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?

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 $-1$.