can anyone explain me what the statement in the bracket means?

Question

Euclidean Parallelism

As shown by the tick marks, lines a and b are parallel. We can prove this because the transversal t produces congruent angles.Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

Every point on line m is located exactly the same minimum distance from line l ('equidistant lines', not including the degenerate case where m = l).

i interpreted it as 'every line is not parallel to itself?
is it correct?

link to the original article:
http://en.wikipedia.org/wiki/Parallel_lines

Answer 1

I take it you are asking about "equidistant lines", which means:

0) draw lines l and m parallel to each other.

a) choose any point on line l, say the point P. Draw a perpendicular line through line l at point P.

b) This perpendicular line will intersect line m at some point, call that point Q. Measure the distance between points P and Q.

c) choose ANOTHER point on line l, call it point P1. Draw a perpendicular line through line l at point P1.

d) This perpendicular line will intersect line m at some point, call that point Q1. Measure the distance between points P1 and Q1.

e) You will find that the distance between P and Q is the same as the distance between P1 and Q1.

The degenerate case is m=l, wherein, the distance between P and Q = distance between P1 and Q1 = 0.

And no, every line IS parallel to itself, since it has the same slope as itself. Fine point of logic.

Answer 2

Yes, in Euclidean geometry both definitions are equivalent, because parallel lines lie in a plane and have no common points (parallel lines are NOT congruent!). So the 2nd definition can be edited the following way: 'Every point on line m is located exactly the same POSITIVE minimum distance from line l'. And the minimum distance between a point and a set of points is defined as a minimum distance between the given point and each of the points in the given set, so when the set is a straight line this is the length of the perpendicular, drawn from the given point to the line.
Parallel straight lines are indeed equidistant in the given sense, but that's true in Euclidean geometry only. If you have a straight line in non-Euclidean space, the points at the same positive distance from it WILL NOT lie on a straight line (i.e. imagine equidistant rail tracks, the left one straight, then the right, oddly enough, will not be a straight line!). The explanation is that the angle sum of a non-Euclidean quadrangle is always less than 360 degrees, so there are no rectangles at all in non-Euclidean geometry and how many more unexpected things are there if you know!

Answer 3

I think it's considered a "degenerate case" because while "the same line" IS a solution to the equidistant lines statement, it does NOT fit another definition: parallel lines are lines in the same plane that do not intersect.

(That definition appears near the bottom of the subsequent section of the Wikipedia article that you referenced, under "Construction").