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What will be the value of $∠ C B D$ , if in the given figure, AB is the diameter of a circle with center O and $O D ∥ C B$ with $∠ B C D = 120 ∘$ ?

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answer is A.

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Detailed Solution

It is given that AB is the diameter of a circle with center O and

O D ∥ C B

with

∠ B C D = 120 ∘

We have to find the value of

∠ C B D

.
We know that ABCD is a cyclic quadrilateral, so the opposite angles will be supplementary.

∠ B C D + ∠ B A D = 180 ∘ ⇒ 120 ∘ + ∠ B A D = 180 ∘ ⇒ ∠ B A D = 180 ∘ − 120 ∘ = 60 ∘

∠ B D A = 90 ∘

[Angle in a semicircle]
In ∆ABD,

∠ B D A + ∠ B A D + ∠ A B D = 180 ∘ ⇒ 90 ∘ + 60 ∘ + ∠ A B D = 180 ∘ ⇒ ∠ A B D = 180 ∘ − 90 ∘ − 60 ∘ = 30 ∘

O D = O A ⇒ ∠ O D B = 90 ∘ − ∠ O D A ⇒ ∠ O D B = 90 ∘ − 60 ∘ = 30 ∘

Since OD||BC, alternate angles are equal.

⇒ ∠ C B D = ∠ O D B = 30 ∘

Therefore, the value of ∠CBD is 3

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Therefore, option 1 is correct.

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