Angle between Intersecting Lines

5.2.4 Angle between Intersecting Lines

Having handled the perpendicular case, we now find the acute angle between two lines that intersect but are not perpendicular.

Theorem. If $\theta$ is the acute angle between non-vertical lines with slopes $m_1$ and $m_2$, then

Proof. Let $\alpha,\beta$ be the inclinations of the two lines, so $\tan\alpha=m_1$, $\tan\beta=m_2$, with $\alpha,\beta \ne 90^\circ$. From the geometry, the angle between the lines is either $\theta = \beta-\alpha$ or $\theta = \pi - (\beta-\alpha)$, depending on configuration. So $\tan\theta = \tan(\beta-\alpha)$ or $\tan\theta = \tan\{\pi-(\beta-\alpha)\} = -\tan(\beta-\alpha)$. Either way,

Since $\theta$ is specifically the acute angle, the lines are not perpendicular here, so $m_1m_2 \ne -1$, i.e. $1+m_1m_2 \ne 0$, and the formula is always well-defined. $\blacksquare$

Worked Example 1. Find the acute angle between lines with slopes $3$ and $-2$. $\tan\theta = \left|\dfrac{3-(-2)}{1+3(-2)}\right| = \left|\dfrac{5}{-5}\right| = 1$, so $\theta = 45^\circ$. …