Difference between parallel and Equal lines

Question

Basically I want to know that when did a pair of parallel lines become equal.

And with the above please tell me the difference between the two

Answer 1

Lines can not be equal because they are not numbers. I think there are same lines.

Two lines defined parallel if they are the same or (if our were placed in the plain) they have no common points.

We need it for the following important property:

If $a||b$ and $b||c$ then $a||c$.

If $a\equiv b$, but $a\not||b$ we obtain that the following is wrong.

$a||b$ and $b||a$ then $a||a$.

If so it's wrong. I don't like it!

Answer 2

Two lines are equal if there are two distinct points contained in both of them. The difference between equal lines and parallel lines is that equal lines consist of precisely the same points, whereas parallel lines have no points in common.

Answer 3

Line are not numbers. They can't be equal. Their value may be equal. Two lines are parallel if they does not meet in any point (when extended). But two lines whose lengths are equal may meet in a point.

Answer 4

I think what you have in mind is, given two lines (say in $\mathbb{R^2}$) that are parallel, how can one describe the point in a sequence of transformations when they coincide.

If I understood your request, then you can define parallel lines (only in this instance) as two lines with the same perpendicular distance between them at every point on one of them. Then they coincide when the distance between them is $0$.