Question

In the given figure, BO and CO are the bisectors of ∠B and ∠C, respectively. If ∠A = 50°, then ∠BOC = ?
(a) 130°
(b) 100°
(c) 115°
(d) 120°

Answer 1

(c) 115°

In $∆ABC$, we have:
$$\begin{gathered} \angle A+\angle B+\angle C=180°\left[\text{Sum of the angles of a triangle}\right] \\ \Rightarrow 50°+\angle B+\angle C=180° \\ \Rightarrow \angle B+\angle C=130° \\ \Rightarrow \frac{1}{2}\angle B+\frac{1}{2}\angle C=65°...\left(i\right) \end{gathered}$$
In $∆OBC$, we have:
$$\begin{gathered} \angle OBC+\angle OCB+\angle BOC=180° \\ \Rightarrow \frac{1}{2}\angle B+\frac{1}{2}\angle C+\angle BOC=180°\left[\text{Using}\left(i\right)\right] \\ \Rightarrow 65°+\angle BOC=180° \\ \Rightarrow \angle BOC=115° \end{gathered}$$