Mathematics · Ch 4 — Straight Lines

Slope of a Line

4.2

Slope of a Line

Inclination of a Line

When a line is drawn in a coordinate plane, it makes two angles with the x‑axis — one acute and one obtuse, which are supplementary (they add up to $180^\circ$ ). The smaller of these, measured from the positive direction of the x‑axis in the anti‑clockwise sense, is called the inclination of the line. We denote it by $\theta$ .

By definition, $0^\circ \le \theta \le 180^\circ$ .

A line parallel to the x‑axis (or coinciding with it) has inclination $0^\circ$ . A vertical line — parallel to or coinciding with the y‑axis — has inclination $90^\circ$ .

Note

Inclination is always measured anti‑clockwise from the positive x‑axis. Clockwise measurement gives the supplementary angle, which is not the inclination.

Slope (or Gradient) of a Line

Definition. If $\theta$ is the inclination of a line $l$ , then $\tan \theta$ is called the slope (or gradient) of the line $l$ . The slope is denoted by $m$ .

m = \tan \theta, \quad \theta \neq 90^\circ

When $\theta = 90^\circ$ , $\tan \theta$ is not defined. Therefore, the slope of a vertical line is not defined.

From this definition, two immediate consequences follow:

The slope of the x‑axis is $\tan 0^\circ = 0$ .
The slope of the y‑axis is not defined (since its inclination is $90^\circ$ ).

Watch out

A common mistake is to say the slope of a vertical line is "infinite" or "undefined" in a casual sense. In coordinate geometry, it is strictly not defined — you cannot assign any real number to it. Never write $m = \infty$ .

Geometric Meaning of Slope

The slope $m = \tan \theta$ tells us how steep the line is and in which direction it tilts.

If $0^\circ < \theta < 90^\circ$ , then $m > 0$ — the line rises as we move to the right.
If $90^\circ < \theta < 180^\circ$ , then $m < 0$ — the line falls as we move to the right. …

Definition 1Slope of a Line

The inclination of a line is the angle $\theta$ that the line makes with the positive direction of the $x$ -axis, measured anticlockwise. By convention, $0^\circ \le \theta \le 180^\circ$ .

Definition. If $\theta$ is the inclination of a line $l$ , then $\tan \theta$ is called the slope (or gradient) of the line $l$ . The slope is denoted by $m$ .

Thus,

m = \tan \theta, \quad \theta \neq 90^\circ.

Two special cases are included in the definition:

The slope of the $x$ -axis (or any line parallel to it) is $0$ , because its inclination is $0^\circ$ and $\tan 0^\circ = 0$ .
The slope of the $y$ -axis (or any vertical line) is not defined, because its inclination is $90^\circ$ and $\tan 90^\circ$ is undefined.

Note

A line makes two supplementary angles with the $x$ -axis. The definition picks the one measured anticlockwise from the positive $x$ -axis. …

Figure 9.2Inclination θ of a line l

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Fig. 9.2 is a simple but essential diagram: it shows a single straight line $l$ drawn through the origin $O$ of a standard $xy$ -coordinate plane. The line rises from the lower-left quadrant to the upper-right quadrant — it is not horizontal and not vertical. The $x$ -axis and $y$ -axis are drawn as full lines crossing at $O$ , with arrows at their positive ends.

The key visual element is the inclination angle $\theta$ . An arc starts at the positive $x$ -axis (the ray pointing right from $O$ ) and sweeps anticlockwise until it meets the line $l$ . This arc is labelled $\theta$ . Because the line $l$ also makes an angle with the negative $x$ -axis (the ray pointing left from $O$ ), a second arc is shown on the left side, labelled $180^\circ - \theta$ . This second arc sweeps anticlockwise from the negative $x$ -axis up to the same line $l$ . The figure thus makes it clear that a line and the $x$ -axis always form two supplementary angles: $\theta$ and $180^\circ - \theta$ .

Note

The figure deliberately shows both angles to reinforce that the inclination is defined as the smaller (or the one measured from the positive direction) — never the larger one. The line itself is the same; only the starting ray changes.

What the diagram teaches is that every non-vertical line has a unique inclination $\theta$ between $0^\circ$ and $180^\circ$ , measured anticlockwise from the positive $x$ -axis. A horizontal line has $\theta = 0^\circ$ ; a vertical line has $\theta = 90^\circ$ . The slope $m$ is then defined as the tangent of this angle:

m = \tan \theta, \quad \theta \neq 90^\circ

Here $m$ is the slope (or gradient) of the line, and $\theta$ is its inclination. The condition $\theta \neq 90^\circ$ is critical: $\tan 90^\circ$ is undefined, so the slope of a vertical line does not exist. For $\theta = 0^\circ$ , $\tan 0^\circ = 0$ , so the slope of the $x$ -axis (or any horizontal line) is zero.

Watch out

A common mistake is to think the slope is $\tan(180^\circ - \theta)$ instead of $\tan \theta$ . The figure's two arcs show that $\theta$ and $180^\circ - \theta$ are supplementary, and $\tan(180^\circ - \theta) = -\tan \theta$ . The slope uses the anticlockwise angle from the positive $x$ -axis — that is $\theta$ , not its supplement. …

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