To determine if two lines are parallel or perpendicular, we need to calculate their slopes and compare them.

Step 1: Calculate the slope of line $e$

Line $e$ passes through the points $(7, 8)$ and $(1, 1)$. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

For line $e$: $m_e = \frac{1 - 8}{1 - 7} = \frac{-7}{-6} = \frac{7}{6}$

Step 2: Calculate the slope of line $f$

Line $f$ passes through the points $(1, 2)$ and $(7, 9)$. Using the same formula: $m_f = \frac{9 - 2}{7 - 1} = \frac{7}{6}$

Step 3: Compare the slopes

• Parallel lines have the same slope. Since $m_e = m_f = \frac{7}{6}$, lines $e$ and $f$ are parallel.
• Perpendicular lines would have slopes that are negative reciprocals of each other, but that is not the case here.

Thus, lines $e$ and $f$ are parallel.

Would you like more details or have any questions about this process? Here are some related questions you might find interesting:

1. How can you determine if two lines are perpendicular?
2. What happens when two lines have equal slopes but different intercepts?
3. Can two non-parallel lines with the same slope exist?
4. What is the equation of a line given a point and slope?
5. How do you find the point of intersection of two lines?

Tip: When lines have the same slope, they never intersect unless they are the same line!