audio read-through Parallel and Perpendicular Lines

Want some other practice with lines?

If you know the slopes of two non-vertical lines, can you easily determine if they're parallel? Perpendicular? Yes!

Parallel lines have the same slope.

Perpendicular lines have slopes that are opposite reciprocals.

This section explores these concepts.

Slopes of Parallel Lines

Two lines in a plane are parallel if they never intersect; they have the same ‘slant’. (For a more rigorous study of parallel lines and related concepts, visit my Geometry course: Parallel Lines.) Slope (or lack of it!) can be used to decide if lines are parallel:

Suppose two distinct lines lie in the same coordinate plane. Then:
the two lines are parallel

if and only if

(they are both vertical)
(they have the same slope)

This statement offers a good opportunity to review the mathematical words ‘if and only if’ and ‘or’. The statement has been visually presented to help you see the mathematical sentence structure more clearly.

Slopes of Perpendicular Lines

Now, let's move on to perpendicular lines. Two lines are perpendicular if they intersect at a $\,90^{\circ}$ angle. For example, the $\,x$-axis and $\,y$-axis are perpendicular.


Suppose two non-vertical lines lie in the same coordinate plane. Let $\,m_1\,$ and $\,m_2\,$ denote the slopes of these lines.

The two lines are perpendicular if and only if their slopes are opposite reciprocals: $$\,\cssId{s54}{m_1 = -\frac{1}{m_2}}\,$$

Equivalently, the two lines are perpendicular if and only if their slopes multiply to $\,-1\,$: $$\,\cssId{s59}{m_1m_2 = -1}\,$$

Make sure you understand the opposite reciprocal equation:

$$ \cssId{s61}{m_1 = -\frac{1}{m_2}} $$
one of the slopes is the opposite of the reciprocal of the other slope
$\displaystyle \cssId{s63}{m_1}$ $\displaystyle \cssId{s65}{=}$ $\displaystyle \cssId{s67}{-}$ $\displaystyle \cssId{s69}{\frac{1}{m_2}}$

For example, what's the reciprocal of $\,2\,$? Answer:  $\frac12$

What's the opposite [of the] reciprocal of $\,2\,$? Answer:  $-\frac12$

So, the numbers $\,2\,$ and $\,-\frac 12\,$ are opposite reciprocals. Also notice that $\,(2)(-\frac 12) = -1\,.$ So, lines with slopes $\,2\,$ and $\,-\frac12\,$ are perpendicular.

It's easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:

characterization of perpendicular lines

The yellow triangle, with base of length $\,1\,$ and right side of length $\,m\,,$ shows that the slope of the first line is: $$\cssId{s82}{\frac{\text{rise}}{\text{run}}} \cssId{s83}{= \frac{m}1} \cssId{s84}{= m}$$

Now, imagine that this yellow triangle is a block of wood that is glued to the line. Rotate this block of wood counter-clockwise by $\,90^{\circ}\,$ (so that the original base is now vertical).

Using the rotated triangle to compute the slope of the new, rotated, line gives: $$\cssId{s89}{\frac{\text{rise}}{\text{run}}} \cssId{s90}{= \frac{1}{-m}} \cssId{s91}{= -\frac1m}$$ Easy! Voila!

Concept Practice