Question

In the given figure, if ∠OAB = 30^∘ and ∠OCB = 57^∘ , then ∠BOC = _______ and ∠AOC = _________ .

Answer 1

Given:
∠OAB = 30^∘ ...(1)
∠OCB = 57^∘ ...(2)


In ∆COB,
OC = OB (radius)
∴ ∠OCB = ∠OBC = 57^∘ (angles opposite to equal sides are equal) ...(3)


∠OCB + ∠OBC + ∠COB = 180° (angle sum property)
⇒ 57^∘ + 57^∘ + ∠COB = 180° (From (3))
⇒ ∠COB = 180° − 114°
⇒ ∠COB = 66° ...(4)


In ∆AOB,
OA = OB (radius)
∴ ∠OAB = ∠OBA = 30^∘ (angles opposite to equal sides are equal) ...(5)


∠OAB + ∠OBA + ∠AOB = 180° (angle sum property)
⇒ 30^∘ + 30^∘ + ∠AOB = 180° (From (5))
⇒ ∠AOB = 180° − 60°
⇒ ∠AOB = 120° ...(6)


Now,
∠AOB = ∠AOC + ∠COB
⇒ 120° = ∠AOC + 66° (From (4) and (6))
⇒ ∠AOC = 120° − 66°
⇒ ∠AOC = 54°


Hence, ∠BOC = 66° and ∠AOC = 54°.