Question

In the given figure,∠AOB = 90° and ∠ABC = 30° , then ∠CAO is equal to ___________.

Answer 1

Given:
∠AOB = 90° ...(1)
∠ABC = 30° ...(2)

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° ...(3)


In ∆ACB,
∠CAB + ∠CBA + ∠ACB = 180° (angle sum property)
⇒ ∠CAB + 30° + 45° = 180° (From (2) and (3))
⇒ ∠CAB + 75°= 180°
⇒ ∠CAB = 180° − 75°
⇒ ∠CAB = 105° ...(4)

Also, in ∆OAB,
OA = OB
∴ ∠OAB = ∠OBA (angles opposite to equal sides are equal) ...(5)

Now,
∠OAB + ∠OBA + ∠AOB = 180° (angle sum property)
⇒ 2∠OAB + 90° = 180° (From (1) and (5))
⇒ 2∠OAB = 180° − 90°
⇒ 2∠OAB = 90°
⇒ ∠OAB = 45° ...(6)


​∠CAO = ∠CAB − ∠OAB
= 105° − 45° (From (4) and (6))
= 60°

Hence, ∠CAO is equal to 60°.