AA.14 Slopes of parallel and perpendicular lines PRP
Line $e$ passes through points $(8,7)$ and $(1,1)$. Line $f$ passes through points $(2,17)$ and $(8$, $10)$. Are line $e$ and line $f$ parallel or perpendicular?
parallel
perpendicular
neither
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Step 1 ：To determine if the lines are parallel, perpendicular, or neither, we need to calculate the slopes of both lines. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. If neither of these conditions are met, the lines are neither parallel nor perpendicular.

Step 2 ：The slope of a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $\frac{y_2-y_1}{x_2-x_1}$.

Step 3 ：Let's calculate the slopes of lines $e$ and $f$.

Step 4 ：The slope of line $e$ is calculated as $\frac{7-1}{8-1}=0.8571428571428571$.

Step 5 ：The slope of line $f$ is calculated as $\frac{17-10}{2-8}=-1.1666666666666667$.

Step 6 ：The slopes of lines $e$ and $f$ are not equal, so the lines are not parallel. The slopes are also not negative reciprocals of each other, so the lines are not perpendicular.

Step 7 ：Therefore, the lines are neither parallel nor perpendicular.

Step 8 ：Final Answer: $\boxed{{\displaystyle \text{neither}}}$