通过平移得到平行线

Parallel lines: constant distance & translation

Non-parallel lines intersect: the distance between the lines becomes 0 at one point.

This never happens for parallel lines. The distance between the lines stays constant, and they never intersect.

In other words, you can translate (move) either line by that constant distance, in some direction, to get the other line. It's just like how when you change the $y$-intercept but keep the slope the same, you get parallel lines.

Graphing Parallel lines & Translations

Consider line $ML$ and line $M'L'$. Line $M'L'$ is simply a parallel translation of line $ML$. In this case, $h$ and $k$ are come arbitrary constants addd to the points $x, y$.

Two lines are parallel if (and only if) they are translates of each other; in other words, if we can translate ("slide") one line onto the other.

For example, the red and blue lines are parallel, because we can horizontally translate the red line $ABC$ (to the right) onto the blue line $XYZ$ (and of course, by going backwards to the left, we can translate $XYZ$ onto $ABC$). Note that $A$ is sent to $X$ (abbreviated $A\to X$), $B\to Y$, and $C\to Z$.

(But this is not the only way to translate the red line onto the blue line! We could slide downward instead, taking $A\to U$, $B\to V$, $C\to W$.)

In notation, we might write $ABC \parallel XYZ$ (note the symmetry of $\parallel$) or $ZX \parallel AB$ or $\text{red line}\parallel\text{blue line}$, etc.

Indeed, since the two lines meet at a point $B$ (we can write $B\in AC$ and $B\in BV$), there's no way to translate one onto the other. For example, we can argue by contrapositive. If there were some translation $T$ taking $AC$ to $BV$, then the translated points $T(A)$, $T(B)$, $T(C)$ would all have to lie on line $BV$. In particular, the direction of the translation would have to be along line $BV$: in order for $B\to T(B)$ to still on line $BV$. But $A$ doesn't lie on line $BV$, so there's no way $A\to T(A)$ (which goes in the same direction as $B\to T(B)$) could lie on $BV$! (This is not too hard to visualize: for example, if you slide $B$ along line $BV$ to, say, the point $V$, then $A$ would go to $U$, which is not on line $BV$.)

If you're picky about rigor, then really we should say something like since $T$ takes $B$ to a point on line $BV$, $T$ takes all points on line $BV$ to other points on line $BV$. So the inverse translation $T^{-1}$ also takes points on line $BV$ to other points on line $BV$. So if we had $T(A) \in BV$, then $T^{-1}(T(A)) = A$ would have to lie on $BV$ as well. But $A$ doesn't lie on $BV$, so we have a contradiction. Hence $T(A)$ does not lie on $BV$.

Warning: This way of describing parallel lines is not a standard definition! But it can be viewed as an "alternative" definition with an advantage: it's intuitive ("hands-on"), and it holds in all dimensions. (If you want some intuition in 3D, pretend two of your pencils are lines, and try translating them around.)

The usual definition is

Two lines in the plane are parallel if they do not intersect.

(Food for thought: Does this second definition also work in 3D? No, since we have skew lines in 3D!)

But instead of being sad about this, we should instead rejoice, because we have two equivalent/alternative definitions of parallel lines (at least, in a plane). In other words, we can combine these definitions to give a useful fact:

Let $\ell_1,\ell_2$ be two lines in the plane. The following are equivalent (if one of them holds, so does the other):

1. $\ell_1,\ell_2$ are translates of each other. (Intuitively, we can slide one onto the other, which intuitively means the lines should always be "the same distance apart", and cannot ever meet/intersect.)

2. $\ell_1,\ell_2$ do not intersect. (Intuitively, their points are all "the same distance" apart, which is intuitively why we can slide one onto the other.)

When these equivalent conditions hold, we say $\ell_1$ and $\ell_2$ are parallel (or $\ell_1 \parallel \ell_2$ for short).

Parallel lines are coplanar lines that do not intersect Because parallel lines do not intersect, we say they have the same slope.

In the image above, lines $x$ and $y$ lie in the same plane and do not intersect. Lines $x$ and $y$ are parallel.

The distance between the parallel lines is always constant no matter what points you are located at.

Looking at the image, we see that the distance between any two points on lines $x$ and $y$ is $3$.

When you translate parallel lines, you simply move them around without changing other properties.

In the figure above, we have lines $x$ and $y$ translated $3$ units apart from each other. The distance between the two lines still remains constant, however no other properties have changed.