Grade by Grade
E. Spatial Sense · Angles, Rectangles, Grids and Measurement

Parallel and Perpendicular Lines

E1.1About 30 minutes
Learning goals
  • I can tell when two lines are parallel.
  • I can tell when two lines are perpendicular.
  • I can find parallel and perpendicular lines in letters, shapes and things at home.
  • I can use symbols on a drawing to show parallel and perpendicular lines.

Words to know

line segment
A part of a line with two end points, like the side of a shape.
parallel
Lines that are always the same distance apart. They never meet, even if you make them longer.
perpendicular
Lines that meet or cross to make a right angle (a square corner).
intersecting
Lines that meet or cross each other.

Let's start with railway tracks

Think of the two rails of a railway track. They run side by side for many kilometres, but they never touch. If they did, the train would fall off! Lines like these are called parallel.

Now think of a plus sign (+). The two lines cross and make four square corners. Lines like these are called perpendicular.

A straight railway track stretches into the distance through fall-coloured trees.
The two rails stay the same distance apart, so they are parallel. Each wooden tie crosses the rails at about a right angle, so the ties are perpendicular to the rails. AI-generated image

Three ways lines can go

  • Parallel lines are always the same distance apart. They never meet, even if you make them longer and longer. On a drawing, we put matching arrow marks on parallel lines.
  • Perpendicular lines meet or cross to make a right angle. On a drawing, we put a small square in the corner.
  • Intersecting lines meet or cross. Perpendicular lines are a special kind of intersecting lines. Some intersecting lines do not make right angles, so they are not perpendicular.
parallelperpendicularintersecting
Left: parallel lines, marked with matching arrows. Middle: perpendicular lines, marked with a small square. Right: lines that intersect but do not make a right angle.

How to check

For parallel lines: measure the distance between the lines in two different places. If the distance is the same, the lines are parallel. If the lines get closer together, they will meet one day, so they are not parallel.

For perpendicular lines: use your right-angle tester from the last lesson. If the corner fits exactly, the lines are perpendicular.

3 cm3 cmparallel4 cm2 cmnot parallel
Left: the gap is 3 cm at both places, so the lines are parallel. Right: the gap is 4 cm at one place and 2 cm at another, so the lines are not parallel.
Worked example

Example 1: Lines in the letter H

Find the parallel and perpendicular line segments in the capital letter H.

  1. The two up-and-down (vertical) segments stay the same distance apart. They are parallel.
  2. The middle bar goes across. It meets each vertical segment at a square corner.
  3. So the middle bar is perpendicular to both vertical segments.

H has one pair of parallel segments, and its middle bar is perpendicular to both of them.

HTLZENV
Capital letters H, T, L, Z, E, N and V drawn with straight segments.
Worked example

Example 2: Lines in the letter Z

Does the letter Z above have parallel lines? Does it have perpendicular lines?

  1. The top bar and the bottom bar both go straight across and stay the same distance apart. They are parallel.
  2. The slanted line meets the top and bottom bars, but not at square corners. Check with a tester: one corner is too narrow, the other is too wide.
  3. So Z has no perpendicular lines.

Z has one pair of parallel segments and no perpendicular segments.

Parallel lines can point any way

Parallel lines do not have to go across or up and down. Two slanted lines can be parallel too, as long as they stay the same distance apart. And perpendicular lines can be tilted, like a plus sign turned into an ×, as long as they still make square corners.

Just for fun

Parallel lines have so much in common. It's a shame they'll never meet!

Hands-on activity

Line detective at home

You need: paper, a pencil, a ruler and your right-angle tester.

  1. Fold a sheet of paper in half and open it. Label one side Parallel and the other side Perpendicular.
  2. Walk around your home and look for lines and edges. Ideas: the rungs of a ladder or bunk-bed ladder, a window frame, floor tiles, the stripes on a towel, the shelves of a bookcase, the lines on lined paper, the edges of a door.
  3. Draw or write each thing on the correct side. Test perpendicular corners with your tester. For parallel lines, measure the gap in two places with your ruler.
  4. Write your full name in capital letters. Use one colour to trace parallel segments and another to circle perpendicular corners.

Math talk: "Why do you think builders make so many things with parallel and perpendicular lines?"

Your turn

1

Look at the letter T above. Are its two segments parallel, perpendicular or neither?

2

Look at the letter E above. How many segments are parallel to each other?

3

Is the long side of the letter E perpendicular to the bars?

4

Look at the letter V above. Are its segments parallel, perpendicular or neither?

5

Look at the letter N above. Which segments are parallel?

6

Two lines are 5 cm apart at one end and 5 cm apart at the other end. Are they parallel?

More practice: warm-up

1

What do we call lines that are always the same distance apart and never meet?

2

What do we call lines that meet at a right angle?

3

What mark shows that two lines on a drawing are perpendicular?

4

Are the lines on a sheet of lined paper parallel or perpendicular to each other?

5

Name a capital letter that has perpendicular segments.

More practice: on your own

1

Look at the letter L above. Does it have parallel segments, perpendicular segments, or both?

2

Which letters above have both parallel and perpendicular segments?

3

Two roads cross at a corner and make 4 square corners. How would you describe the roads?

4

Ava says: "These two lines are not parallel because they are slanted." Is she right?

5

Two lines are 6 cm apart at the top of a page and 2 cm apart at the bottom. Will they ever meet if you make them longer?

Challenge

1

Draw a shape with exactly one pair of parallel sides and no right angles. What could it look like?

2

Line A is perpendicular to line B. Line C is also perpendicular to line B. How are lines A and C related? Draw it to check.

3

Kai draws two lines that cross. One of the four angles they make is a right angle. Are the lines perpendicular? What are the other three angles?

Check your understanding

Quiz

1.

Which pair of lines never meet and stay the same distance apart?

2.

Two lines cross and make square corners. They are...

3.

Which of these is made of only parallel segments?

4.

Two lines are 4 cm apart at one end and 1 cm apart at the other end. They are...

5.

What does a small square drawn where two lines meet tell you?

For parents and teachers

Big idea: As part of E1.1 (properties of rectangles), Grade 4 students learn the words parallel (always the same distance apart, never meeting) and perpendicular (meeting at a right angle). They will use these words in the next lesson to describe the sides of rectangles.

Common difficulties:

  • Thinking only horizontal lines can be parallel. Show slanted parallel lines, like the slats of a fence on a hill.
  • Calling any crossing lines perpendicular. They must make a right angle. Use the paper tester.
  • Thinking lines that do not touch on the page are parallel. If the gap is shrinking, they would meet if extended.

How to help: Point out parallel and perpendicular lines on walks: road lines, fences, sidewalk cracks, hydro poles and wires, a crosswalk. Ask "How could we check?"