Question

In the given figure, if ∠ACB = 40^∘ , then ∠AOB = ______ and ​∠OAB = _____________

Answer 1

Given:
∠ACB = 40^∘ ...(1)


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
⇒ ∠AOB = 2(40°) (From (1))
⇒ ∠AOB = 80° ...(2)


In ∆AOB,
OA = OB (radius)
∴ ∠OAB = ∠OBA (angles opposite to equal sides are equal) ...(3)


∠OAB + ∠OBA + ∠AOB = 180° (angle sum property)
⇒ 2∠OAB + 80° = 180° (From (2) and (3))
⇒ 2∠OAB = 180° − 80°
⇒ 2∠OAB = 100°
⇒ ∠OAB = 50°

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Hence, ∠AOB = 80° and ​∠OAB = 50°.