In the given figure, ∠OAB = 30° and ∠OCB = 57°, find ∠BOC and ∠AOC.

In △AOB,

⇒ OA = OB [Radius of same circle]

⇒ ∠OBA = ∠BAO = 30° [Angles opposite to equal sides are equal]

Also,

⇒ ∠OBA + ∠BAO + ∠AOB = 180° [By angles sum property of triangle]

⇒ 30° + 30° + ∠AOB = 180°

⇒ ∠AOB = 180° - 60° = 120°

In △OCB,

OC = OB [Radius of same circle]

⇒ ∠OBC = ∠OCB = 57° [Angles opposite to equal sides are equal]

Also,

⇒ ∠OCB + ∠OBC + ∠BOC = 180° [By angles sum property of triangle]

⇒ 57° + 57° + ∠BOC = 180°

⇒ ∠BOC = 180° - 114° = 66°

From figure,

⇒ ∠AOB = ∠AOC + ∠BOC

⇒ 120° = ∠AOC + 66°

⇒ ∠AOC = 120° - 66° = 54°.

Hence, ∠AOC = 54° and ∠BOC = 66°.