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$A B C D$ is a cyclic quadrilateral such that $A B$ is a diameter of the circle circumscribing it and if $∠ A D C = 140 °$ , then $∠ B A C$ is equal to:

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60 °

50 °

40 °

30 °

answer is B.

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Detailed Solution

Given,

A B C D

is a cyclic quadrilateral with

A B

as the diameter of the circle. Also,

∠ A D C = 140 °

We have to find:

∠ B A C

.
Then,

∠ A D C + ∠ A B C = 180 °

(Since the sum of the opposite angles of a quadrilateral is

180 °

∠ A B C = 180 ° − 140 ° = 40 °

Also,

∠ A C B = 90 °

Since a diameter's angle at the circle's circumference is

90 °

Δ A B C

we have,

∠ A C B = 90 °

and

∠ A B C = 40 °

.
Hence,

∠ C A B = 180 ° − (∠ A C B + ∠ A B C)

⇒

∠ C A B = 180 ° − (90 ° + 40 °)

⇒

∠ B A C = 50 °

Hence, the correct option is 2.

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