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Inclination & Angle Between Two Lines

The angle of inclination

\theta

of a line is the angle it makes with the positive x-axis, measured anti-clockwise, where

0° \le \theta < 180°

Theorem 1📌 Learn for exam

Gradient from Inclination

m = \tan\theta

If $m > 0$ , $\theta$ is acute ( $0° < \theta < 90°$ ). If $m < 0$ , $\theta$ is obtuse ( $90° < \theta < 180°$ ) — take the calculator's $\tan^{-1}$ answer and add 180° to land in the correct range.

▸ Two Lines — drag each handlewatch θ₁, θ₂ and the angle between them update

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Worked Example

Inclination from a gradient

Find the angle of inclination of a line with gradient $m = 2$ .

Worked Example

Inclination from two points

Find the angle of inclination of the line through $A(-2,3)$ and $B(4,-5)$ .

Theorem 2📌 Learn for exam

Angle Between Two Lines

\tan\alpha = \left|\frac{m_1 - m_2}{1+m_1m_2}\right|

This gives the acute angle between two lines directly from their gradients — no need to find each inclination angle first.

Worked Example

Angle between two lines

Find the angle between the lines $y=3x+1$ and $y=-x+5$ .

Tip

A vertical line has undefined gradient and inclination exactly 90°. If one of your two lines is vertical, use

\theta_1 - \theta_2

directly instead of the tanα formula (which breaks when

m

is undefined).

Analytical Geometry

Inclination & Angle Between Two Lines

Inclination from a gradient

Inclination from two points

Angle between two lines