Slope from Inclination

Slope in terms of the angle θ the line makes with the positive x-axis. θ ∈ [0°, 180°), θ ≠ 90°.

The inclination of a line is the angle $\theta$ it makes with the positive direction of the $x$-axis, measured anticlockwise.

From the definition of slope as the tangent of this angle:

$$\boxed{m = \tan\theta}$$

Domain restriction

$\theta = 90°$ gives a vertical line, for which $\tan 90°$ is undefined — consistent with vertical lines having no slope.

For $\theta \in [0°, 90°)$: $m \geq 0$ (line rises left to right or is horizontal).

For $\theta \in (90°, 180°)$: $m < 0$ (line falls left to right).

Consistency with the two-point formula

If $P(x_1,y_1)$ and $Q(x_2,y_2)$ lie on a line of inclination $\theta$, then from the right triangle formed:

$$\tan\theta = \frac{y_2 - y_1}{x_2 - x_1} = m$$

Both definitions are equivalent.