In figure, ∠AOB = 90° and ∠ABC = 30°. Then, ∠CAO s equal to—

Given ∠AOB = 90° and ∠ABC = 30°

We know that angles opposite to equal sides are equal.

⇒∠OAB = ∠ABO

By angle sum property,

⇒∠OAB + ∠ABO + ∠BOA = 180°

⇒ 2 ∠OAB + ∠BOA = 180°

⇒ 2 ∠OAB + 90° = 180°

⇒ 2 ∠OAB = 180° - 90° = 90°

⇒∠OAB = 90°/ 2

∴ ∠OAB = 45°

We know that angles subtended by an arc at the centre of the circle is double the angle subtended by it to any other part of the circle.

∴ ∠C = 45°

By angle sum property,

⇒∠ABC + ∠BCA + ∠ACB = 180°

⇒ 30° + 45° + ∠CAB = 180°

⇒ 75° + ∠CAB = 180°

⇒∠CAB = 180° - 75°

∴ ∠CAB = 105°

Also ∠CAB = ∠CAO + ∠OAB

⇒ 105° = ∠CAO + 45°

⇒∠CAO = 105° - 45°

∴ ∠CAO = 60°