To determine which lines are perpendicular, we need to look at their slopes. Two lines are perpendicular if the product of their slopes is $-1$.

Let's analyze the equations one by one.

Line 1: $y = 2$

• This is a horizontal line, and horizontal lines have a slope of 0.

Line 2: $y = \frac{1}{5}x - 3$

• This is in slope-intercept form, $y = mx + b$, where $m$ is the slope. So, the slope of this line is $\frac{1}{5}$.

Line 3: $x = -4$

• This is a vertical line, and vertical lines have an undefined slope.

Line 4: $y + 1 = -5(x + 2)$

• First, rewrite it in slope-intercept form: $y + 1 = -5(x + 2) \implies y + 1 = -5x - 10 \implies y = -5x - 11$
• The slope of this line is $-5$.

Checking for Perpendicularity

• Two lines are perpendicular if the product of their slopes is $-1$.

1. Line 1 and Line 2:

   • Slope of Line 1 = 0 (horizontal line)
   • Slope of Line 2 = $\frac{1}{5}$
   • $0 \times \frac{1}{5} = 0$, so they are not perpendicular.

2. Line 1 and Line 3:

   • Line 1 has a slope of 0 (horizontal line).
   • Line 3 has an undefined slope (vertical line).
   • Horizontal and vertical lines are always perpendicular, so Line 1 and Line 3 are perpendicular.

3. Line 1 and Line 4:

   • Slope of Line 1 = 0 (horizontal line)
   • Slope of Line 4 = $-5$
   • $0 \times (-5) = 0$, so they are not perpendicular.

4. Line 2 and Line 3:

   • Slope of Line 2 = $\frac{1}{5}$
   • Line 3 has an undefined slope (vertical line).
   • Horizontal lines and vertical lines are always perpendicular, so Line 2 and Line 3 are perpendicular.

5. Line 2 and Line 4:

   • Slope of Line 2 = $\frac{1}{5}$
   • Slope of Line 4 = $-5$
   • $\frac{1}{5} \times (-5) = -1$, so Line 2 and Line 4 are perpendicular.

6. Line 3 and Line 4:

   • Line 3 has an undefined slope (vertical line).
   • Slope of Line 4 = $-5$.
   • Vertical and non-vertical lines are not perpendicular.

Conclusion

• Line 1 and Line 3 are perpendicular.
• Line 2 and Line 4 are perpendicular.

Thus, the correct answer is "Lines 1 and 3 are perpendicular, and Lines 2 and 4 are perpendicular."

Would you like further details or any clarification?

Here are some related questions for further study:

1. How do you determine the slope of a line from its equation in point-slope form?
2. What is the general condition for two lines to be parallel?
3. How can you find the equation of a line given its slope and a point on the line?
4. What are the differences between horizontal, vertical, and sloped lines?
5. How do you graphically check for perpendicularity between two lines?

Tip: Always remember that the slope of a vertical line is undefined, and the slope of a horizontal line is 0.