C is a point on the minor arc AB a circle with centre O. If `angle AOB = 100^(@)`, what is `angle ACB` ?

Options

A. `80^(@)`
B. `90^(@)`
C. `100^(@)`
D. `130^(@)`

Answer

To find the angle \( ACB \) given that \( \angle AOB = 100^\circ \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a circle with center \( O \).
- \( C \) is a point on the minor arc \( AB \).
- The angle at the center \( \angle AOB = 100^\circ \).

2. **Use the Property of Angles in a Circle**:
- The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.
- This means that if \( \angle AOB = 100^\circ \), then the angle \( ADB \) (where \( D \) is any point on the major arc \( AB \)) can be found using the formula:
\[ \angle AOB = 2 \times \angle ADB \]

3. **Calculate \( \angle ADB \)**:
- From the property mentioned, we can set up the equation:
\[ 100^\circ = 2 \times \angle ADB \]
- Solving for \( \angle ADB \):
\[ \angle ADB = \frac{100^\circ}{2} = 50^\circ \]

4. **Identify the Cyclic Quadrilateral**:
- The points \( A, B, C, D \) form a cyclic quadrilateral. In a cyclic quadrilateral, the sum of the opposite angles is \( 180^\circ \).
- Therefore, we have:
\[ \angle ADB + \angle ACB = 180^\circ \]

5. **Substitute the Known Angle**:
- We already found \( \angle ADB = 50^\circ \). Now we can substitute this into the equation:
\[ 50^\circ + \angle ACB = 180^\circ \]

6. **Solve for \( \angle ACB \)**:
- Rearranging the equation gives:
\[ \angle ACB = 180^\circ - 50^\circ = 130^\circ \]

### Conclusion:
Thus, the measure of \( \angle ACB \) is \( 130^\circ \).