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The image shows a geometry question with two pentagons sharing one side (Y-T). The problem specifies:

- Pentagon **RSTYZ** is a regular polygon.
- The measure of angle \( \angle RST = 108^\circ \).
- You're asked to find \( \angle STU \), given that \( \angle UTY = 115^\circ \).

### Step-by-step solution:
1. **Understanding Regular Pentagons**:
   In a regular pentagon, each interior angle is calculated by:
   \[
   \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
   \]
   where \( n = 5 \) for a pentagon. Plugging in the value:
   \[
   \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
   \]
   So, each angle in the pentagon is \( 108^\circ \), matching the given \( \angle RST = 108^\circ \).

2. **Using the given angles**:
   - \( \angle UTY = 115^\circ \) is an exterior angle. Exterior angles and interior angles of a polygon are supplementary:
     \[
     \text{Interior angle of pentagon } = 180^\circ - 115^\circ = 65^\circ
     \]
     So, \( \angle TYU = 65^\circ \).

3. **Finding \( \angle STU \)**:
   Since \( YT \) is a straight line and \( \angle TYU = 65^\circ \), the sum of \( \angle STU \) and \( \angle TYU \) must be \( 180^\circ \):
   \[
   \angle STU = 180^\circ - 65^\circ = 115^\circ
   \]

Therefore, the measure of \( \angle STU \) is **115°**, which corresponds to option **D**.

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Do you have any questions or need further details?

Here are 5 related questions you can consider:
1. How do you calculate the interior angles of any regular polygon?
2. What is the sum of the exterior angles of a polygon?
3. Can you explain how interior and exterior angles of polygons relate?
4. How do you find the number of sides of a polygon if you know one interior angle?
5. What would happen if \( \angle UTY \) was smaller than \( 115^\circ \)?

**Tip:** In regular polygons, the interior and exterior angles always sum to \( 180^\circ \), which helps in solving many geometry problems!

In the figure below, pentagon RSTYZ is a regular polygon and m∠RST = 108°. If m∠UTY = 115°, what is m∠STU?

Solve the geometry problem involving a regular pentagon RSTYZ where m∠RST = 108° and m∠UTY = 115°. Learn how to use the properties of regular polygons and supplementary angles to find m∠STU. Suitable for students in Grades 8-10.

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