# In figure,

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°

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