# Parallel lines, parallel to itself

In a lecture the professor drew two lines, a and b, parallel to each other. The professor then asked if line a is parallel to itself.

The students said no because the definition of a parallel line is that it never cuts another line and a cuts itself all the way through.

The professor said that this is wrong, of course a is parallel to itself.

Is the professor asuming that the lines are transparent and thus do not cut each other? Are the students using the wrong definition of parallelism? Who is right, and why?

• I think lines are parallel iff the distance between them is the same at every point, that is, $a$ is parallel to $b$ whenever the distance from a given point $A$ on $a$ to the nearest point on $b$ is invariant of $A$. In case of two equal lines, this distance is always $0$, so yeah, two equal lines are parallel. Also, the tag you added is completely irrelevant to your post. – vrugtehagel Jan 28 '16 at 16:09
• The professor is using the reflexive variant of the definition of parallel lines. – N. F. Taussig Jan 28 '16 at 17:13

## 2 Answers

The student was using, perhaps unconsciously, Euclid's definition of parallel. Please see Definition 23.

The professor was using another definition of parallel, which has nicer structural properties. (Parallelism, under the professor's definition, is an equivalence relation. Also, Euclid's definition of parallelism of lines does not extend nicely to higher dimensions.)

It would be interesting to check which definition is the more widely used one in the schools.

• Thanks for your answer! This was the bioinformatics department in Uppsala. It was in the context of mathematics (classification) – Lennart Jan 28 '16 at 21:24
• You are welcome. From the "modern" point of view, allowing equality is better, since then it can be identified with same slope. But it would be wrong to say the student's point of view was wrong, indeed it is the classical point of view. – André Nicolas Jan 28 '16 at 21:31

If the minimum distance $d$ (between two straight lines) perpendicular to either line is constant, the lines are parallel. Holds even when $d=0$.