In the below diagram, O is the center of the circle, AC is the diameter and if ∠APB=1200, then which of the following is the measure of ∠BQC?

Consider the given figure,

Here, quadrilateral

APBC

is a cyclic quadrilateral.

AC

is the diameter and

∠ APB = 120^{0}

. Sum of opposite angles of a cyclic quadrilateral are supplementary.
Hence,

∠ APB + ∠ ACB = 180^{0}

∠ ACB = 180^{0} - ∠ APB

Since,

∠ APB = 120^{0}

\Rightarrow ∠ ACB = 180^{o} - 120^{o}

\Rightarrow ∠ ACB = 60^{0}

The angle subtended by the diameter of a circle at the circumference is

90^{o}

.
Here,

AC

is a diameter and hence

∠ ABC = 90^{0}

.
Thus, in

ΔABC,

by angle sum property,

∠ CAB + ∠ ABC + ∠ ACB = 180^{o}

⟹

∠ CAB + 90^{o} + 60^{o} = 180^{o}

⟹

∠ CAB = 180^{0} - 90^{0} - 60^{0}

⟹

∠ CAB = 30^{0}

Now, quadrilateral

ABQC

is another cyclic quadrilateral.
So,

⟹ ∠ CAB + ∠ CQB = 180^{0}

...[Opposite angles of a cyclic quadrilateral]

⟹ ∠ CQB = 180^{0} - ∠ CAB

⟹ ∠ CQB = 180^{o} - 30^{o}

⟹ ∠ CQB = 150^{0}

Hence, the measure of

∠ CQB

150^{0}

.
Therefore, option 2 is correct.

In the below diagram, $O$ is the center of the circle, $AC$ is the diameter and if $∠ APB = 120^{0}$ , then which of the following is the measure of $∠ BQC$ ?