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To solve this problem, we need to find the slopes of each line and then analyze their relationships.

### Part (a): Finding the Slope of Each Line

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### 1. Slope of Line 1
Given points: \((0, 2)\) and \((2, -2)\)
\[
m_1 = \frac{-2 - 2}{2 - 0} = \frac{-4}{2} = -2
\]

#### 2. Slope of Line 2
Given points: \((-1, -1)\) and \((1, -5)\)
\[
m_2 = \frac{-5 - (-1)}{1 - (-1)} = \frac{-5 + 1}{1 + 1} = \frac{-4}{2} = -2
\]

#### 3. Slope of Line 3
Given points: \((-3, 0)\) and \((-5, -4)\)
\[
m_3 = \frac{-4 - 0}{-5 - (-3)} = \frac{-4}{-2} = 2
\]

### Part (b): Determining the Relationship Between the Lines

- **Lines 1 and 2** have the same slope (\(m_1 = m_2 = -2\)), so they are **parallel**.
- **Lines 1 and 3** have slopes \(m_1 = -2\) and \(m_3 = 2\). Since \(m_1 \cdot m_3 = -2 \times 2 = -4\), they are **neither parallel nor perpendicular**.
- **Lines 2 and 3** have slopes \(m_2 = -2\) and \(m_3 = 2\). Since \(m_2 \cdot m_3 = -2 \times 2 = -4\), they are **neither parallel nor perpendicular**.

### Summary of Answers
1. **Slope of Line 1** = -2
2. **Slope of Line 2** = -2
3. **Slope of Line 3** = 2

**Relationships**:
- **Line 1 and Line 2**: Parallel
- **Line 1 and Line 3**: Neither
- **Line 2 and Line 3**: Neither

Would you like more details, or do you have any questions?

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Here are 5 related questions to further understand the concept:

1. What does it mean for two lines to be perpendicular in terms of slopes?
2. How would you find the equation of each line given a point and its slope?
3. How can you use slope-intercept form to represent these lines?
4. What conditions are necessary for three lines to be concurrent?
5. How would you calculate the angle between two non-parallel lines using their slopes?

**Tip**: When two lines are perpendicular, the product of their slopes is \(-1\).

Suppose we are given the following:

Line 1 passes through (0, 2) and (2, -2).
Line 2 passes through (-1, -1) and (1, -5).
Line 3 passes through (-3, 0) and (-5, -4).

(a) Find the slope of each line.

(b) For each pair of lines, determine whether they are parallel, perpendicular, or neither.

Calculate the slopes of lines passing through given points and determine if they are parallel, perpendicular, or neither. This example problem illustrates key concepts in analytic geometry, including slope calculation and relationships between lines, suitable for Grades 8-10.

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