# Angle between Straight Lines in Plane

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## 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.

Then from Slope of Straight Line is Tangent of Angle with Horizontal:

\(\displaystyle \tan \psi_1\) | \(=\) | \(\displaystyle m_1\) | |||||||||||

\(\displaystyle \tan \psi_2\) | \(=\) | \(\displaystyle m_2\) |

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 $m_2$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.9$: Angle $\psi$ between Two Lines having Slopes $m_1$ and $m_2$