Question

In the given diagram O is the center of the circle and CD is a tangent. $\angle$CAB and $\angle$ACD are supplementary to each other $\angle$OAC = ${30}^{\circ }$. Find the value of $\angle$OCB:

• A. 30°
• B. 20°
• C. 60°
• D. None of these

Answer 1

The correct option is A

30°

OA = OC

$\angle$OAC = $\angle$OCA = ${30}^{\circ }$

Therefore $\angle$AOC = ${180}^{\circ }$ - $\angle$OAC - $\angle$OCA = ${180}^{\circ }$ – ${30}^{\circ }$ – ${30}^{\circ }$ = ${120}^{\circ }$

$\angle$ ABC = $\frac{1}{2}$ $\angle$AOC = $\frac{1}{2}$ $\angle$${120}^{\circ }$ = ${60}^{\circ }$

$\angle$CAB and $\angle$ACD are supplementary $\Rightarrow$ AB is parallel to CD
$\angle$ABC = $\angle$BCD (Alternate interior angle)

∠BCD = ${60}^{\circ }$

∠OCB = ${30}^{\circ }$