In a triangle ABC , OB and OC are the bisector of angle `angleB` and `angleC` respectively . `angleBAC = 60^@` . The angle `angleBOC` will be :

Options

A. `150^@`
B. `120^@`
C. `100^@`
D. `90^@`

Answer

To find the angle \( \angle BOC \) in triangle \( ABC \) where \( OB \) and \( OC \) are the bisectors of angles \( B \) and \( C \) respectively, and \( \angle BAC = 60^\circ \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have triangle \( ABC \).
- \( \angle BAC = 60^\circ \).
- \( OB \) is the bisector of \( \angle B \).
- \( OC \) is the bisector of \( \angle C \).

2. **Understand the Angles in Triangle**:
- In any triangle, the sum of the angles is \( 180^\circ \).
- Therefore, we can express \( \angle B + \angle C \) as:
\[ \angle B + \angle C = 180^\circ - \angle A = 180^\circ - 60^\circ = 120^\circ \]

3. **Let \( \angle B = b \) and \( \angle C = c \)**:
- From the previous step, we have:
\[ b + c = 120^\circ \]

4. **Using the Angle Bisector Theorem**:
- The angle bisector divides the angle into two equal parts. Thus:
\[ \angle OBA = \frac{b}{2} \quad \text{and} \quad \angle OCA = \frac{c}{2} \]

5. **Finding \( \angle BOC \)**:
- The angle \( \angle BOC \) can be calculated using the formula:
\[ \angle BOC = 180^\circ - \left( \angle OBA + \angle OCA \right) \]
- Substituting the values:
\[ \angle BOC = 180^\circ - \left( \frac{b}{2} + \frac{c}{2} \right) = 180^\circ - \frac{b+c}{2} \]
- Since \( b + c = 120^\circ \):
\[ \angle BOC = 180^\circ - \frac{120^\circ}{2} = 180^\circ - 60^\circ = 120^\circ \]

6. **Final Answer**:
- Therefore, the angle \( \angle BOC \) is \( 120^\circ \).