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To solve this problem, we need to calculate the slopes of lines \( PQ \) and \( RS \), then determine whether they are parallel, perpendicular, or neither by comparing the slopes.

### Step 1: Slope Formula
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

### Step 2: Calculate the slope of line \( PQ \)
The coordinates of points \( P \) and \( Q \) are:
- \( P(-9, -4) \)
- \( Q(-7, -1) \)

Using the slope formula:
\[
m(PQ) = \frac{-1 - (-4)}{-7 - (-9)} = \frac{-1 + 4}{-7 + 9} = \frac{3}{2}
\]

### Step 3: Calculate the slope of line \( RS \)
The coordinates of points \( R \) and \( S \) are:
- \( R(-2, 5) \)
- \( S(-6, -1) \)

Using the slope formula:
\[
m(RS) = \frac{-1 - 5}{-6 - (-2)} = \frac{-1 - 5}{-6 + 2} = \frac{-6}{-4} = \frac{3}{2}
\]

### Step 4: Determine the relationship between the lines
- If the slopes are equal, the lines are **parallel**.
- If the product of the slopes is \( -1 \), the lines are **perpendicular**.
- Otherwise, the lines are **neither**.

Since \( m(PQ) = \frac{3}{2} \) and \( m(RS) = \frac{3}{2} \), the lines \( PQ \) and \( RS \) are **parallel**.

### Final Answer:
- \( m(PQ) = \frac{3}{2} \)
- \( m(RS) = \frac{3}{2} \)
- Type of lines: **Parallel**

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Do you have any questions or would you like more details?

Here are some related questions for further practice:
1. What would the slopes be if \( PQ \) and \( RS \) were perpendicular?
2. How would the result change if the slope of \( PQ \) was negative?
3. Can two lines with undefined slopes be parallel?
4. What happens to the relationship of lines if their slopes are both zero?
5. What does it mean geometrically when two lines are neither parallel nor perpendicular?

**Tip:** When comparing slopes, remember that equal slopes indicate parallel lines, while slopes that multiply to \( -1 \) indicate perpendicular lines.

Use slope to determine if lines PQ and RS are parallel, perpendicular, or neither. Given points: P(-9, -4), Q(-7, -1), R(-2, 5), S(-6, -1).

Learn how to determine if lines PQ and RS are parallel, perpendicular, or neither using the slope formula with the points P(-9, -4), Q(-7, -1), R(-2, 5), and S(-6, -1). This problem covers key concepts in coordinate geometry and slopes of lines, suitable for students in Grades 8-10.

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