Angle Between Two Straight Lines

Two straight lines in a plane are either parallel, coincident, or intersecting at exactly one point; when they intersect, they form two angles at that point — one acute (or a right angle) and its supplement, obtuse — which always add to $180^\circ$. By convention, 'the angle between two lines' means the acute angle between them (or the right angle, if the lines are perpendicular).

Let $y=m_1x+c_1$ and $y=m_2x+c_2$ have inclinations $\theta_1,\theta_2$ with the $x$-axis, so $m_1=\tan\theta_1,\ m_2=\tan\theta_2$. If $\varphi$ is the angle between the lines, then $\varphi=\theta_2-\theta_1$, and applying the tangent-difference identity,

$\tan\varphi=\tan(\theta_2-\theta_1)=\frac{\tan\theta_2-\tan\theta_1}{1+\tan\theta_1\tan\theta_2}=\frac{m_2-m_1}{1+m_1m_2} \quad\Longrightarrow\quad \varphi=\tan^{-1}\left(\frac{m_2-m_1}{1+m_1m_2}\right).$ …