Find the acute angle between the pair of lines $\;$ $2x + y + 4 = 0$ and $y - 3x = 7$
Let $\theta$ be the angle between the lines $\;$ $2x + y + 4 = 0$ $\;\;\; \cdots \; (1)$ $\;$ and
$y - 3x = 7$ $\;\;\; \cdots \; (2)$
Equation $(1)$ can be written as $\;$ $y = -2x + 4$
$\therefore \;$ Slope of equation $(1)$ is $\;$ $m_1 = -2$
Equation $(2)$ can be written as $\;$ $y = 3x + 7$
$\therefore \;$ Slope of equation $(2)$ is $\;$ $m_2 = 3$
Now, $\;$ $\theta = \tan^{-1} \left(\dfrac{m_1 - m_2}{1 + m_1 m_2}\right)$
i.e. $\;$ $\theta = \tan^{-1} \left(\dfrac{-2 - 3}{1 + \left(-2\right) \times 3}\right)$
i.e. $\;$ $\theta = \tan^{-1} \left(\dfrac{-5}{-5}\right)$
i.e. $\;$ $\theta = \tan^{-1} \left(1\right) = 45^\circ$