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To solve this problem, we need to compare the slopes of the line segments \( DF \) and \( FH \) based on the given graph and the geometric relationships.

### Step-by-Step Solution:

1. **Find the slope of line segment \( DF \):**
   - The slope formula is:
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   - From the graph, the coordinates of points \( D \) and \( F \) appear to be approximately:
     - \( D(-9, -6) \)
     - \( F(3, 2) \)
   - So, the slope of \( DF \) is:
   \[
   \frac{2 - (-6)}{3 - (-9)} = \frac{2 + 6}{3 + 9} = \frac{8}{12} = \frac{2}{3}
   \]

2. **Find the slope of line segment \( FH \):**
   - The coordinates of points \( F \) and \( H \) appear to be approximately:
     - \( F(3, 2) \)
     - \( H(9, 6) \)
   - So, the slope of \( FH \) is:
   \[
   \frac{6 - 2}{9 - 3} = \frac{4}{6} = \frac{2}{3}
   \]

3. **Compare the slopes of \( DF \) and \( FH \):**
   - The slope of \( DF \) is \( \frac{2}{3} \), and the slope of \( FH \) is also \( \frac{2}{3} \). Therefore, the slopes are equal.

4. **Answer the question about ratios:**
   - The slope of line segment \( DF \) is equal to the slope of line segment \( FH \) because the **ratio of the vertical change (rise) is equal to the ratio of the horizontal change (run)** between corresponding points on each segment.

Would you like any further clarification or details about the slopes?

### Related Questions:
1. What does it mean when two line segments have the same slope?
2. How can we use the slope formula to determine if two lines are parallel?
3. Can the slope of a line be a negative value, and what does that indicate?
4. How can you find the slope of a line if you are given an equation in standard form?
5. What is the geometric interpretation of a slope being zero?

### Tip:
To quickly determine if two line segments are parallel, simply check if their slopes are equal!

The slope of line segment DF is equal to the slope of line segment FH because the ratio of vertical change is equal to the ratio of horizontal change.

Explore how the slopes of line segments DF and FH are equal by comparing the ratios of their vertical and horizontal changes. This problem focuses on the geometric concept of slope and its application in identifying parallel lines. Ideal for students in Grades 8-10.

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