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Parallel and perpendicular lines

Geometry · level 3, lesson 3 of 6

A question to start

Look at a train track. The two rails run side by side for hundreds of kilometres, over hills and round long bends, and they never touch. If they came even a little closer together, the train would fall off.

Now stand between the rails and look far down the track. The rails seem to meet in a point in the distance! Do they really meet? And if not, why do they look as if they do?

The idea

Lines that go on forever

In geometry, a line is perfectly straight, has no thickness, and goes on forever in both directions. We can only ever draw a piece of it, so imagine your drawing carrying on past the edge of the paper, past the walls, past the Moon.

Take any two lines lying flat on the same sheet of paper (mathematicians say "in the same plane": a plane is a perfectly flat surface that goes on forever). There are only two things that can happen:

  • they cross at exactly one point, or
  • they never cross, however far you carry them on.

They cannot cross twice. Two straight lines that met at two points would have to be the same line.

Parallel lines

Two lines in the same plane that never meet are called parallel. Railway tracks, the lines on writing paper, the opposite edges of a door, the top and bottom of this page: all parallel.

What makes them never meet? They point in exactly the same direction. If one line tipped even a tiny bit more than the other, the gap between them would slowly shrink on one side, and far enough along they would cross.

That gives us a test you can do with a ruler: parallel lines stay the same distance apart everywhere. Measure the gap (straight across, at a square corner to the lines) in two different places. If the gap is the same, the lines are parallel.

In drawings, we show parallel lines by putting the same small arrow mark on each of them. If a drawing has two different pairs of parallel lines, the second pair gets double arrows.

parallel perpendicular crossing, not at a right angle

There is a short way to write "is parallel to": two upright strokes, ∥. If line \(AB\) is parallel to line \(CD\), we write \(AB \parallel CD\). (Lines are often named by two points on them, written with capital letters.)

Perpendicular lines

Two lines that cross at a right angle (90°) are called perpendicular (say "per-pen-DIK-you-lar"). The corners of a window, the legs of a table and the floor, the lines of a noughts-and-crosses grid: all perpendicular.

When two lines are perpendicular, all four angles where they cross are right angles. Why? The four angles make a full turn, 360°. If one of them is 90°, the one next to it along the same straight line is \(180 - 90 = 90\) degrees, and so on all the way round.

We show perpendicular lines with the small square at the crossing. The short way to write "is perpendicular to" is an upside-down T: ⊥. So \(AB \perp CD\) means line \(AB\) is perpendicular to line \(CD\).

Lines that cross at any other angle are just called intersecting lines. ("Intersect" means "cut across each other".)

Parallel and perpendicular together

Look at a ladder. The two long sides, the rails, are parallel. Each rung is perpendicular to both rails.

rungs ⊥ rails, rails ∥ rails 3 cm 3 cm the same gap everywhere: parallel

This shows a fact you can trust: if two lines are both perpendicular to the same line, they are parallel to each other. Picture standing on a rung and looking along it. Both rails go straight "up" from the rung, at a right angle. They both start out in exactly the same direction, so they will never meet.

And the other way round: if a line is perpendicular to one of two parallel lines, it is perpendicular to the other too. That is why one rung can meet both rails at a right angle.

This pair of facts is why streets in many cities are laid out on a grid: all the north-south streets are parallel to each other, all the east-west streets are parallel to each other, and every north-south street is perpendicular to every east-west street.

Why the rails look as if they meet

Back to the question at the start. The rails never meet: if they did, trains would derail. They only look as if they meet because things far away look smaller. The gap between the rails is the same size everywhere, but a gap far away looks tiny, so the rails seem to come together. Artists use this on purpose: drawing parallel lines that meet at a vanishing point makes a flat picture look deep. It is a trick of the eye, not a fact about the lines.

Drawing parallel and perpendicular lines

A perpendicular line, with a set square (a triangle-shaped ruler with a right-angle corner) or your folded-paper right-angle tester from lesson 1:

  1. Lay one edge of the right-angle corner along your line.
  2. Slide it to the point where you want the new line.
  3. Draw along the other edge of the right angle.

A parallel line, with a ruler and a set square:

  1. Lay one edge of the set square along your line.
  2. Put a ruler firmly against another edge of the set square.
  3. Hold the ruler still and slide the set square along it.
  4. Draw along the same edge you started with. It has moved, but it has not turned, so it points the same way.

