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In the figure, arc $ABC$ of the circle subtends angle of $130 °$ at the centre O. If AB is produced to D, then $∠ CBD$ will be:

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60 °

65 °

70 °

130 °

answer is B.

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Detailed Solution

It is given that arc

ABC

of the circle subtends angle of

130 °

at the centre O.
Question Image

From figure,

Reflex ∠ A O C + ∠ A O C = 360 ° ⇒ Reflex ∠ A O C + 130 ° = 360 ° ⇒ Reflex ∠ A O C = 230 °

As we know that the angle subtended by an arc at the center of the circle is double the angle subtended by it at any other point on the circle.
So,

Reflex ∠ A O C = 2 × ∠ A B C

⇒ ∠ A B C = 12 × Reflex ∠ A O C ⇒ ∠ A B C = 12 × 230 ° = 115 °

Now, we know the sum of all angles lying on a straight line is

180 ° .

⇒ ∠ C B D + ∠ A B C = 180 ° ⇒ ∠ C B D + 115 ° = 180 ° ⇒ ∠ C B D = 65 °

The value of

∠ C B D = 65 °

.
Hence, the correct option is 2.

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