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

Properties of Rectangles: Angles, Sides and Lines of Symmetry

E1.1 A1About 40 minutes
Learning goals
  • I can describe a rectangle by its right angles, parallel sides and perpendicular sides.
  • I can find the lines of symmetry of a rectangle and a square.
  • I can explain why a square is a special rectangle.
  • I can use properties to decide whether a shape is a rectangle.

Words to know

property
Something that is always true about a shape, like "has 4 right angles".
rectangle
A shape with 4 straight sides and 4 right angles.
square
A rectangle whose 4 sides are all the same length.
line of symmetry
A line that splits a shape into two halves that match exactly when you fold along it.
diagonal
A line segment that joins two corners of a shape that are not next to each other.
quadrilateral
Any shape with 4 straight sides.

Remember from Grade 2

You learned that a line of symmetry splits a shape into two matching halves. You tested it by folding, or with a small mirror.

Now you know about right angles, parallel lines and perpendicular lines too. Today you will use all of these words to describe one shape very carefully: the rectangle.

Let's start with a mystery

A friend says: "I'm thinking of a shape with 4 sides." Can you guess it? Not yet! It could be a square, a rectangle, a kite or many other shapes.

To know for sure, we need properties: facts that are always true about a shape. Mathematicians describe shapes by their properties so that everyone pictures the same thing.

The properties of a rectangle

Every rectangle has:

  1. 4 straight sides (it is a quadrilateral).
  2. 4 right angles. Every corner is a square corner.
  3. 2 pairs of parallel sides. The top and bottom are parallel. The left and right sides are parallel.
  4. 4 pairs of perpendicular sides. At each corner, the two sides that meet are perpendicular. There are 4 corners, so there are 4 pairs.
  5. Opposite sides of equal length. The top and bottom are the same length. The two other sides are the same length.
  6. 2 lines of symmetry (more if it is a square).
rectangle4 right angles2 pairs ofparallel sides4 pairs ofperpendicular sides
A rectangle. The small squares show the 4 right angles. Single arrows mark one pair of parallel sides, and double arrows mark the other pair.

Lines of symmetry in a rectangle

Cut out a paper rectangle (not a square). Fold it so the left and right sides meet. The halves match! That fold is a line of symmetry. Now open it and fold so the top and bottom meet. Another line of symmetry!

What about folding from corner to corner, along a diagonal? Try it. The corners stick out and the halves do not match. So the diagonal of a rectangle is not a line of symmetry (unless the rectangle is a square).

rectangle: 2 linessquare: 4 lines
The dashed lines are lines of symmetry. A rectangle that is not a square has 2. A square has 4, because its diagonals also work.
fold on a diagonalfold down the middle
Left: folding a long rectangle along a diagonal. The folded half sticks out past the other half, so the diagonal is not a line of symmetry. Right: folding down the middle makes the halves match.
Worked example

Example 1: Is a square a rectangle?

Use properties to decide if a square is a rectangle.

  1. A rectangle needs 4 straight sides. A square has 4 straight sides. ✔
  2. A rectangle needs 4 right angles. A square has 4 right angles. ✔
  3. A rectangle needs 2 pairs of parallel sides. A square has them. ✔
  4. A square has one extra property: all 4 sides are the same length. That does not break any rectangle rule.

Yes! A square is a special rectangle with 4 equal sides. Because its sides are equal, it also has 2 extra lines of symmetry along its diagonals, for 4 in all.

Worked example

Example 2: Describe a door

Describe the front of a door using rectangle properties.

  1. Count the sides and corners: 4 straight sides and 4 corners.
  2. Check each corner with a right-angle tester: all 4 are right angles.
  3. The left and right edges are parallel. The top and bottom edges are parallel.
  4. At each corner, the edges are perpendicular: 4 pairs.
  5. Fold it in your mind: top to bottom matches, and left to right matches. 2 lines of symmetry. (We are only looking at the outline, not the doorknob!)

The outline of a door is a rectangle: 4 right angles, 2 pairs of parallel sides, 4 pairs of perpendicular sides and 2 lines of symmetry.

Tilted rectangles are still rectangles

If you turn a rectangle so it stands on a corner, it still has 4 right angles and 2 pairs of parallel sides. It is still a rectangle. Properties do not change when a shape is turned.

