In the given figure, O is the center of the circle, ∠AOB = 90° and ∠ABC = 30°, the measure of ∠CAO is :

1. 45°

2. 90°

3. 60°

4. 75°

Given:

∠AOB = 90°

∠ABC = 30°

We know, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle.

Thus, ∠AOB = 2∠ACB

⇒ 90° = 2∠ACB

⇒ ∠ACB = 45°

In ∆ACB,

⇒ ∠CAB + ∠CBA + ∠ACB = 180° (angle sum property)

⇒ ∠CAB + 30° + 45° = 180°

⇒ ∠CAB + 75°= 180°

⇒ ∠CAB = 180° − 75°

⇒ ∠CAB = 105°

In ∆OAB,

OA = OB (Radius of the same circle)

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

Now,

⇒ ∠OAB + ∠OBA + ∠AOB = 180° (angle sum property)

⇒ 2x + 90° = 180°

⇒ 2x = 180° − 90°

⇒ 2x = 90°

⇒ x = 45°

⇒ ∠OAB = 45°.

From figure,

⇒ ​∠CAO = ∠CAB − ∠OAB

= 105° − 45°

= 60°

Hence, Option 3 is the correct option.