Question:

The line through the points $\displaystyle (4,3)$ and $\displaystyle (-6,0)$ intersects the line through $\displaystyle (0,0)$ and $\displaystyle (-1,5)$. Find the angles of intersection.

Attempt:

Slope of line: $\displaystyle (4,3)$ and $\displaystyle (-6,0)$

$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0-3}{-6-4} = \frac{3}{10}$

Slope of line: $\displaystyle (0,0)$ and $\displaystyle (-1,5)$

$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5-0}{-1-0} = -5$

Angles of Intersection:

$\displaystyle m_1 = \frac{3}{10}$ , $\displaystyle m_2 = -5$

$\displaystyle \tan\phi = \frac{m_1 + m_2}{1 + m_1\cdot m_2} = \frac{\left(\frac{3}{10}\right) + (-5)}{1 + \left(\frac{3}{10}\right)\cdot(-5)} = \frac{47}{5}$

$\displaystyle \tan\phi = \frac{47}{5}$

$\displaystyle \phi = \tan^{-1}\left(\frac{47}{5}\right)$

$\displaystyle \phi = 84^o$

$\displaystyle \psi = 180^o - 84^o$

$\displaystyle \psi = 96^o$

So, the Angles of Intersection are $\displaystyle 84^o$ and $\displaystyle 96^o$. Am I right?