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

Arc AB subtends ∠APB at center and ∠ACB at the point C on the circle.

∴ ∠APB = 2∠ACB

⇒ ∠APB = 2∠ACB
⇒ 130° = 2∠ACB
⇒ ∠ACB = 65°.

From figure,

∠ACB + ∠BCD = 180°. (∵ both angles form a linear pair)

⇒ 65° + ∠BCD = 180°
⇒ ∠BCD = 180° - 65°
⇒ ∠BCD = 115°.

In circle with center Q,

⇒ ∠BQD + Reflex ∠BQD = 360°
⇒ x° + Reflex ∠BQD = 360°
⇒ Reflex ∠BQD = 360° - x°.

Arc BD subtends reflex ∠BQD at center and ∠BCD at the point C on the circle.

∴ Reflex ∠BQD = 2∠BCD

⇒ 360° - x° = 2 × 115°
⇒ 360° - x° = 230°
⇒ x° = 360° - 230°
⇒ x° = 130°.

Hence, the value of x = 130.