Hi Ben,

We got two responses to your question.

All your answers can be justified. It depends on how you define the angle between two lines -- one definition insists that the lines intersect in a single point. Another definition defines it as the angle between direction vectors that are parallel to the line -- this definition not only allows for parallel lines, but also for lines in three dimensions that are not in the same plane; it also allows for directed angles (that is, angles that can take on any value from minus infinity to infinity). In summary, you can say that the angle between parallel lines is undefined, or it can be either 0 or 180 degrees, or any multiple of 180 degrees.

Chris

This is an interesting question. There are many definitions of Angle on the internet, and the one you are using is reasonable.

First notice that for two intersecting lines, there are two angles which are defined (actually four, but pairs of 'opposite angles' are equal). So it is reasonable to have two answers.

Now for two equal lines, there are again two angles defined: 0 and 180. This is a form of limiting or asymptotic behaviour: as the angle of intersection of two lines goes to 0 (or 180 degrees), the two lines coincide. Sometimes people put the 'directions' of the lines (as arrows of length 1) out from the origin to the unit circle, and look at that angle. For the two coinciding lines, these two directions (vectors if you have seen that word) are either pointing to the same spot (angle 0) or pointing in opposite directions (angle 180 degrees).

For two parallel lines I would say you have a choice. If the definition you want to use is about intersecting lines, then there is no angle. However, what your friends say is a good and useful extension of the concept. Think about the 'directions' as arrows out from the center of the circle. Again the two directions either coincide or point in opposite directions. In terms of the limiting behaviour, you can think of the two lines as each pivoting about some fixed point (a different one for each line). Now as you rotate one of them, the point of intersection moves further and further away, and the angle becomes smaller and smaller (or larger and larger if you look at the other angle). In the end, the angle becomes either 0 or 180 degrees (depending on which one you are watching).

Notice as the angle goes to 0, the point of intersection also goes off the plane 'to infinity'. So geometers do talk about the lines 'intersecting at infinity'. If you look at a drawing or picture of a straight railway track, it 'looks like' the two parallel tracks intersect at the horizon, with a very small angle. This is the image which fits here. The horizon is 'at infinity' and the angle has been distorted a bit, but is becoming 0.

It is nice to know that the usual ways of talking about angles can, if you choose, be extended to these extreme situations. Something similar can also happen for the 'angle between two planes' in space.

In case you find this to be obscure, and not likely to be of any practical use, I can mention that this type of geometry is a good language for talking about things like whether a bridge is rigid or is in danger of falling down. At least every one used it a century ago (even engineers). Unfortunately, these days most people do not study this geometry (in North America) so many people have trouble talking carefully about these types of problems. They end up with sentences like - if the lines intersect then, if they are parallel then .... .
It is so much simpler to talk, and to think, when you can include both situations in a single sentence and a single kind of thinking.

Walter