Slope and the angle between lines

What you will be able to do

• find the slope of a line from two of its points;
• link the slope to the angle of inclination, with $m = \tan\theta$;
• see why a vertical line has no slope;
• find the angle between two lines from their slopes.

How steep is a line?

A ramp rises $1\ \text{m}$ for every $4\ \text{m}$ of ground it crosses. Its steepness is $\dfrac{1}{4}$. The same idea measures any straight line on the coordinate plane.

Take two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ on a line, with $x_1 \neq x_2$. Going from $P$ to $Q$ the line rises by $y_2 - y_1$ while it runs across by $x_2 - x_1$.

The slope of the line is $m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{rise}}{\text{run}}$.

The slope is the same for any two points of the line. Triangles built on the line in this way are all similar, so their sides keep the same ratio. If you swap $P$ and $Q$, both the rise and the run change sign, and the slope stays the same.

• A line that climbs to the right has a positive slope.
• A line that falls to the right has a negative slope.
• A horizontal line has rise $0$, so its slope is $0$.

The angle of inclination

The angle of inclination $\theta$ of a line is the angle from the positive $x$-axis, turning anticlockwise, to the line. It lies in the range $0^\circ \leq \theta < 180^\circ$.

In the right triangle of the figure, $\tan\theta = \dfrac{\text{rise}}{\text{run}}$. So

A few values are worth knowing. A line at $45^\circ$ has $m = 1$. A line at $60^\circ$ has $m = \sqrt{3}$. A line at $30^\circ$ has $m = \dfrac{1}{\sqrt{3}}$. A line at $135^\circ$ has $m = -1$, because $\tan 135^\circ = -\tan 45^\circ$.

An acute angle gives a positive slope, and an obtuse angle gives a negative slope. To go back from a negative slope, find the acute angle for $|m|$ and subtract it from $180^\circ$.

A vertical line has $\theta = 90^\circ$. Its run is $0$, so the slope would need a division by zero, and $\tan 90^\circ$ is not defined. We say a vertical line has no slope. Its equation is of the form $x = a$.

The angle between two lines

Two lines cross at a point and make two angles, $\theta$ and $180^\circ - \theta$. Let the lines have slopes $m_1$ and $m_2$ and inclinations $\theta_1$ and $\theta_2$, with $\theta_2 > \theta_1$.

The angle between the lines is $\theta = \theta_2 - \theta_1$. Use the tangent of a difference:

The result is positive for the acute angle and negative for the obtuse one. To ask only for the acute angle, take the absolute value:

Two special cases come out of the formula. If $m_1 = m_2$, then $\tan\theta = 0$ and the lines are parallel. If $1 + m_1 m_2 = 0$, the denominator is $0$ and $\theta = 90^\circ$, so the lines are perpendicular. The next lesson uses both.

A worked example

A line through two points. Find the slope and the angle of inclination of the line through $A(1, 5)$ and $B(4, -1)$.

1. The slope. $m = \dfrac{-1 - 5}{4 - 1} = \dfrac{-6}{3} = -2$.
2. The acute angle. $\tan\alpha = 2$ gives $\alpha = 63.4^\circ$.
3. The inclination. The slope is negative, so $\theta = 180^\circ - 63.4^\circ = 116.6^\circ$.

The angle between two lines. Find the acute angle between lines of slopes $2$ and $5$.

1. The formula. $\tan\theta = \left|\dfrac{5 - 2}{1 + 2 \times 5}\right| = \dfrac{3}{11}$.
2. The angle. $\theta = 15.3^\circ$.

Practise with new numbers

Try it in Python

The program finds the slope and the angle of inclination of a line from two points, and the angle between two slopes. It returns None for a vertical line.

Common mistakes

• Dividing the run by the rise. The slope is $\dfrac{y_2 - y_1}{x_2 - x_1}$, with the $y$ values on top.
• Mixing the order. Subtract the points in the same order on the top and the bottom.
• Saying a vertical line has slope $0$. A horizontal line has slope $0$. A vertical line has no slope.
• Using $\theta = \arctan m$ for a negative slope. That gives a negative angle. Add $180^\circ$ for the angle of inclination.
• Forgetting the absolute value. Without it, the formula may give the obtuse angle.

Remember these

• $m = \dfrac{y_2 - y_1}{x_2 - x_1} = \tan\theta$, where $\theta$ is the angle of inclination.
• A positive slope means an acute angle, a negative slope an obtuse angle, and a vertical line has no slope.
• The acute angle between two lines has $\tan\theta = \left|\dfrac{m_2 - m_1}{1 + m_1 m_2}\right|$.
• The obtuse angle between them is $180^\circ - \theta$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the slope of the line through $(-1, 6)$ and $(3, -2)$.

2. What angle does a line of slope $\sqrt{3}$ make with the positive $x$-axis?

3. A line has slope $-1$. Find its angle of inclination.

4. Why does a vertical line have no slope?

5. Find the acute angle between the $x$-axis and the line of slope $1$.