The question provided involves two similar polygons, which means their corresponding sides are proportional and their corresponding angles are equal. Let's walk through the steps based on the given information and figures.

Breakdown:

1. Corresponding Angles:

   • Angle $D$ in the left figure corresponds to angle $I$ in the right figure.
   • Since the two polygons are similar, corresponding angles are equal. So, $m \angle I = 32^\circ$.

2. Corresponding Sides:

   • Side $AB$ in the left figure corresponds to side $FG$ in the right figure.
   • Side $CD$ in the left figure corresponds to side $HI$ in the right figure.

3. Proportional Sides:

   • Since the polygons are similar, the ratio of the lengths of corresponding sides is proportional.
   • We have the ratio: $\frac{\text{length of } AB}{\text{length of } FG} = \frac{\text{length of } CD}{\text{length of } HI}$ Substituting the known values: $\frac{80}{x} = \frac{48}{50}$ where $x$ is the unknown length of side $HI$.

4. Solving for $x$: Cross-multiply to solve for $x$: $80 \times 50 = 48 \times x$ $4000 = 48x$ $x = \frac{4000}{48} \approx 83.33$

Conclusion:

The length of side $HI$ is approximately $83.33$.

Do you have any questions or would you like more details on any part of the solution?

Related Questions:

1. What is the rule for finding corresponding sides in similar polygons?
2. How would the solution change if the figures were not similar?
3. Why is the ratio of corresponding sides important in similar figures?
4. How can you check if two polygons are similar based on side lengths alone?
5. How does angle similarity relate to side proportionality?

Tip: When working with similar polygons, always ensure you identify the correct corresponding sides and angles before applying proportionality.