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In given figure, BC is a diameter of the circle and $∠ BAO = 60^{\circ}$ . Then, which of the following is equal to $∠ ADC$ ?

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30^{\circ}

45^{\circ}

60^{\circ}

120^{\circ}

answer is C.

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Detailed Solution

Consider the given question,
Question Image

Given that,

∠ BAO = 60^{\circ}

.
In

△ AOB

OA = OB

[Radius]
Then,

∠ OAB = ∠ OBA = 60^{0}

.
We know that,
Exterior angle of a triangle is equal to sum of the two opposite angles of the triangle.
Then,

\Rightarrow ∠ AOC = ∠ OAB + ∠ OBA

\Rightarrow ∠ AOC = 60^{0} + 60^{0}

\Rightarrow ∠ AOC = 120^{0}

We know that,
Angle subtended at the centre of the circle by an arch is twice the angle subtended at the circumference by the same arc.
Thus,

∴ ∠ ADC = \frac{1}{2} ∠ AOC

\Rightarrow ∠ ADC = \frac{120^{\circ}}{2}

\Rightarrow ∠ ADC = 60^{0}

Hence, the measure of

∠ ADC

60^{0}

.
Therefore, option 3 is correct.

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