Straight Lines - Slope of a Line

The inclination of a line is the angle $\theta$ made by the line with the positive direction of the x-axis measured in the anti-clockwise direction. Visually, if the line tilts to the right, $\theta$ is acute ($0 < \theta < 90^{\circ}$), and if it tilts to the left, $\theta$ is obtuse ($90^{\circ} < \theta < 180^{\circ}$).

The slope or gradient of a non-vertical line is denoted by $m$ and is defined as $m = \tan \theta$, where $\theta$ is the inclination. For a horizontal line, $\theta = 0^{\circ}$ so $m = 0$; for a vertical line, $\theta = 90^{\circ}$ and the slope is undefined.

The slope of a line passing through two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as the ratio of the vertical change (rise) to the horizontal change (run). On a graph, this is represented by the steepness of the segment connecting the two points.

Two non-vertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$). Visually, parallel lines run in the same direction and never meet, regardless of how far they are extended.

Two non-vertical lines are perpendicular if and only if the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$). Geometrically, this means the lines intersect at a right angle ($90^{\circ}$).

Three points $A, B,$ and $C$ are collinear (lie on the same straight line) if the slope of segment $AB$ is equal to the slope of segment $BC$. If you plot these points, they will form a single continuous path without any bends.

The acute angle $\theta$ between two intersecting lines with slopes $m_1$ and $m_2$ is determined by the relative difference in their steepness. This relationship is captured using the tangent of the angle between them.