Angle between Straight Lines in Plane

Theorem
Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given by the equations:

Then the angle $\psi$ between $L_1$ and $L_2$ is given by:


 * $\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$

Proof

 * Angle-between-Straight-Lines.png

Let $\psi_1$ and $\psi_2$ be the angles that $L_1$ and $L_2$ make with the $x$-axis respectively.

Then by the definition of slope:

and so:

Also presented as
Some sources retain the form:
 * $\tan \psi = \dfrac {\tan \psi_2 - \tan \psi_1} {1 + \tan \psi_1 \tan \psi_2}$