If O is the incentre of a triangle ABC and `angle AOC=124^@`, find the `angle ABC`.

Options

A. 56
B. 112
C. 68
D. can't be determined

Answer

To solve the problem, we need to find the angle \( \angle ABC \) given that \( O \) is the incenter of triangle \( ABC \) and \( \angle AOC = 124^\circ \).

### Step-by-Step Solution:

1. **Understanding the Incenter**:
The incenter \( O \) of triangle \( ABC \) is the point where the angle bisectors of the triangle intersect. It is also the center of the circle inscribed in the triangle.

2. **Using the Angle Relationship**:
The angle \( \angle AOC \) can be expressed in terms of the angles of the triangle. Specifically, the relationship is given by:
\[ \angle AOC = 90^\circ + \frac{\angle ABC}{2} \]

3. **Substituting the Given Value**:
We know that \( \angle AOC = 124^\circ \). We can substitute this value into the equation:
\[ 124^\circ = 90^\circ + \frac{\angle ABC}{2} \]

4. **Isolating \( \angle ABC \)**:
To find \( \angle ABC \), we first subtract \( 90^\circ \) from both sides:
\[ 124^\circ - 90^\circ = \frac{\angle ABC}{2} \]
This simplifies to:
\[ 34^\circ = \frac{\angle ABC}{2} \]

5. **Multiplying to Solve for \( \angle ABC \)**:
Now, we multiply both sides by 2 to find \( \angle ABC \):
\[ \angle ABC = 34^\circ \times 2 = 68^\circ \]

6. **Final Answer**:
Therefore, the measure of angle \( \angle ABC \) is \( 68^\circ \).