If the `/_ABC `and `/_ACB ` of triangle ABC are `80^@ and 60^@` respectively and the Incentre of the triangle is at point "I", then calculate angle BIC.

Options

A. `55^@`
B. `80^@`
C. `110^@`
D. `120^@`

Answer

To find the angle \( \angle BIC \) in triangle \( ABC \) where \( \angle ABC = 80^\circ \) and \( \angle ACB = 60^\circ \), we can follow these steps:

### Step 1: Calculate \( \angle A \)
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can find \( \angle A \) using the formula:

\[ \angle A + \angle ABC + \angle ACB = 180^\circ \]

Substituting the known values:

\[ \angle A + 80^\circ + 60^\circ = 180^\circ \]

### Step 2: Simplify the equation
Now, simplify the equation:

\[ \angle A + 140^\circ = 180^\circ \]

### Step 3: Solve for \( \angle A \)
Subtract \( 140^\circ \) from both sides:

\[ \angle A = 180^\circ - 140^\circ = 40^\circ \]

### Step 4: Use the formula for \( \angle BIC \)
The formula for calculating \( \angle BIC \) when the incenter is inside the triangle is:

\[ \angle BIC = 90^\circ + \frac{\angle A}{2} \]

### Step 5: Substitute \( \angle A \) into the formula
Now, substitute \( \angle A = 40^\circ \) into the formula:

\[ \angle BIC = 90^\circ + \frac{40^\circ}{2} \]

### Step 6: Calculate \( \angle BIC \)
Calculate \( \frac{40^\circ}{2} \):

\[ \frac{40^\circ}{2} = 20^\circ \]

Now, add this to \( 90^\circ \):

\[ \angle BIC = 90^\circ + 20^\circ = 110^\circ \]

### Final Answer
Thus, the angle \( \angle BIC \) is \( 110^\circ \).

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