https://file.aiquickdraw.com/mathbot/file/17256429742604pvj6y1w.jpg

Referring to the figure, decide whether the lines are
a. parallel
b. perpendicular
c. neither

To determine whether the lines in the image are parallel, perpendicular, or neither, let's analyze their slopes.

### Analyzing the Slopes:
1. **Identify the lines**: There are three lines labeled in the graph.
   - Line \( EF \)
   - Line \( FG \)
   - Line \( HI \)
   
2. **Calculate the slopes**:
   - The slope of a line is calculated by using the formula \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the \( y \)-coordinate, and \( \Delta x \) is the change in the \( x \)-coordinate.

3. **Slopes of the lines**:
   - **Line \( EF \)**: This line appears to be vertical. The slope of a vertical line is undefined.
   - **Line \( FG \)**: This line is diagonal, with a positive slope. To determine the exact slope, count the rise over run.
   - **Line \( HI \)**: This line is also diagonal but appears to have a different angle compared to \( FG \).

### Conclusion:
- **Parallel lines** have the same slope. 
- **Perpendicular lines** have slopes that are negative reciprocals of each other (e.g., \( m_1 \times m_2 = -1 \)).
- **Neither** if none of these conditions apply.

Since line \( EF \) is vertical (undefined slope), it's not parallel or perpendicular to any of the other lines. The other two lines, \( FG \) and \( HI \), do not appear to be parallel (since their slopes are not identical) or perpendicular (since their slopes are not negative reciprocals).

Thus, the lines in the figure are **neither parallel nor perpendicular**.

Would you like more details on this? Here are some related questions you might be interested in:

1. How can we calculate the exact slopes of \( FG \) and \( HI \)?
2. What are the conditions for lines to be perpendicular in detail?
3. How would the graph change if one of the lines were parallel to the x-axis?
4. Can we have perpendicular lines if one is vertical or horizontal?
5. What does it mean for two lines to be coincident?

**Tip**: Always check the direction and relative angles of lines carefully to distinguish between parallel, perpendicular, and neither.

Referring to the figure, decide whether the lines are a. parallel b. perpendicular c. neither

Learn how to determine whether lines in a graph are parallel, perpendicular, or neither using slope analysis. This problem involves calculating slopes, identifying line types, and understanding the relationships between different lines. Suitable for students in Grades 8-10.

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