Some lines, no matter how far you follow them, never cross. Two straight lines that run in exactly the same direction and stay forever apart are called parallel. They are one of the most useful ideas in all of geometry: once you know two lines are parallel, a whole family of angle facts falls into place.
Picture the two rails of a railway track. They run side by side, always the same distance apart, never meeting — even as they race off to the horizon. That is exactly what "parallel" means.
In the everyday geometry of a flat page, two parallel lines never meet — however far you extend them. But draw two lines of longitude on a globe: both cross the equator at a perfect right angle, so near the equator they look perfectly parallel. Now follow them north… they crash straight into each other at the North Pole (and again at the South Pole)! On a curved surface the familiar rule breaks down.
This is your very first glimpse of non-Euclidean geometry — the strange curved-space mathematics that, centuries later, Einstein needed to describe gravity.
Mathematicians have a short symbol for "is parallel to". If line
On a diagram you can't always tell parallel lines apart from ones that only look parallel, so we mark them: a matching pair of little arrowheads (or single/double ticks) is drawn on each line to say "these two are parallel". When you see the same arrow mark on two lines, you may take it as given that they never meet.
Once you start looking, parallel lines are everywhere:
In each case two (or more) lines keep a constant gap and never run into one another.
Two straight lines that are not parallel must eventually meet — if not on your page, then somewhere beyond its edge if you kept drawing. The point where they cross is called the point of intersection, and we say the lines intersect.
There is one especially important way for lines to meet: at a perfect square corner —
a right angle of
Two facts make parallel lines so powerful:
Drag the slider to swing the crossing line. However you tilt it, the matching angles
That second fact is the doorway to a whole set of angle theorems. The Alternate Interior Angles Theorem is the first one we'll meet, and it lives in the very next concept up the tree.
The neat angle facts for parallel lines — the "F" angles (corresponding), the "Z" angles (alternate) and the "C" angles (co-interior) — only hold when the two lines really are parallel. If the lines are not parallel, those rules simply do not apply, and using them gives wrong answers.
So before you reach for any of them, you must be told the lines are parallel, or see the matching arrow marks on the diagram. Assuming two lines are parallel just because they happen to look it is one of the most common exam mistakes of all.