∠OCD = 90° (CD is a tangent and OC is radius at the point of tangency) ∠OAC = ∠OCA = 30° (Since, they are opposite angles to equal line segments, OA = OC) ∠ACD = ∠ACO + ∠OCD = 30° + 90° = 120° ∠BAC = 180° - 120° = 60° (Given ∠CAB and ∠ACD are supplementary) In triangle AOC ∠OAC + ∠ OCA + ∠AOC = 180° (Sum of angles of a triangle) ∠AOC= 120° Hence, ∠ABC =
AOC
2
=
120
2
=60° (Angle subtended by the chord at the circle is half that subtended at the centre) Thus, in triangle ABC ∠CAB + ∠CBA + ∠ACO + ∠OCB= 180 60 + 60 + 30 + ∠OCB = 180 ∠OCB = 30°