Given: ∠OAB = 20° and ∠OCB = 50°

Here,

In ΔAOB

OA = OB (radius)

∠OAB = ∠OBA (angles opposite to equal sides are equal)

∴ ∠ OBA = 20°

Now, by angle sum property

∠AOB + ∠OBA + ∠OAB = 180°

∠AOB + 20° + 20° = 180°

∠AOB = 180° – 20° – 20°

∠AOB = 140°

Now, Consider Δ BOC

OC = OB (radius)

∠OCB = ∠OBC (angles opposite to equal sides are equal)

∴ ∠ OBA = 50°

Now, by angle sum property

∠COB + ∠OBC + ∠OCB = 180°

∠COB + 50° + 50° = 180°

∠COB = 180° – 50° – 50°

∠COB = 80°

Here,

∠AOB = ∠AOC + ∠COB

140° = ∠AOC + 80°

∠AOC = 140° – 80°

∠AOC = 60°

∴ ∠AOC = 60°