If one of the angles of a triangles is `120^@` , then the angle between the interior bisectors of the other two angles is

Options

A. `90^@`
B. `30^@`
C. `150^@`
D. `60^@`

Answer

To solve the problem, we need to find the angle between the interior bisectors of the other two angles in a triangle where one angle is \(120^\circ\).

### Step-by-Step Solution:

1. **Identify the Angles in the Triangle**:
Let the triangle be \(ABC\) where \(\angle A = 120^\circ\). Let the other two angles be \(\angle B\) and \(\angle C\). We can denote these angles as:
\[ \angle B = 2x \quad \text{and} \quad \angle C = 2y \]

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in a triangle is always \(180^\circ\). Therefore, we can write:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the known values:
\[ 120^\circ + 2x + 2y = 180^\circ \]

3. **Simplify the Equation**:
Rearranging the equation gives:
\[ 2x + 2y = 180^\circ - 120^\circ \]
\[ 2x + 2y = 60^\circ \]

4. **Divide by 2**:
Dividing the entire equation by 2:
\[ x + y = 30^\circ \]

5. **Consider Triangle BOC**:
Now, we need to find the angle between the internal bisectors of angles \(B\) and \(C\). Let this angle be \(\angle BOC\). In triangle \(BOC\), the sum of angles is also \(180^\circ\):
\[ \angle OBC + \angle OCB + \angle BOC = 180^\circ \]
Here, \(\angle OBC = x\) and \(\angle OCB = y\).

6. **Substitute Known Values**:
Substituting the values we have:
\[ x + y + \angle BOC = 180^\circ \]
Since we found \(x + y = 30^\circ\), we can substitute:
\[ 30^\circ + \angle BOC = 180^\circ \]

7. **Solve for \(\angle BOC\)**:
Rearranging gives:
\[ \angle BOC = 180^\circ - 30^\circ = 150^\circ \]

### Final Answer:
The angle between the interior bisectors of the other two angles is \(150^\circ\).