angle between lines

Question

The angle between two lines is given by

$ \tan(\theta) = \big|\frac{m_2-m_1}{1+m_1m_2}\big| $

where $m_1$ and $m_2$ are the slopes of the two lines in question.

What is confusing me is the reverse problem. When we try to find the slope of the lines making an acute angle with a line of given slope, say a line of slope $m_1$ is given to us and we are to find the values of $m_2$ making angle $\theta$, we can deduce the two slope values of the two lines from this very formula. But doesn't this formula calculate two angles for the same pair of lines? In short, I would like to know how to interpret the solutions of the above equation for $m_2$ graphically.

Answer 1

Yes, you're correct in seeing that there are two angles. They are $\theta$ and $\pi - \theta$. This is because, $tan(\pi-\theta)=-tan(\theta)$

If you remember,the angle $\pi$ corresponds to 180$^{\circ}$.

So, in the image below, the angles $\angle e$ and $\angle f$ are both the angles between the lines, and $\angle e$ = $180^{\circ} - \angle f$ = $tan(\pi) - \angle f$

Answer 2

Where I'm from, the convention for angles between two undirected lines refers to the acute angle between them.

The negative angle you find using a calculator should be between $-90^\circ$ and $0^\circ$.

One way to think about it is instead of this range, you have an angle $\theta$ which corresponds to an angle between $90^\circ$ and $180^\circ$. Then one of your angles is acute and one obtuse (assuming your lines are not perpendicular, of course), so you take the acute angle by convention.