E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Quadrilaterals: Sides, Angles and Diagonals

E1.1 A1About 45 minutes

Learning goals

I can list the properties of squares, rectangles, rhombuses, parallelograms, trapezoids and kites.
I can describe the diagonals of each kind of quadrilateral: are they equal, perpendicular, or do they bisect each other?
I can use properties to name a quadrilateral and to find missing lengths.

Words to know

quadrilateral
A polygon with 4 straight sides and 4 vertices.

property
Something that is always true about a kind of shape, like "all 4 sides are equal".

diagonal
A line segment that joins two vertices that are not next to each other. Every quadrilateral has 2 diagonals.

bisect
To cut into two equal parts. Diagonals that bisect each other cross at the middle of both.

perpendicular
Meeting at a right angle (90°).

parallel
Always the same distance apart, so the lines never meet.

Remember from Grade 4 and Grade 5

In Grade 4 you learned that properties are facts that are always true about a shape. You found the properties of rectangles: 4 right angles, opposite sides equal and parallel, and 2 lines of symmetry. In Grade 5 you met parallelograms: four-sided shapes with 2 pairs of parallel sides.

Let's start with a mystery shape

Your friend is thinking of a quadrilateral. You ask: "Are all its sides equal?" "No." "Does it have right angles?" "Yes, four of them." You guess: a rectangle!

Math detectives use properties to name shapes. Today you will add a new kind of clue: the diagonals. A diagonal joins two corners that are not next to each other. Every quadrilateral has 2 diagonals, and they cross inside the shape (in all the shapes in this lesson). How they cross tells you a lot about the shape.

A square, a rectangle and a rhombus with their diagonals drawn as dashed lines. Tick marks show equal sides. The small square at the centre of the square and the rhombus shows that their diagonals cross at a right angle.

A parallelogram, a trapezoid and a kite with their diagonals drawn as dashed lines. Sides with the same number of tick marks are equal. The kite's diagonals cross at a right angle.

Three questions to ask about diagonals

Are they the same length?
Are they perpendicular? (Do they cross at a right angle?)
Do they bisect each other? (Do they cross at the middle of both diagonals?)

The table below collects the properties. Read it slowly, one row at a time, and look back at the pictures as you go.

Shape	Sides	Angles	Diagonals
Square	4 equal sides; 2 pairs of parallel sides	4 right angles	Equal length, perpendicular, bisect each other
Rectangle	Opposite sides equal; 2 pairs of parallel sides	4 right angles	Equal length, bisect each other
Rhombus	4 equal sides; 2 pairs of parallel sides	Opposite angles equal	Perpendicular, bisect each other
Parallelogram	Opposite sides equal; 2 pairs of parallel sides	Opposite angles equal	Bisect each other
Trapezoid	Exactly 1 pair of parallel sides	No rule for all trapezoids	No rule for all trapezoids
Kite	2 pairs of equal sides that are next to each other	1 pair of opposite angles equal	Perpendicular; one diagonal is bisected by the other

Properties of quadrilaterals. In this unit a trapezoid has exactly one pair of parallel sides. (Some books also count parallelograms as trapezoids.)

Family connections

Some shapes belong to more than one family.

A square has all the properties of a rectangle (4 right angles), so every square is a special rectangle.
A square also has all the properties of a rhombus (4 equal sides), so every square is a special rhombus.
Rectangles and rhombuses both have 2 pairs of parallel sides, so they are special parallelograms.

That's why the square has the longest list of properties: it collects them all!

Worked example

Example 1: Name the shape from its diagonals

A quadrilateral's diagonals are the same length and bisect each other, but they are not perpendicular. What is it?

Diagonals that bisect each other belong to a parallelogram, a rectangle, a rhombus or a square.
Equal diagonals leave the rectangle and the square.
A square's diagonals are perpendicular. These are not, so it is not a square.

A rectangle (one that is not a square).

Worked example

Example 2: Use a property to find a length

The diagonals of rectangle ABCD cross at point M. The distance from A to M is 5 cm. How long is each diagonal, and how far is it from B to M?

A rectangle's diagonals bisect each other, so M is the middle of diagonal AC. AC = 5 + 5 = 10 cm.
A rectangle's diagonals are equal, so BD is also 10 cm.
M is the middle of BD too, so BM = 10 ÷ 2 = 5 cm.

Both diagonals are 10 cm, and BM = 5 cm.

Worked example

Example 3: Kite or rhombus?

Ama draws two diagonals that cross at a right angle. One is cut exactly in half, but the other is not. What shape does she get when she joins the ends?

