https://file.aiquickdraw.com/mathbot/file/1727354662684u4vfijow.jpg

Let's analyze the question in the image. The problem provides the following information:

- \( PQRST \) is a regular pentagon.
- \( QPU \), \( RVU \), and \( STU \) are straight lines.

You are asked to find the value of \( x \), which is the angle at \( U \).

### Step-by-step analysis:

1. **Regular Pentagon Properties**:
   Since \( PQRST \) is a regular pentagon, the interior angle of a regular pentagon can be calculated using the formula:
   \[
   \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
   \]
   where \( n \) is the number of sides. For a pentagon (\( n = 5 \)):
   \[
   \text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
   \]
   Therefore, each internal angle of the pentagon is \( 108^\circ \).

2. **Angle at Point \( V \)**:
   Since \( QPU \), \( RVU \), and \( STU \) are straight lines, this tells us that the angles along these lines must be supplementary or add up to 180° when appropriate. The marked angle at point \( V \) (which appears to be labeled \( 108^\circ \)) indicates that this is one of the internal angles of the pentagon.

3. **Using the Linear Pair Property**:
   The angle \( x \) is located on the straight line \( STU \), and the angle at \( V \) is part of this line as well. Since \( RVU \) is straight, the sum of the angle \( x \) and the angle at \( V \) (which is 108°) must equal 180° (since they form a straight line).
   
   Thus, we have the equation:
   \[
   x + 108^\circ = 180^\circ
   \]
   Solving for \( x \):
   \[
   x = 180^\circ - 108^\circ = 72^\circ
   \]

4. **Dividing the Angle at \( V \)**:
   The angle \( x \) is divided in half by the line segment \( ST \), so the value of \( x \) is half of \( 72^\circ \):
   \[
   x = \frac{72^\circ}{2} = 36^\circ
   \]

### Final Answer:
The value of \( x \) is \( 36^\circ \). Hence, the correct answer is:

**Option D: \( 36^\circ \)**.

Would you like more details or further clarification?

Here are five related questions to expand your understanding:
1. How would you calculate the exterior angle of a regular pentagon?
2. What is the total sum of the interior angles of a pentagon?
3. How do straight angles relate to supplementary angles in geometric problems?
4. If the figure were a hexagon instead of a pentagon, how would that change the calculation?
5. How can you apply the properties of regular polygons to real-life objects?

**Tip**: When dealing with regular polygons, always remember that each internal angle depends only on the number of sides and can be calculated with a simple formula.

In the diagram, PQRST is a regular polygon. QPU, RVU, and STU are straight lines. Calculate the value of x.

Learn how to calculate the angle x in a regular pentagon, where QPU, RVU, and STU are straight lines. Explore concepts like interior angles of polygons and supplementary angles with a step-by-step solution. Suitable for Grades 9-11 geometry students.

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