Question

In the given figure, if O is the circumcentre of ∠ABC, then find the value of ∠OBC + ∠BAC.

Answer 1

Since, O is the circumcentre of $△ABC$, So, O would be centre of the circle passing through points A, B and C.

$\angle ABC$= 90° (Angle in the semicircle is 90°.)

$\Rightarrow \angle OAB+\angle OBC=90°$ .....(1)

As OA = OB (Radii of the same circle)

$$\begin{gathered} \therefore \angle OAB=\angle OBA\left(\mathrm{Angle}\mathrm{opposite}\mathrm{to}\mathrm{equal}\mathrm{sides}\mathrm{are}\mathrm{equal}\right) \\ \mathrm{or},\angle BAC=\angle OBA \\ \mathrm{From}\left(1\right) \\ \mathit{\angle }BAC+\angle OBC=90° \end{gathered}$$