In figure, O and O’ are centers of two circles intersecting at B and C.
Question:

In figure, O and O’ are centers of two circles intersecting at B and C. ACD is a straight line, find x.

Solution:

By degree measure theorem

∠AOB = 2∠ACB

⇒ 130° = 2∠ACB

⇒ ∠ACB = 130°/2 = 65°

∴ ∠ACB + ∠BCD = 180° [Liner a pair of angles]

⇒ 65° + ∠BCD = 180°

⇒ ∠BCD = 180° − 65° = 115° By degree measure theorem reflex

∠BOD = 2∠BCD

⇒ reflex ∠BOD = 2 × 115° = 230°

Now, reflex ∠BOD + ∠BO′D = 360° [Complex angle]

⇒ 230° + x = 360°

⇒ x = 360° − 230°

∴ x = 130°