Perpendicular diagonals fit a square, a rhombus or a kite.
In a square and a rhombus, both diagonals are cut in half (they bisect each other).
Here only one is cut in half. That is the kite's property.

A kite.

Just for fun

Riddle: I have 4 equal sides, but I'm not a square. My diagonals cross at right angles, but they aren't the same length. What am I?

A rhombus: a square that leaned over to tie its shoe!

Prove it, don't guess it

A shape can look like a square and still have sides of 5 cm and 5.2 cm. Mathematicians check with tools: a ruler for lengths, a protractor or a square corner for right angles. Saying "I measured, and here's why" is called reasoning and proving. It's what real mathematicians do.

Hands-on activity

Diagonal detectives

You need: 2 pencils the same length, 1 shorter pencil (or a straw cut shorter by a grown-up), paper, a ruler and a square corner (the corner of a sheet of paper or a book).

Lay two equal pencils on the paper so they cross at their middles, at a slant (not at a right angle). Mark a dot at each of the 4 ends. Join the dots with a ruler. Which shape did you make? (A rectangle.)
Move the same two pencils so they still cross at their middles, but at a right angle (check with the square corner). Mark and join the ends. (A square.)
Now use one long pencil and the shorter one. Cross them at both middles at a right angle. (A rhombus.)
Keep the right angle, but slide the short pencil so it crosses the long pencil near one end, still at the short pencil's middle. (A kite.)
Cross the long and short pencils at both middles, but not at a right angle. (A parallelogram.)

Math talk: "Which rule about the diagonals made each shape?" Talk about it with your grown-up.

Your turn

Which two quadrilaterals always have 4 right angles?

Which quadrilaterals in the table always have perpendicular diagonals?

The diagonals of a parallelogram cross at M. One diagonal is 12 cm long. How far is it from M to each end of that diagonal?

True or false: every square is a rhombus.

True or false: every rhombus is a square.

A quadrilateral has 2 pairs of parallel sides. Its diagonals are the same length, but they are not perpendicular. Name it.

More practice: warm-up

How many diagonals does a quadrilateral have?

What is a diagonal?

Which quadrilateral has exactly one pair of parallel sides?

What does bisect mean?

How many equal sides does a rhombus have?

More practice: on your own

A kite has sides of 5 cm, 5 cm, 8 cm and 8 cm. Where do the equal sides go?

One diagonal of a rectangle is 14 cm. How long is the other diagonal? Why?

At what angle do the diagonals of a square cross?

One angle of a parallelogram is 70°. What is the angle opposite it?

Sam says, "My quadrilateral's diagonals are perpendicular, so it must be a rhombus." Is he right?

Challenge

A quadrilateral's diagonals bisect each other, are perpendicular and are the same length. Name the shape and explain.

Is every quadrilateral with perpendicular diagonals a kite or a rhombus? Try this: draw a 6 cm line across and a 4 cm line up and down that cross at a right angle, 1 cm from the left end of the 6 cm line and 1 cm from the top of the 4 cm line. Join the 4 ends. Is it a kite?

In rectangle ABCD, the diagonals cross at E. AE = 4 cm and side AB = 6 cm. Find the lengths of AC and BD, and the perimeter of triangle ABE.

Check your understanding

Quiz

Which quadrilateral always has diagonals that are the same length?

A quadrilateral has exactly one pair of parallel sides. What is it?

What is always true about the diagonals of a rhombus?

Which statement is true?

A parallelogram's diagonals are 10 cm and 16 cm. They cross at M. How far is M from each end of the 16 cm diagonal?

In which shape is exactly one diagonal cut in half by the other?

For parents and teachers

Big idea: The Ontario curriculum (E1.1) asks Grade 6 students to create lists of the geometric properties of quadrilaterals, including the properties of the diagonals, rotational symmetry and line symmetry. This lesson covers sides, angles and diagonals; the next lesson covers symmetry.

Common difficulties:

Judging by looks. Encourage your child to measure with a ruler and check corners with the corner of a sheet of paper.
"A square isn't a rectangle." Many children think each shape has only one name. Go back to the property list: a square has every property of a rectangle, so it belongs to that family too.
Mixing up "perpendicular" and "bisect". Perpendicular is about the angle where diagonals cross; bisect is about where they cross (at the middle).
Trapezoid definitions. This unit uses "exactly one pair of parallel sides". Some books count parallelograms as trapezoids too; if your child's teacher uses the other definition, go with the teacher.

How to help: Hunt for quadrilaterals around the house (windows, tiles, a kite-shaped cookie cutter, a road sign) and ask "How do you know?" Praise clear reasons, not just correct names.