https://file.aiquickdraw.com/mathbot/file/1730807224966gz1eg02q.jpg

To solve this problem, we need to analyze the information provided in the diagram and use geometric principles to find the measure of \( m \overset{\frown}{WV} \).

### Steps to Solve the Problem

1. **Identify Given Information**:
   - \( m \overset{\frown}{WV} \) is the arc we need to find.
   - The angle \( \angle WTV \), which is subtended by the arc \( WV \), has a measure of \( 5x + 10 \).
   - The arc \( WS \) has a measure of \( 12x - 2 \).
   - The formula provided at the bottom suggests that this is a problem where we can solve for \( x \) first and then use it to find the required angle.

2. **Understand the Relationship**:
   - The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the intercepted arc.
   - Therefore, \( m \angle WTV = \frac{1}{2} \times m \overset{\frown}{WV} \).

3. **Set Up an Equation**:
   - From the inscribed angle theorem, we have:
     \[
     5x + 10 = \frac{1}{2} m \overset{\frown}{WV}
     \]
   - Solving for \( m \overset{\frown}{WV} \):
     \[
     m \overset{\frown}{WV} = 2(5x + 10)
     \]
     \[
     m \overset{\frown}{WV} = 10x + 20
     \]

4. **Determine \( x \)**:
   - We need to use any additional information to find \( x \). However, based on the image, it seems we might need more context or equations to solve for \( x \).
   - If there are relationships or values for other arcs or angles in the problem not visible here, they could help determine \( x \).

5. **Calculate \( m \overset{\frown}{WV} \)**:
   - Once \( x \) is determined, substitute back to find \( m \overset{\frown}{WV} \).

Could you provide additional details if there are any further instructions or values given? Alternatively, I can guide you through how to solve for \( x \) if there's more information on the other arcs or angles. 

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### Further Questions for Practice
1. How does the inscribed angle theorem relate to other circle properties?
2. What happens if the angle is outside the circle instead of inside?
3. How would you approach this problem if the central angle was given instead?
4. Can you find the length of the arc if the radius of the circle is provided?
5. What other properties of circles can be used to solve problems with unknown angles?

**Tip**: Always look for relationships between angles and arcs when dealing with circles; many problems can be simplified with these fundamental theorems.

Find the measure of the arc \( m \overset{\frown}{WV} \) in the given circle diagram.

Learn how to find the measure of arc WV in a circle using the inscribed angle theorem. This problem involves identifying relationships between inscribed angles and intercepted arcs, setting up equations, and solving for unknown variables. Ideal for high school students studying geometry.

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