Q. 14.5( 15 Votes )
(i) In Figure (1), 
is the center of the circle. If 
and 
find 
(ii) In figure (2), 
and 
are three points on the circle with center 
such that 
and 
Find 
Answer :
(i) Join OB.
∠OAB = ∠OBA = 40°[Because OB = OA]
∠OCB = ∠OBC = 30°[Because OB = OC]
∠ABC = ∠OBA + ∠OBC
⇒ ∠ABC = 40°+ 30°
⇒ ∠ABC = 70°
∠AOC = 2 × ∠ABC
⇒ ∠AOC = 2 × ∠ABC
⇒ ∠AOC = 2 × 70°
⇒ ∠AOC = 140°
(ii) 
∠BOC = 360° - (∠AOB + ∠AOC) [Sum of all angles at a point = 360°]
⇒ ∠BOC = 360° - (90° + 110°)
⇒ ∠BOC = 360° - 200°
⇒ ∠BOC = 160°
We know that ∠BOC = 2 × ∠BAC
⇒ ∠BAC = (1/2) × ∠BOC
⇒ ∠BAC = (1/2) × 160°
⇒ ∠BAC = 80°
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