Math talk: convince a friend

To prove a shape is a rectangle, check every corner for a right angle. To prove it is not a rectangle, you only need to find one corner that is not a right angle.

Just for fun

Q: What did the square say to the rectangle?

A: "Relax, I'm a rectangle too. I just like my sides all the same length!"

Hands-on activity

Rectangle investigation

You need: 3 or 4 sheets of paper of different sizes (a sticky note, a sheet of printer paper, an envelope or a recipe card), scissors, a ruler and your right-angle tester.

Grown-up note: help with scissors, or cut the shapes yourself.

  1. Check that each piece is a rectangle. Test all 4 corners.
  2. Measure the sides of each one with a ruler. Are the opposite sides the same length?
  3. Fold each one to find its lines of symmetry. Draw along the folds with a pencil.
  4. Try folding each one along a diagonal. Does it ever work?
  5. Make a square: fold one corner of a sheet of printer paper down to the opposite edge so the top edge lines up with the side edge, and cut off the extra strip. Find all its lines of symmetry.
  6. Make a table: shape, number of right angles, pairs of parallel sides, lines of symmetry.

Math talk: "Which piece had the most lines of symmetry? Why?"

ABCD
Shapes A, B, C and D for the questions below. Shape B has two right angles marked.

Your turn

1

How many right angles does every rectangle have?

2

How many pairs of parallel sides does a rectangle have?

3

How many lines of symmetry does a rectangle that is not a square have? How many does a square have?

4

Look at shape A above. It has 2 pairs of parallel sides. Is it a rectangle?

5

Look at shape B. It has 2 right angles. Is it a rectangle?

6

Look at shape C. Is it a rectangle?

7

Look at shape D. Is it a rectangle? How do you know?

More practice: warm-up

1

How many sides does a rectangle have?

2

What kind of angle is every corner of a rectangle?

3

In a rectangle, are the opposite sides parallel or perpendicular?

4

In a rectangle, are two sides that meet at a corner parallel or perpendicular?

5

Is a square a rectangle?

More practice: on your own

1

A rectangle is 8 cm long on its top side. How long is its bottom side? Why?

2

Why does a rectangle have 4 pairs of perpendicular sides?

3

Jon folds a paper rectangle (not a square) along a diagonal. The corners stick out. What does this tell him?

4

A rectangle has two sides that are 5 cm and two sides that are 5 cm. What special name does it have, and how many lines of symmetry?

5

Rosa says: "My shape has 4 sides and 2 pairs of parallel sides, so it must be a rectangle." Is she right?

Challenge

1

Can a 4-sided shape have exactly 3 right angles? Think about what the 4th corner must be.

2

A rectangle's long sides are each 12 cm and its short sides are each 5 cm. What is its perimeter? Which properties did you use?

3

Write a riddle about a rectangle using at least three properties. Then give the answer.

Check your understanding

Quiz

1.

How many right angles does a rectangle have?

2.

How many lines of symmetry does a long rectangle (not a square) have?

3.

Which shape has 4 lines of symmetry?

4.

In a rectangle, the two sides that meet at a corner are...

5.

Which statement is true?

6.

A shape has 4 sides and 2 pairs of parallel sides, but no right angles. Is it a rectangle?

For parents and teachers

Big idea: The Ontario curriculum (E1.1) asks Grade 4 students to identify the geometric properties of rectangles: the number of right angles (4), parallel sides (2 pairs), perpendicular sides (4 pairs, one at each corner) and lines of symmetry (2, or 4 for a square). Learning that a square is a special kind of rectangle is an important step in thinking about shapes by their properties instead of their look.

Common difficulties:

  • Thinking the diagonal is a line of symmetry. Have your child fold a paper rectangle along the diagonal and see the corners stick out.
  • Saying a square is not a rectangle. Go through the rectangle rules one by one with a square.
  • Not recognizing a tilted rectangle. Turn a rectangle on its corner and ask again.
  • Counting perpendicular pairs. Point to each corner: one pair per corner.

How to help: Use real rectangles (cereal boxes, books, picture frames, envelopes). Ask your child to "prove" it is a rectangle using properties. Paper folding is the best tool for symmetry.