB = 50° – 20° = 30°

ΔCAB,

∠CBA = 180° – 110° – 30° = 40°

∴  ∠OBC = ∠CBA + ∠OBA

= 40° + 20°

= 60°