C is a point on the minor arc AB of the circle with centre O. Given `angle ACB = x^(@) and angle AOB = y^(@)`. Calculate x, if ACBO is a parallelogram.

Options

A. `120^(@)`
B. `180^(@)`
C. `140^(@)`
D. `110^(@)`

Answer

To solve the problem, we need to analyze the properties of the angles in the circle and the parallelogram formed by points A, B, C, and O.

### Step-by-Step Solution:

1. **Understanding the Parallelogram**:
In a parallelogram, opposite angles are equal. Therefore, in parallelogram ACBO, we have:
\[ \angle ACB = \angle AOB \]

2. **Using the Inscribed Angle Theorem**:
The inscribed angle theorem states that the angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the circumference. Thus, we have:
\[ \angle AOB = 2 \times \angle ACB \]
This means:
\[ y = 2x \]
where \( y \) is the angle AOB and \( x \) is the angle ACB.

3. **Setting Up the Equation**:
From the relationship we established, we can express \( x \) in terms of \( y \):
\[ x = \frac{y}{2} \]

4. **Final Calculation**:
If we know the value of \( y \) (angle AOB), we can directly calculate \( x \) using the equation derived above. If \( y \) is not given, we cannot find a numerical value for \( x \) without additional information.

### Conclusion:
Thus, the angle \( x \) can be calculated as:
\[ x = \frac{y}{2} \]