In the figure, given below, P and Q are the centers of two circles intersecting at B and C. ACD is a straight line. Calculate the numerical value of x.

We know that,

Angle at the center is double the angle at the circumference subtended by the same chord.

⇒ ∠APB = 2∠ACB

⇒ ∠ACB = $\dfrac{1}{2}$ ∠APB

⇒ ∠ACB = $\dfrac{1}{2} \times$ 150° = 75°.

From figure,

⇒ ∠ACB + ∠BCD = 180° [As ACD is a straight line]

⇒ 75° + ∠BCD = 180°

⇒ ∠BCD = 180° - 75° = 105°.

Also,

⇒ Reflex ∠BQD = 2∠BCD [Angle at the center is double the angle at the circumference subtended by the same chord.]

⇒ (360° - x) = 2 x 105°

⇒ x = 360° - 210° = 150°.

Hence, x = 150°.