TITLE: How would we know which slope to assign $m_1$ and to which $m_2$, while finding angle between two non vertical lines?
QUESTION [0 upvotes]: If we have to compute the angle between two lines when their slopes are given, then for using the formula $\tan θ=\dfrac{m_2-m_1}{1+m_2\cdot m_1}$, how would we know which slope to assign $m_1$ and to which $m_2$?
In my textbook question was given like: find the angle from the line with slope $-\frac{7}{3}$ to the line $\frac{5}{2}$ so I can't figure out which one of them should be $m_1$ and which should be $m_2 $ ?

REPLY [0 votes]: The angle between two lines of slope $m_1$ and $m_2$ is $$\tan \theta = \left|\dfrac{ m_1 - m_2}{1+m_1m_2} \right|$$
Now if $~\frac{ m_1 - m_2}{1+m_1m_2}~$ is positive then the angle between the lines is acute and if $~\frac{ m_1 - m_2}{1+m_1m_2}~$  is negative then the angle between the lines is obtuse.
Geometrically, the two different angles represent the same situation with respect to the lines. For example if $~m_1 = 1/2~$ and $~m_2 = -1/3~,$ then $~\tan θ~$ comes out to be $~1~,$ which means $~θ = 45^\circ~.$ But if we take $~m_1= -1/3~$ and $~m_2= 1/2~,$ then $~\tan θ~$ would come out to be $~-1~,$ which means $~θ = 135^\circ~.$ For more understandable about the fact take a look at the next figure,

So, we needn’t worry about this issue much. But to keep things clean, we use a modulus sign on the formula of $~\tan\theta~$ i.e., the formula for the angle between two lines.
{"set_name": "stack_exchange", "score": 0, "question_id": 4176130}