I is the incentre of `Delta ABC of angle A = 46^@` , then `angle BIC` = ?

Options

A. `93^@`
B. `113^@`
C. `124^@`
D. `134^@`

Answer

To find the angle \( \angle BIC \) in triangle \( ABC \) where \( I \) is the incenter and \( \angle A = 46^\circ \), we can use the following steps:

### Step-by-Step Solution:

1. **Understand the Incenter**:
The incenter \( I \) of a triangle is the point where the angle bisectors of the triangle intersect. It is equidistant from all sides of the triangle.

2. **Use the Formula for \( \angle BIC \)**:
The formula to find \( \angle BIC \) in terms of the angles of triangle \( ABC \) is:
\[ \angle BIC = 90^\circ + \frac{\angle A}{2} \]

3. **Substitute the Value of \( \angle A \)**:
Given that \( \angle A = 46^\circ \), we can substitute this value into the formula:
\[ \angle BIC = 90^\circ + \frac{46^\circ}{2} \]

4. **Calculate \( \frac{46^\circ}{2} \)**:
\[ \frac{46^\circ}{2} = 23^\circ \]

5. **Add the Values**:
Now, add \( 90^\circ \) and \( 23^\circ \):
\[ \angle BIC = 90^\circ + 23^\circ = 113^\circ \]

6. **Conclusion**:
Therefore, the measure of \( \angle BIC \) is \( 113^\circ \).

### Final Answer:
\[ \angle BIC = 113^\circ \]