Question

O is the centre of the circle. If $\angle$BAC= ${50}^{\circ }$, find ∠OBC.

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Answer 1

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Given: In a circle with centre at O $\angle$BAC = ${50}^{\circ }$.

To find: $\angle$ OBC = ?

Procedure: $\angle$ BAC = ${50}^{\circ }$
∠ BOC = 2$\angle$BAC = 2(${50}^{\circ }$) =${100}^{\circ }$
(Arc BC subtends $\angle$ BOC at the centre and $\angle$ BAC at remaining part of circle)
In $△$OBC, OB = OC = radius
$\angle$OBC = $\angle$ OCB (Opposite angles of equal sides of a Δ)
Now, $\angle$ OBC + $\angle$ OCB + $\angle$ BOC = ${180}^{\circ }$ (Sum of angles in a triangle)
$\angle$ OBC + $\angle$ OCB + ${100}^{\circ }$= ${180}^{\circ }$
$\angle$ OBC + $\angle$OBC =${180}^{\circ }$ – ${100}^{\circ }$

2$\angle$OBC = ${80}^{\circ }$

∴ $\angle$ OBC = ${40}^{\circ }$