Question : In the given figure, O is the centre of the circle, $\angle DAB=110^{\circ}$ and $\angle BEC=100^{\circ}$. What is the value (in degrees) of $\angle OCB$?
Option 1: 5
Option 2: 10
Option 3: 15
Option 4: 20
Correct Answer: 10
Solution :
Given, $\angle DAB=110^{\circ}$ and $\angle BEC=100^{\circ}$
Since the angle subtended by the chord at the centre is double the angle subtended by the chord at the circumference,
reflex$\angle BOC = 2\angle BEC = 200^\circ$
So, $\angle BOC = 360^\circ-200^\circ = 160^\circ$
By angle sum property in $\triangle BOC$,
$\angle BOC + \angle OCB+ \angle OBC =180^\circ$
⇒ $160^\circ+ 2\angle OCB =180^\circ$ ($\angle OCB = \angle OBC$ since OB and OC are radii of same circle)
⇒ $\angle OCB = 10^\circ$
Hence, the correct answer is 10.
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