Circle Angle Theorem Explanation

The question relates the angle subtended by an arc at the center of a circle to the angle subtended by the same arc at any point on the circumference. The key theorem states that the angle at the center is double the angle at the circumference.

Applying the Circle Theorem

We are given:

• Points A, B, and C lie on a circle with center O.
• The angle at the circumference, $\angle \text{ACB} = 47.5^{\circ}$.
• We need to find the angle at the center, $\angle \text{AOB}$, on the minor arc $\widehat{\text{AB}}$.

Calculating Angle AOB

According to the circle theorem:

$\angle \text{AOB} = 2 \times \angle \text{ACB}$

Substitute the given value:

$\angle \text{AOB} = 2 \times 47.5^{\circ}$

$\angle \text{AOB} = 95^{\circ}$

Conclusion

The angle $\angle \text{AOB}$ on the minor $\widehat{\text{AB}}$ is $95^{\circ}$.