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.

Writing it down

Mathematicians have a short symbol for "is parallel to". If line a is parallel to line b, we write:

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.

Parallel lines all around you

Once you start looking, parallel lines are everywhere:

• the two rails of a train track, or the lanes painted on a road;
• the ruled blue lines on a sheet of writing paper;
• the opposite edges of a window, a door, a book, or a phone screen;
• the shelves of a bookcase, stacked one above another.

In each case two (or more) lines keep a constant gap and never run into one another.

When lines are not parallel

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 90^\circ. Lines that cross at a right angle are called perpendicular. So "intersecting" is the general case (they meet at some angle), and "perpendicular" is the special case (they meet at exactly 90^\circ).

The key property

Two facts make parallel lines so powerful:

• They stay the same distance apart. Measure straight across from one line to the other anywhere along their length and you always get the same gap.
• A crossing line reveals tidy angles. When a third line cuts across two parallel lines, the angles it forms come out in neat, equal patterns — exactly the same at both crossings.

Drag the slider to swing the crossing line. However you tilt it, the matching angles \theta at the two crossings stay equal — that is the tidy pattern.

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.

Test yourself