# 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:

 $\displaystyle L_1: \ \$ $\displaystyle y$ $=$ $\displaystyle m_1 x + c_1$ $\displaystyle L_2: \ \$ $\displaystyle y$ $=$ $\displaystyle m_2 x + c_2$

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

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

 $\displaystyle \tan \psi_1$ $=$ $\displaystyle m_1$ $\displaystyle \tan \psi_2$ $=$ $\displaystyle m_2$

and so:

 $\displaystyle \tan \psi$ $=$ $\displaystyle \map \tan {\psi_2 - \psi_1}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\tan \psi_2 - \tan \psi_1} {1 + \tan \psi_1 \tan \psi_2}$ Tangent of Difference $\displaystyle$ $=$ $\displaystyle \dfrac {m_2 - m_1} {1 + m_1 m_2}$ Definition of $m_1$ and $m_2$

$\blacksquare$