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# Proving Relationships of Parallel and Perpendicular Lines

## Proving Relationships of Parallel and Perpendicular Lines 1.2 - Solution

Parallel lines have the same slope but different $y\text{-}$intercepts. Perpendicular lines have slopes that are opposite reciprocals of one another. Therefore, to determine if the lines are parallel, perpendicular, or neither, we first need to find their individual slopes using the Slope Formula.

Line Points $\dfrac{y_2-y_1}{x_2-x_1}$ Slope
$p$ $({\color{#0000FF}{\text{-} 2,4}}) \, \& \, ({\color{#009600}{2,3}})$ $\dfrac{{\color{#009600}{3}}-{\color{#0000FF}{4}}}{{\color{#009600}{2}}-({\color{#0000FF}{\text{-} 2}})}$ $\text{-}\dfrac{1}{4}$
$q$ $({\color{#0000FF}{\text{-} 3,1}}) \, \& \, ({\color{#009600}{3,\text{-} 1}})$ $\dfrac{{\color{#009600}{\text{-} 1}}-{\color{#0000FF}{1}}}{{\color{#009600}{3}}-({\color{#0000FF}{\text{-} 3}})}$ $\text{-}\dfrac{1}{3}$
$\ell$ $({\color{#0000FF}{0,\text{-} 2}}) \, \& \, ({\color{#009600}{2,4}})$ $\dfrac{{\color{#009600}{4}}-({\color{#0000FF}{\text{-} 2}})}{{\color{#009600}{2}}-{\color{#0000FF}{0}}}$ $3$
$m$ $({\color{#0000FF}{\text{-} 2,0}}) \, \& \, ({\color{#009600}{0,6}})$ $\dfrac{{\color{#009600}{6}}-{\color{#0000FF}{0}}}{{\color{#009600}{0}}-({\color{#0000FF}{\text{-} 2}})}$ $3$

Lines $\ell$ and $m$ have the same slope, but different $y\text{-}$intercepts. Therefore, they are parallel. Meanwhile, the slopes of lines $q\,\&\, \ell$ and $q\,\&\,m$ are opposite reciprocals, so they are perpendicular. The line $p$ is neither parallel nor perpendicular to lines $q,$ $\ell,$ and $m.$