A circle with centre O is given and C is a point on its minor arc AB. If `angle AOB = 100^@` then `angle ACB` is equal to which of the following?

Options

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

Answer

To solve the problem, we need to find the value of angle ACB given that angle AOB is 100 degrees.

### Step-by-Step Solution:

1. **Understanding the Circle and Angles**:
We have a circle with center O, and points A and B lie on the circumference of the circle. The angle AOB is given as 100 degrees.

2. **Identifying the Minor and Major Arcs**:
Since C lies on the minor arc AB, we can also identify the major arc AB. The total angle around point O is 360 degrees. Therefore, the angle corresponding to the major arc AB can be calculated as:
\[ \text{Angle of Major Arc AOB} = 360^\circ - \text{Angle AOB} = 360^\circ - 100^\circ = 260^\circ \]

3. **Finding the Angle ACB**:
The angle ACB is an inscribed angle that subtends the same arc AB. The inscribed angle theorem states that an inscribed angle is half of the central angle that subtends the same arc. Therefore:
\[ \text{Angle ACB} = \frac{1}{2} \times \text{Angle AOB} = \frac{1}{2} \times 100^\circ = 50^\circ \]

4. **Conclusion**:
Thus, the measure of angle ACB is 50 degrees.

### Final Answer:
Angle ACB = 50 degrees.