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To find the measure of angle \( \angle SVU \) in the circle diagram, let's analyze the given information:

1. **Circle Geometry**: Point \( V \) is at the center of the circle, and points \( U, S, \) and \( T \) are on the circumference of the circle. \( VU \) and \( VS \) are radii.

2. **Angles in the Circle**: 
   - \( \angle TVU = 60^\circ \) (central angle).

3. **Arc Measures and Angles**:
   - The arc opposite to \( \angle SVU \) is defined by \( UT \).
   - The measure of the central angle \( \angle TVU = 60^\circ \) tells us the arc \( TU \) is also \( 60^\circ \) (because the central angle subtended by an arc is equal to the measure of the arc).

4. **Finding \( m\angle SVU \)**:
   - \( \angle SVU \) is an **inscribed angle** subtended by arc \( SU \).
   - The measure of an inscribed angle is half the measure of the arc it subtends.
   
   Since the central angle \( \angle TVU = 60^\circ \) subtends arc \( TU \), and \( TU \) is an arc, the corresponding arc \( SU \) is \( 360^\circ - 60^\circ = 300^\circ \) (the entire circle's measure is \( 360^\circ \)).

Thus, the inscribed angle \( \angle SVU \) is:

\[
m\angle SVU = \frac{1}{2} \times m\text{arc } SU = \frac{1}{2} \times 300^\circ = 150^\circ.
\]

### Conclusion:

\[
m\angle SVU = 150^\circ.
\]

Would you like further details or have any questions?

### 5 Relative Questions:

1. What is the relationship between central and inscribed angles in a circle?
2. How do you find the measure of an arc using a central angle?
3. What are the properties of angles formed by tangents and chords in a circle?
4. How do you calculate the measure of an inscribed angle subtended by a semicircle?
5. What are the formulas for finding angles in cyclic quadrilaterals?

### Tip:

When working with circles, always remember that the measure of a central angle is equal to the measure of the arc it subtends, while an inscribed angle is half the measure of the arc it subtends.

Learn how to find the measure of angle SVU in circle geometry with a detailed step-by-step solution. This problem involves circle geometry concepts, including central angles, inscribed angles, and arc measures. Suitable for high school students studying geometry.

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