A line, a ray and a segment

While we are naming things: a line goes on forever both ways. A ray starts at a point and goes on forever one way, like a beam of light from a torch. A line segment is the piece between two end points, and it has a length you can measure. The arms of an angle are really rays. The sides of a shape are segments. When we say two segments are parallel, we mean the lines they lie on are parallel.

Worked examples

1. Lines in capital letters. Find parallel and perpendicular pairs in the letters H, E, Z and N.

  • H: the two uprights are parallel. The crossbar is perpendicular to both of them (just like a rung on a ladder).
  • E: the three bars are all parallel to each other. Each bar is perpendicular to the upright.
  • Z: the top and bottom bars are parallel. The slanting stroke is neither parallel nor perpendicular to them: it meets them at acute angles.
  • N: the two uprights are parallel. The slant meets them at acute angles, so no perpendicular pair.

2. A rectangle. A rectangle has 4 sides: top, bottom, left, right. How many pairs of parallel sides and how many pairs of perpendicular sides?

  • Parallel: top ∥ bottom, left ∥ right. 2 pairs.
  • Perpendicular: each corner is a right angle, and each corner joins one pair of sides: top-left, top-right, bottom-left, bottom-right. 4 pairs.

3. Testing for parallel. Two lines are 4 cm apart at the left end of the page and 4 cm apart at the right end. A second pair is 4 cm apart at the left and 3 cm apart at the right. Which pair is parallel?

The first pair: the gap stays the same. The second pair gets closer together towards the right, by 1 cm across the page. If you carried them on past the right edge, the gap would keep shrinking until they crossed. They are not parallel.

4. On a grid of streets. Elm Street runs north-south. Oak Street runs east-west. Pine Street runs north-south. Which pairs are parallel and which perpendicular?

  • Elm and Pine both run north-south: parallel.
  • Oak runs east-west, at a right angle to north-south: Oak ⊥ Elm and Oak ⊥ Pine.

Try it

  1. Name two things in your kitchen that have parallel edges, and two that have perpendicular edges.
  2. Which of these capital letters have a pair of perpendicular lines: T, V, L, X, F?
  3. How many right angles are there where two perpendicular lines cross?
  4. Line \(a\) is perpendicular to line \(b\). Line \(c\) is also perpendicular to line \(b\). What can you say about \(a\) and \(c\)?
  5. Line \(p\) is parallel to line \(q\). Line \(r\) is perpendicular to \(p\). What is the angle between \(r\) and \(q\)?
  6. A square has 4 sides. How many pairs of parallel sides does it have? How many pairs of perpendicular sides?
  7. Two lines are 5 cm apart at one place and 7 cm apart at another. Are they parallel? Will they ever meet?
  8. In a room, think of a line along the floor by the front wall, and a line along the ceiling by the left wall. They never meet. Are they parallel?
Answers
  1. Many answers are right. Parallel: opposite edges of a fridge door, the two long edges of a chopping board. Perpendicular: the edge of a table and its leg, the side and bottom of a cereal box.
  2. T, L, F. T and L each have a square corner; F has two bars that are each perpendicular to the upright. V meets at an acute angle. X crosses at a slant in most fonts (if the strokes crossed at a right angle it would count, but they usually do not).
  3. Four. Once one angle is 90°, its neighbour on the straight line is \(180 - 90 = 90°\), and so on round.
  4. \(a\) and \(c\) are parallel: both are perpendicular to the same line \(b\), like two rails on one rung.
  5. 90°. A line perpendicular to one of two parallel lines is perpendicular to the other.
  6. 2 pairs of parallel sides (top and bottom, left and right) and 4 pairs of perpendicular sides (one pair at each corner).
  7. Not parallel: the gap changes. They get closer on the 5 cm side, so somewhere beyond that side they cross.
  8. No. They point in different directions (one runs across the room, the other runs front to back), so they are not parallel. They never meet because they are at different heights: they are not in the same flat plane. Lines like these are called skew lines. They can only happen in three dimensions.

Common mistake

Thinking "never meet on the page" means parallel. Two short segments can stop before they meet, yet point in different directions. Imagine carrying them on forever: parallel lines must keep exactly the same gap. Measure the gap in two places to be sure.

Thinking perpendicular lines must be upright and flat. A line slanting up to the right can be perpendicular to one slanting up to the left, as long as they meet at 90°. Turn the page: perpendicular lines stay perpendicular however you hold them.

Where it leads