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. ${30}^{\circ }$
• B. ${20}^{\circ }$
• C. ${60}^{\circ }$
• D. ${50}^{\circ }$

Answer 1

The correct option is A

${30}^{\circ }$

Since, OA = OC (Radii)

$\angle$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
[The angle at the centre is twice the angle at the circumference.]
= $\frac{1}{2}\times {120}^{\circ }$ = ${60}^{\circ }$

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

$\angle BCD={60}^{\circ }$

From the figure,
$\angle ACB=\angle ACO+\angle OCB\Rightarrow 60°=30°+\angle OCB\Rightarrow \angle OCB=60°-30°=30°$