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Two lines that intersect in the plane form two angles: one acute (between $0$ and $\pi/2$, inclusive) and another obtuse (between $\pi/2$ and $\pi$, inclusive).

When we speak of "the angle between two lines", is there a standard convention for which of the above two we're referring to?

I think it is (or "should" be) the acute angle, but can't seem to find any definitive reference for this.

So far the closest I've found is this passing remark by Camille:

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

  • In general the angle between two lines is $\alpha\in[0,\pi)$ – Martín Vacas Vignolo Nov 03 '21 at 02:34
  • Acute means between $0$ and $\pi/2$. – markvs Nov 03 '21 at 02:40
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    It's fair to establish a convention that we choose the angle $[0, \tfrac{\pi}{2}]$ by default, but it would be best to name that convention in whatever context you're working with these angles. – Sammy Black Nov 03 '21 at 02:40
  • @markvs: Oops thanks correected –  Nov 03 '21 at 02:44
  • When we have a line in a Cartesian plane with $x,y$ coordinates, we often speak of the angle the line makes with the $x$ axis, and we don't mean for this angle value to be the same for the lines $y=x$ and $y=-x$. That's not exactly an "angle between two lines" but I think it is close enough to cause confusion if you're not careful. – David K Nov 03 '21 at 02:47
  • There is no one preferred angle between two intersecting lines. Two such lines form two angles which sum up to $\pi$. Otherwise it would be hard to define obtuse (acute) triangles. – markvs Nov 03 '21 at 02:51
  • @markvs: Don't three points define any triangle? –  Nov 03 '21 at 02:56
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    Although it is more of an English convention than a mathematical one, when someone says "the thing" but there are multiple things, it means "the thing that is relevant to the conversation". – DanielV Nov 03 '21 at 03:02
  • @user24096:I was writing about obtuse (acute) triangles. But also internal (external) angles of a triangle are angles between lines containing its sides (two for each vertex). – markvs Nov 03 '21 at 03:06
  • For what it's worth, here and here is my position on the matter. – ryang Nov 03 '21 at 04:32
  • @DavidK To be fair, in the context of "the angle that $y=-x$ makes with the $x$-axis", the $x$-axis is being considered a directed line, so this example is actually closer to 'angle between vectors' than to 'angle between lines', and thus the answer is understood to be $135º$ rather than acute. – ryang Nov 03 '21 at 04:38
  • @ryang There is some kind of directionality implied, yes, but it's implicit. That's the potentially confusing part. On the other hand, looking at the original problem statement for your first example, it says "the equation of the straight line", which is a strange way to say "the two equations of the two straight lines" that satisfy the other conditions of the problem. – David K Nov 03 '21 at 13:01
  • I think I've sometimes seen "the acute angle between two lines", which seems like a good way to disambiguate the statement at the cost of just one extra word (although if a right angle is a possibility I might try "non-obtuse" even though it sounds clumsier than "acute"). – David K Nov 03 '21 at 13:05

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It's more common to speak of the angle between two vectors by using, for instance, the dot product formula: $$ \cos \theta = \frac{\vec{v} \cdot \vec{w}}{\lvert\vec{v}\rvert \, \lvert\vec{w}\rvert}. $$ Of course, every vector generates a line by taking its span (all possible scaled copies of it) $$ \operatorname{span} \{\vec{v}\} = \{ c\vec{v} \mid c \in \Bbb{R} \}, $$ but we lose the direction information that the vector had: $$ \operatorname{span} \{-\vec{v}\} = \operatorname{span} \{\vec{v}\} $$

As the dot product formula shows, if you negate either of the vectors, the cosine is negated, so the supplementary angle $\pi - \theta$ gives the angle between them.

Sammy Black
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    The question was about angles between two lines. It goes back to at least Euclid. You are saying that it is better to ask a different question. Note that there are very many different dot products even in $\mathbb R^2$. – markvs Nov 03 '21 at 02:55