Question

In the figure given below, P and Q are centres of two circles, intersecting at B and C, and ACD is a straight line.

If $\angle APB={150}^{\circ }\text{and}\angle BQD=x^{\circ }$, find the value of x.

Answer 1

ANSWER:
We know that the angle subtended by an arc of a circle at the centre is double the angle subtended by it on the
remaining part of the circle.
Here, arc AEB subtends ∠ APB at the centre and ∠ ACB at C on the circle.
∴ ∠ APB = 2∠ ACB
⇒ ∠ ACB=150°/2=75°
...(1)
Since ACD is a straight line, ∠ ACB + ∠ BCD = 180 °
⇒ ∠ BCD = 180 °− 75 °
⇒ ∠ BCD = 105 °
...(2)
Also, arc BFD subtends reflex ∠ BQD at the centre and ∠ BCD at C on the circle.
∴ reflex ∠ BQD = 2∠ BCD
⇒ reflex ∠ BQD=2*105°=210°
...(3)
Now,
reflex ∠ BQD + ∠ BQD = 360 °
⇒ 210 °+ x = 360 °
⇒ x = 360 °− 210 °
⇒ x = 150 °
Hence, x = 150 °.