In the given diagram, O is the centre of the circle and CD is a tangent. $∠ C A B$ and $∠ A C D$ are supplementary to each other. $∠ O A C = 30^{\circ}$ . Find the value of $∠ O C B$ .

30^{\circ}

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20^{\circ}

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60^{\circ}

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None of these

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Solution

The correct option is A $30^{\circ}$
$∠ O C D = 90^{\circ} ∠ O A C = ∠ O C A = 30^{\circ} ∠ A C D = ∠ A C O + ∠ O C D = 30^{\circ} + 90^{\circ} = 120^{\circ} ∴ ∠ B A C = 180^{\circ} - 120^{\circ} = 60^{\circ}$
$\Rightarrow ∠ B C D = 60^{\circ} (∠ B C D = ∠ B A C)$
$\Rightarrow ∠ O C B = ∠ O C D - ∠ B C D = 90^{\circ} - 60^{\circ} = 30^{\circ}$

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In the given diagram, O is the centre of the circle and CD is a tangent. ∠CAB and ∠ACD are supplementary to each other. ∠OAC=30∘. Find the value of ∠OCB.

The correct option is A 30∘∠OCD=90∘∠OAC=∠OCA=30∘∠ACD=∠ACO+∠OCD=30∘+90∘=120∘∴∠BAC=180∘−120∘=60∘ ⇒∠BCD=60∘ (∠BCD=∠BAC) ⇒∠OCB=∠OCD−∠BCD=90∘−60∘=30∘

In the given diagram, O is the centre of the circle and CD is a tangent. $∠ C A B$ and $∠ A C D$ are supplementary to each other. $∠ O A C = 30^{\circ}$ . Find the value of $∠ O C B$ .