Inclination of a Line

In mathematics, lines are modeled as having no thickness and extending infinitely in opposite directions.

Their directions are commonly described in two ways: by slope or by inclination. The slope is the ratio of rise to run between any two points on a line.

The inclination is the angle formed with the positive x-axis, measured counterclockwise.

Slope and inclination are connected through the tangent function, which relates the slope to its corresponding angle.

When two lines intersect, the angle of intersection is described as the smaller, acute angle between them.

The intersecting lines and the x-axis form a triangle. Analyzing its interior angles shows that the angle of intersection equals the difference in the lines' inclinations.

Applying a trigonometric identity gives the tangent of this angle, which can also be expressed using the slopes of the lines.

Slopes and inclinations also appear in real-world applications, such as roof pitch, which is usually defined by slope or angle and shows how steep the roof is.

Roof pitch helps water or snow slide off easily, protecting the house from damage.