Question

O is the centre of the circle. If $\angle$BAC = 50°, find $\angle$OBC.

Answer 1

Given : In a circle with centre at O and $\angle$BAC = 50°.
To find : $\angle$OBC = ?


Proof : $\angle$BAC = 50°
$\angle$BOC = 2 $\angle$BAC = 2 (50°) = 100° [Arc BC subtends $\angle$BOC at the centre and $\angle$BAC at remaining part of circle]
In $△$OBC, OB = OC = radius
$\Rightarrow$ $\angle$OBC = $\angle$OCB (Opposite angles of equal sides of a $△$)

Now, $\angle$OBC + $\angle$OCB + $\angle$BOC = 180° [Sum of angles of a triangle]
$\Rightarrow$ $\angle$OBC + $\angle$OCB + 100º = 180°
$\Rightarrow$ $\angle$OBC + $\angle$OBC = 180° – 100°
2$\angle$OBC = 80°
$\therefore \angle$OBC = 40°.