If I is the incentre of `triangle`ABC and `angle`A = `60^@`, then the value of `angle`BIC is

Options

A. `100^@`
B. `120^@`
C. `150^@`
D. `110^@`

Answer

To find the value of angle \( BIC \) in triangle \( ABC \) where \( I \) is the incenter and \( \angle A = 60^\circ \), we can follow these steps:

### Step 1: Understand the properties of the triangle
The sum of the angles in any triangle is \( 180^\circ \). Therefore, we can write the equation for triangle \( ABC \):
\[ \angle A + \angle B + \angle C = 180^\circ \]
Given that \( \angle A = 60^\circ \), we can substitute this value into the equation:
\[ 60^\circ + \angle B + \angle C = 180^\circ \]

### Step 2: Simplify the equation
Now, we can isolate \( \angle B + \angle C \):
\[ \angle B + \angle C = 180^\circ - 60^\circ = 120^\circ \]

### Step 3: Use the properties of the incenter
The incenter \( I \) of triangle \( ABC \) is the point where the angle bisectors of the triangle intersect. Therefore, we know:
\[ \angle BIC = 90^\circ + \frac{1}{2} \angle A \]
Substituting the value of \( \angle A \):
\[ \angle BIC = 90^\circ + \frac{1}{2} \times 60^\circ \]

### Step 4: Calculate \( \angle BIC \)
Now, we can calculate \( \angle BIC \):
\[ \angle BIC = 90^\circ + 30^\circ = 120^\circ \]

### Conclusion
Thus, the value of \( \angle BIC \) is:
\[ \angle BIC = 120^\circ \]

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