Angle Between Two Lines

When two non-parallel lines intersect, they form two pairs of equal vertically-opposite angles that are supplementary to each other; this section finds those angles directly from the two lines' slopes.

Setting up. Let two lines $L_1, L_2$ have inclinations $\theta_1,\theta_2$ and slopes $m_1=\tan\theta_1$, $m_2=\tan\theta_2$ respectively, and let $\theta$ be the angle measured from $L_1$ round to $L_2$ at their point of intersection. Since an exterior angle of the triangle formed by the two lines and the x-axis equals the sum of the two interior opposite angles, $\theta_2 = \theta_1+\theta$, i.e. $\theta=\theta_2-\theta_1$.

Derivation. Using the tangent-subtraction identity,

The two lines actually form two angles at their intersection, $\theta$ and its supplement $180^\circ-\theta$, and $\tan(180^\circ-\theta)=-\tan\theta$. So the magnitude of either angle is recovered by taking the absolute value:

— written with $m_1-m_2$ in the numerator purely by convention, since the sign is discarded by the modulus anyway. Once $\tan\theta$ is known, $\theta=\tan^{-1}\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$ gives the acute angle between the lines; the obtuse angle is its supplement, $180^\circ-\theta$.

Corollary 1 — condition for parallel lines. Two lines are parallel exactly when they never meet, i.e. the angle between them is $0^\circ$, i.e. $\tan\theta=0$. From the formula this happens exactly when the numerator vanishes:

(equivalently: two distinct lines with the same inclination never cross).

Corollary 2 — condition for perpendicular lines. Two lines are perpendicular when the angle between them is $90^\circ$, where $\tan\theta$ is undefined — which, from the formula, happens exactly when the denominator vanishes (while the numerator does not):

$L_1 \perp L_2 \iff 1+m_1m_2=0 \iff m_1m_2=-1.$ …