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To solve for the measure of angle \( \angle UWV \), we can break down the problem based on the diagram and the given algebraic expressions for adjacent angles.

### Step-by-step process:

1. **Identify angle relationships**:
   - We are given two angles around point \( W \). 
   - One angle, \( \angle VWQ \), is given as \( (6x + 24)^\circ \).
   - Another angle, \( \angle SWU \), is given as \( (3x + 30)^\circ \).
   - These two angles form a straight line, meaning they are **supplementary**. Therefore, their sum should be \( 180^\circ \).

2. **Set up the equation**:
   Since \( \angle VWQ + \angle SWU = 180^\circ \), we have:
   \[
   (6x + 24) + (3x + 30) = 180
   \]

3. **Solve for \( x \)**:
   Simplify the equation:
   \[
   9x + 54 = 180
   \]
   Subtract 54 from both sides:
   \[
   9x = 126
   \]
   Divide by 9:
   \[
   x = 14
   \]

4. **Find the measure of \( \angle UWV \)**:
   Now, substitute \( x = 14 \) into the expression for \( \angle UWV = 3x + 30 \):
   \[
   \angle UWV = 3(14) + 30 = 42 + 30 = 72^\circ
   \]

Thus, the measure of \( \angle UWV \) is \( 72^\circ \).

Would you like further details on any of the steps? Here are five related questions to deepen your understanding:

1. How would the solution change if the angles weren't supplementary?
2. Can you describe another scenario where adjacent angles form a linear pair?
3. How does the concept of supplementary angles apply to parallel lines and transversals?
4. What if you were asked to find the measure of angle \( \angle VWQ \) instead? How would you proceed?
5. How would you solve this problem if it involved angles around a point (which sum to 360 degrees)?

**Tip**: When dealing with angle problems, always check for supplementary, complementary, and vertical angles—they can simplify your calculations greatly!

Learn how to find the measure of angle UWV using algebra and the supplementary angle theorem. This problem includes detailed step-by-step calculations and is suitable for students in Grades 7-10.

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