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The image you uploaded contains a geometry problem involving two similar polygons, as well as the application of proportional relationships between their sides.

### Here's a breakdown of the solution process:
1. **Angle Correspondence**:
   - The two polygons are similar, meaning their corresponding angles are equal.
   - Since \( m \angle D = 32^\circ \) in the left polygon, the corresponding angle in the right polygon is \( \angle I \), which also measures \( 32^\circ \).
   
2. **Side Correspondence**:
   - \( \overline{AB} \) from the left polygon corresponds to \( \overline{FG} \) from the right polygon.
   - \( \overline{CD} \) from the left polygon corresponds to \( \overline{HI} \) from the right polygon.

3. **Proportions Between Corresponding Sides**:
   - Since the polygons are similar, the sides are proportional. We can set up the following ratio:
   \[
   \frac{\text{length of } \overline{AB}}{\text{length of } \overline{FG}} = \frac{\text{length of } \overline{CD}}{\text{length of } \overline{HI}}
   \]
   - Substituting the known values:
   \[
   \frac{80}{50} = \frac{48}{x}
   \]
   - Solving for \( x \) (the length of \( \overline{HI} \)) by cross-multiplying:
   \[
   80 \times x = 50 \times 48 \quad \Rightarrow \quad 80x = 2400 \quad \Rightarrow \quad x = \frac{2400}{80} = 30
   \]
   Thus, the length of \( \overline{HI} \) is \( 30 \) units.

### Final Answer:
The length of side \( \overline{HI} \) in the right figure is 30 units.

Let me know if you'd like more details or clarifications!

#### Here are five related questions:
1. What would happen if the polygons were not similar?
2. How are proportions used in other geometric problems involving similar shapes?
3. What are the key differences between similar and congruent polygons?
4. How can we verify that two polygons are similar based solely on side lengths?
5. What are some applications of similar polygons in real-world problems?

#### Tip:
Always ensure that you match corresponding angles and sides when working with similar polygons to avoid incorrect proportions.

Assume that the figures shown to the right are similar. Given the lengths of sides and measures of angles in the left figure, what information is known about the right figure?

Explore how to solve for unknown side lengths in similar polygons by using proportions. This example problem involves corresponding angles, proportional sides, and the application of the similar polygons theorem. Suitable for students in Grades 8-10.

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