# Angle of Intersection of Two Lines

Let ${l_1}$ and ${l_2}$ be two coplanar and non-parallel lines with inclination ${\alpha _1}$ and ${\alpha _2}$ respectively, as shown in the given diagram. The angle of intersection of lines ${l_1}$ and ${l_2}$ is the angle $\theta$ through which line ${l_1}$ is rotated counter-clockwise about the point of intersection so that it coincides with ${l_2}$. The angle $\theta$ is the angle of the intersection of lines ${l_1}$ and ${l_2}$ measured from ${l_2}$ to ${l_1}$. The angle $\psi$ is also the angle of intersection of lines ${l_1}$ and ${l_2}$ measured from ${l_2}$ to ${l_1}$. If the lines are not perpendicular, then one angle between them is an acute angle.

Theorem: The angle $\theta$ of the intersection of two non-vertical lines ${l_1}$ and ${l_2}$ from ${l_2}$ to ${l_1}$ is given by $\tan \theta = \frac{{{m_2} – {m_1}}}{{1 + {m_1}{m_2}}}$, where ${m_1}$ and ${m_2}$ are the slopes of lines ${l_1}$ and ${l_2}$ respectively. Proof: Let ${l_1}$ and ${l_2}$ be two coplanar and non-parallel lines with inclination ${\alpha _1}$ and ${\alpha _2}$ respectively, as shown in the given diagram. It is clear from the diagram that
$\begin{gathered} {\alpha _1} + \theta = {\alpha _2} \\ \Rightarrow \theta = {\alpha _2} – {\alpha _1} \\ \Rightarrow \tan \theta = \tan \left( {{\alpha _2} – {\alpha _1}} \right) \\ \Rightarrow \tan \theta = \frac{{\tan {\alpha _2} – \tan {\alpha _1}}}{{1 + \tan {\alpha _1}\tan {\alpha _2}}}\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Since ${\alpha _1}$ and ${\alpha _2}$ are the inclination of lines ${l_1}$ and ${l_2}$ respectively, their slopes are ${m_1} = \tan {\alpha _1}$, ${m_2} = \tan {\alpha _2}$. Putting these values of $\tan {\alpha _1}$, $\tan {\alpha _2}$ in equation (i), we have
$\Rightarrow \boxed{\tan \theta = \frac{{{m_2} – {m_1}}}{{1 + {m_1}{m_2}}}}$

If the lines ${l_1}$ and ${l_2}$ are perpendicular, then $\theta = {90^ \circ }$, and using above formula, we have
$\begin{gathered} 1 + {m_1}{m_2} = \frac{{{m_2} – {m_1}}}{{\tan \theta }} \\ \Rightarrow 1 + {m_1}{m_2} = \frac{{{m_2} – {m_1}}}{{\tan {{90}^ \circ }}} \\ \Rightarrow 1 + {m_1}{m_2} = \frac{{{m_2} – {m_1}}}{\infty }\,\,\, \Rightarrow 1 + {m_1}{m_2} = 0 \\ \end{gathered}$
$\boxed{{m_1}{m_2} = – 1}$

This is the condition for two lines to be perpendicular.