I is the incentre of a triangle ABC. If `/_ABC = 65^@` and `/_ACB = 55^@`, then the value of `/_BIC `is

Options

A. `130^@`
B. `120^@`
C. `140^@`
D. `110^@`

Answer

To find the value of angle \( \angle BIC \) in triangle \( ABC \) where \( I \) is the incenter, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Angles**:
- We are given \( \angle ABC = 65^\circ \) and \( \angle ACB = 55^\circ \).

2. **Calculate \( \angle BAC \)**:
- The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can find \( \angle BAC \) using the formula:
\[ \angle BAC = 180^\circ - \angle ABC - \angle ACB \]
- Substituting the values:
\[ \angle BAC = 180^\circ - 65^\circ - 55^\circ = 180^\circ - 120^\circ = 60^\circ \]

3. **Use the Incenter Angle Formula**:
- The angle \( \angle BIC \) can be calculated using the formula:
\[ \angle BIC = 90^\circ + \frac{\angle BAC}{2} \]
- Substituting the value of \( \angle BAC \):
\[ \angle BIC = 90^\circ + \frac{60^\circ}{2} = 90^\circ + 30^\circ = 120^\circ \]

4. **Conclusion**:
- Therefore, the value of \( \angle BIC \) is \( 120^\circ \).

### Final Answer:
\[ \angle BIC = 120^\circ \]