The Relationship Between the Angle of Inclination and Slope

Angle of inclination. For a straight line, the angle of inclination $\theta$ is the angle the line makes with the $x$-axis, measured in the counter-clockwise (positive) direction from the positive $x$-axis.

Slope (gradient). The slope of a line, usually denoted $m$, is a single number measuring the line's direction and steepness. It can be computed in any of three equivalent ways:

1. From the inclination: $m=\tan\theta$. When $\theta=\pi/2$ (a vertical line), $\tan(\pi/2)$ is undefined, so a vertical line has no (undefined) slope.
2. From two points $(x_1,y_1),(x_2,y_2)$ on the line with $x_2\ne x_1$: $m=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}$ — the change in $y$ divided by the corresponding change in $x$ ('rise over run').
3. From the general form $ax+by+c=0$: $m=-\dfrac{a}{b}$ ($b\ne0$; $m$ is undefined when $b=0$, i.e. for a vertical line $x=$ const.).

The slope can be positive (line rises left-to-right, $0\le\theta<\pi/2$), negative (line falls left-to-right, $\pi/2<\theta<\pi$), zero (horizontal, $\theta=0$), or undefined (vertical, $\theta=\pi/2$). …