Math Core

Angle measures are one way to tell whether lines are parallel. On the coordinate plane there is a faster way: compare slopes. With slopes you can decide whether two lines are parallel or perpendicular using nothing but arithmetic, which turns many geometry questions into quick calculations.

Slope review

The slope of a nonvertical line through $(x_1, y_1)$ and $(x_2, y_2)$ is

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}.

A line rising from left to right has positive slope; a falling line has negative slope. A horizontal line has slope $0$ , and a vertical line has undefined slope, because its run is $0$ . When a line is written as $y = mx + b$ , its slope is $m$ . If it's written another way, such as $4x - 2y = 6$ , solve for $y$ first: $y = 2x - 3$ , so the slope is $2$ .

Parallel lines

Picture a slope triangle on a line: go right $3$ , up $2$ . If you slide (translate) that line anywhere in the plane, every slope triangle slides with it, so the new line still goes right $3$ , up $2$ . Parallel lines are exactly the lines you can get this way, so they have the same slope.

Both lines have slope 2/3: from each marked point, right 3 and up 2 reaches the next.Open in grapher →

Slopes of parallel lines

Two distinct nonvertical lines are parallel if and only if they have the same slope.

Any two distinct vertical lines are parallel.

The word distinct matters. The equations $y = 2x + 1$ and $4x - 2y = -2$ have the same slope, but the second one simplifies to $y = 2x + 1$ as well, so they describe the same line, not two parallel lines.

Perpendicular lines

Now rotate a line $90^\circ$ counterclockwise about a point on it. A slope triangle with run $a$ and rise $b$ turns into a triangle with run $-b$ and rise $a$ . (Rotating the point $(a, b)$ by $90^\circ$ about the origin gives $(-b, a)$ .) So the original slope is $\dfrac{b}{a}$ , and the slope of the perpendicular line is

\frac{a}{-b} = -\frac{a}{b}.

Flip the fraction, then change its sign. Multiplying the two slopes gives $\dfrac{b}{a} \cdot \left(-\dfrac{a}{b}\right) = -1$ .

Slopes 2 and −1/2 multiply to −1, so the lines are perpendicular. They meet at (2, 3).Open in grapher →

Slopes of perpendicular lines

Two nonvertical lines are perpendicular if and only if their slopes are opposite reciprocals, meaning their product is $-1$ :

m_1 \cdot m_2 = -1.

A vertical line and a horizontal line are always perpendicular.

Some opposite-reciprocal pairs: $3$ and $-\dfrac{1}{3}$ ; $-\dfrac{4}{5}$ and $\dfrac{5}{4}$ ; $1$ and $-1$ .

Worked example: Classifying pairs of lines

Decide whether each pair of lines is parallel, perpendicular or neither.

$y = 3x - 2$ and $6x - 2y = 10$
$2x + 5y = 5$ and $y = \dfrac{5}{2}x + 1$
$y = 4x$ and $y = -4x + 7$

Solutions.

Solve the second equation for $y$ : $-2y = -6x + 10$ , so $y = 3x - 5$ . Both slopes are $3$ and the $y$ -intercepts differ ( $-2$ and $-5$ ), so the lines are parallel.
Solve the first equation for $y$ : $5y = -2x + 5$ , so $y = -\dfrac{2}{5}x + 1$ . The slopes are $-\dfrac{2}{5}$ and $\dfrac{5}{2}$ , and $-\dfrac{2}{5} \cdot \dfrac{5}{2} = -1$ , so the lines are perpendicular.
The slopes are $4$ and $-4$ . They aren't equal, and their product is $-16$ , not $-1$ . The lines are neither. (Changing the sign without flipping isn't enough.)

Worked example: Lines through given points

Line $\overleftrightarrow{AB}$ passes through $A(-2, 1)$ and $B(2, 7)$ . Line $\overleftrightarrow{CD}$ passes through $C(1, -3)$ and $D(7, -7)$ . Are the lines parallel, perpendicular or neither?

m_{AB} = \frac{7 - 1}{2 - (-2)} = \frac{6}{4} = \frac{3}{2}, \qquad m_{CD} = \frac{-7 - (-3)}{7 - 1} = \frac{-4}{6} = -\frac{2}{3}.

Since $\dfrac{3}{2} \cdot \left(-\dfrac{2}{3}\right) = -1$ , the lines are perpendicular.

Worked example: Is it a right triangle?

Triangle $PQR$ has vertices $P(1, 1)$ , $Q(4, 5)$ and $R(8, 2)$ . Is it a right triangle? If so, which angle is the right angle?

Find the slope of each side:

m_{PQ} = \frac{5 - 1}{4 - 1} = \frac{4}{3}, \qquad m_{QR} = \frac{2 - 5}{8 - 4} = -\frac{3}{4}, \qquad m_{PR} = \frac{2 - 1}{8 - 1} = \frac{1}{7}.

$\dfrac{4}{3} \cdot \left(-\dfrac{3}{4}\right) = -1$ , so $\overline{PQ} \perp \overline{QR}$ . The two sides meet at $Q$ , so $\angle Q$ is a right angle and $\triangle PQR$ is a right triangle.

Worked example: Finding an unknown coordinate

The line through $(1, k)$ and $(4, 5)$ is perpendicular to the line $3x - 2y = 7$ . Find $k$ .

First find the slope of the given line: $-2y = -3x + 7$ , so $y = \dfrac{3}{2}x - \dfrac{7}{2}$ , with slope $\dfrac{3}{2}$ . A perpendicular line has slope $-\dfrac{2}{3}$ . Set the slope formula equal to it:

\begin{aligned} \frac{5 - k}{4 - 1} &= -\frac{2}{3} \\ 5 - k &= -2 \\ k &= 7 \end{aligned}

Check: the slope from $(1, 7)$ to $(4, 5)$ is $\dfrac{5 - 7}{3} = -\dfrac{2}{3}$ . ✓

Common mistake

An opposite reciprocal needs both steps: flip the fraction and change the sign. The perpendicular slope to $\dfrac{2}{3}$ is $-\dfrac{3}{2}$ , not $\dfrac{3}{2}$ and not $-\dfrac{2}{3}$ . Also watch the special case: a line perpendicular to a horizontal line (slope $0$ ) is vertical, with undefined slope, since $0$ has no reciprocal.

Tip

Slopes let you classify quadrilaterals on the coordinate plane. If both pairs of opposite sides have equal slopes, the figure is a parallelogram. If, in addition, adjacent sides have slopes that multiply to $-1$ , it is a rectangle.

Practice

Practice 1

What is the slope of any line parallel to $y = -4x + 9$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the slope of any line perpendicular to $y = \dfrac{3}{5}x - 2$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

What is the slope of any line perpendicular to $4x - 3y = 12$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Classify the lines $2x + y = 7$ and $x - 2y = 4$ .

Practice 5

Line $j$ passes through $(0, 3)$ and $(4, 5)$ . Line $k$ passes through $(-1, -2)$ and $(3, 0)$ . Classify the lines.

Practice 6

The line through $(k, 2)$ and $(3, 8)$ is parallel to $y = 3x - 1$ . Find $k$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Triangle $JKL$ has vertices $J(-2, 1)$ , $K(1, 4)$ and $L(4, 1)$ . Which angle, if any, is a right angle?

Practice 8

A parallelogram is a quadrilateral with both pairs of opposite sides parallel, and a rectangle is a parallelogram with four right angles. Quadrilateral $ABCD$ has vertices $A(0, 0)$ , $B(4, 1)$ , $C(5, 5)$ and $D(1, 4)$ . Which best describes it?