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Question

# In the figure above ABCD is a cyclic quadrilateral and O is the centre of the circle B, O and D are collinear points If ∠ABC=70∘ and ∠OAD=50∘, then find ∠OCB.

A
15
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B
20
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C
30
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D
45
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Solution

## The correct option is C 30∘Given that:∠ABC=70∘,∠OAD=50∘To find:∠OCB=?Solution:∠ADC+∠ABC=180∘ (Sum of opposite angles of a cyclic quadrilateral is 180∘.)⇒∠ADC+70∘=180∘⇒∠ADC=110∘∠AOC=2∠ABC (Angle at the centre is twice of angle subtended by the chord at the circumference.)∠AOC=2×70∘=140∘Now, In quadrilateral AOCD,∠AOC+∠OAD+∠ADC+∠DCO=360∘or, 140∘+50∘+110∘+∠DCO=360∘or, ∠DCO=360∘−300∘=60∘Since, B,O and D are collinear∠DCB=90∘or, ∠DCO+∠OCB=90∘or, ∠OCB=90∘−60∘or, ∠OCB=30∘Hence, C is the correct option.

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