(I know there are previous posts here on finding the parameters of a triangle, but those questions are based on pure geometry, while mine is based on coordinate geometry and slopes, so I guess asking a separate question is okay.)
Apparently, if $m_1$, $m_2$, and $m_3$ are the slopes of three lines $L_1=0$, $L_2=0$, and $L_3=0$, where $m_1\gt m_2\gt m_3$, then the interior angles of the triangle, say $\Delta ABC$, formed by the lines are given by; $$\tan{A} =\frac{m_1-m_2}{1+m_1m_2}\\\tan{B} =\frac{m_2-m_3}{1+m_2m_3} \\\tan{C} =\frac{m_3-m_1}{1+m_1m_3} $$ How is it possible to find the correct interior angles this way in every case? As I understand it, each of the tan values could represent either the acute or the obtuse angles between the lines, but this theorem has the slopes in the numerators subtracted in just the right way so as to somehow produce the correct sign for the tangents of the angles formed.
All I could figure out was that if you subtracted the smaller slope from the larger in this type of difference-of-angles tan equation, you would get:
A positive value for two lines whose slopes are both negative or both positive, and
If the lines have opposite signs for their slopes, a positive value if the acute angle between the lines is horizontal, and a negative value if the acute angle opens up vertically.
If correct, I suspect that can be used to prove the result, but I'm not sure how. Could someone help me prove the theorem? (The proof doesn't necessarily have to incorporate my attempt if it's cumbersome or if there's a better way; apart from helping with the proof, verifying if my attempt was correct would be enough.)