The quadrilateral family

Quadrilaterals are named by their properties: parallel sides, equal sides and right angles. A shape that has all of another's properties belongs to it too: every square is a rectangle and a rhombus, and every rectangle is a parallelogram.

Is a square a rectangle? People argue about it, but mathematicians agree: yes. A name is a list of properties, and a square has every property on the rectangle's list.

A quadrilateral has four sides. Its name comes from its properties:

Name	Properties
trapezium	at least one pair of parallel sides
parallelogram	two pairs of parallel sides (opposite sides equal)
rhombus	four equal sides (so two pairs of parallel sides)
rectangle	four right angles (so two pairs of parallel sides)
square	four equal sides and four right angles
kite	two pairs of equal sides next to each other

The board marks parallel sides with matching arrows, equal sides with matching ticks, and right angles with a small square. When a shape has every property of another, it belongs to that family too: a square is also a rectangle, a rhombus and a parallelogram.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Square or rectangle?

What does a shape need to be a rectangle?

Is a square a rectangle?No: a rectangle has two long sides and two short sidesOnly a big squareYes

TryTry it on the board and compare the result with your prediction.

A rectangle that is not a square

Watch for the arrows (parallel sides), ticks (equal sides) and right-angle marks.

Try it yourselfYour edits are saved in this browser.

Drag the corners to make a rectangle that is not a square. What else does the board say it is? Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

A family of properties

Here is how to name a quadrilateral: check the properties, from the most special.

Worked exampleBefore each step, say what you think comes next.

$\text{4 equal sides, 4 right angles} \Rightarrow \text{square}$ Check the most special names first.
$\text{4 right angles} \Rightarrow \text{rectangle}$ Right angles but sides not all equal.
$\text{4 equal sides} \Rightarrow \text{rhombus}$ Equal sides but no right angles.
$\text{2 pairs parallel} \Rightarrow \text{parallelogram}$ Then parallelogram, trapezium (at least one pair), kite.

Now match each name to its property.

Read Sam's reasoning. Does an extra property take a name away?

Match each quadrilateral to its defining property.

trapezium

parallelogram

rhombus

rectangle

kite

Sam decides whether a square is a rectangle. Click the line where his reasoning goes wrong.

Which is always true?Every kite is a rhombusEvery rhombus is a parallelogramEvery rectangle is a squareEvery parallelogram is a rhombus

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

All four sides the same length, but no right angles.

Another hint

Try corners (1, 1), (6, 1), (9, 5) and (4, 5): each side is 5.

Need a hint?

Two pairs of parallel sides, with no right angles and the sides not all equal.

Need a hint?

Two short equal sides next to each other at the top, two long equal sides next to each other at the bottom.

Another hint

Try (5, 1), (8, 5), (5, 7), (2, 5).

Try it yourselfYour edits are saved in this browser.

Make a rhombus that is not a square.

Use the board. This ticks when you get there.

Make a parallelogram that is not a rectangle and not a rhombus.

Use the board. This ticks when you get there.

Make a kite that is not a rhombus.

Use the board. This ticks when you get there.

Edge case: a quadrilateral has four equal sides and one right angle. What is it?It cannot existA kiteA rhombus that is not a squareA square

PractiseA few on your own, to check you have it.

Naming quadrilaterals

A quadrilateral has two pairs of parallel sides, no right angles, and sides that are not all equal. What is its most special name?RectangleTrapeziumParallelogramRhombus

Which names fit a square?Square, rectangle, rhombus, parallelogram, trapezium and kiteSquare and rectangle onlySquare and rhombus onlyOnly square

One angle of a parallelogram is 70°. What is the angle next to it, in degrees?

Review: an isosceles triangle has two angles of 35°. What is the third, in degrees?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A right-angled trapezium

Then: how many right angles can such a trapezium have, and why not three?

Try it yourselfYour edits are saved in this browser.

Need a hint?

Make the top and bottom parallel, then make one side straight up.

Another hint

Keep the other side slanted so there is only one parallel pair.

Make a trapezium that is not a parallelogram, with at least one right angle.

Use the board. This ticks when you get there.

For teachers

Students drag quadrilaterals until the board names them, learn the defining properties of the trapezium, parallelogram, rhombus, rectangle, square and kite, and reason about the family (a square is a rectangle and a rhombus) (AC9M7SP02, node 2). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can name a quadrilateral from its properties.
I can explain why a square is also a rectangle and a rhombus.
I can use parallel sides to find angles in a parallelogram.

Before class

Do the Run board yourself first. Students need parallel lines and co-interior angles (Parallel lines unit).

The one idea: a name is a list of properties, and a shape with every property on a list belongs to that family. This is the unit's hinge: students who hold the everyday meaning ("a rectangle is longer than wide") cannot reason about the family (Sam's mistake).

Definitions: this site follows the Australian Curriculum glossary, where a trapezium has at least one pair of parallel sides, so every parallelogram (and rectangle, rhombus and square) is a trapezium too. Some textbooks say exactly one pair; mention it if yours does.

Running the lesson

Predict: take a vote; most say no. Keep the vote visible until after Investigate.
Run: ask what the board added to its name as the rectangle appeared.
Investigate: build a family tree on the board, from quadrilateral down to square.
Modify: for the edge case, ask what co-interior angles say about the next corner.
Make: ask why a trapezium can't have exactly three right angles.

The hard part (on purpose)

Accepting that one shape can have several names. Ask students to read a square's properties against each name's list, out loud, instead of telling them the answer.

When students get stuck

A student says a square is not a rectangle (Sam's mistake). Ask: What does a shape need to be a rectangle? Does a square have it? Fix: Four right angles: yes. Extra properties don't remove a name.
A student names a rhombus a diamond and stops there. Ask: What do its sides and its parallel pairs say? Fix: Four equal sides: rhombus (and so a parallelogram).
A student cannot make a kite on the board. Ask: Which sides should be equal: opposite ones or neighbours? Fix: Neighbours: two short ones together and two long ones together.

Stretch and support

Stretch: Draw a Venn diagram of rectangles and rhombuses. What goes in the overlap? Where do parallelograms go?

Support: Use a property checklist (parallel pairs, equal sides, right angles) for every shape before naming it.

Where it leads

Next: Polygons (node 3): regular or irregular, convex or concave.

Where it leads: properties of diagonals and congruence proofs for quadrilaterals (Years 8 to 10).

What each step is for

square or rectangle?: Committing to whether a square is a rectangle surfaces the everyday meaning of 'rectangle' (longer than it is wide) against the property definition.
a rectangle that is not a square: Dragging an irregular quadrilateral into a rectangle shows the properties appearing one by one (parallel arrows, right angles) and the board's 'also' names.
a family of properties: Each name is a list of properties, and having all of one name's properties puts a shape in that family; Sam lets an extra property exclude a square from the rectangles.
one change at a time: One change at a time on the board: a rhombus that is not a square, a parallelogram that is neither, a kite, then the edge case of four equal sides with one right angle.
naming quadrilaterals: Naming from properties, a family question and an angle from parallel sides, mixed; a square excluded from the rectangles is the planted slip; the review keeps isosceles triangles fresh.
a right-angled trapezium: Building a trapezium with a right angle that is not a parallelogram combines a parallel pair with a right angle while avoiding a second parallel pair.

Now you can

Name a quadrilateral from its properties.
Explain why a square is also a rectangle and a rhombus.
Use parallel sides to find angles in a parallelogram.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson9. PolygonsA polygon is named by its number of sides (pentagon 5, hexagon 6, octagon 8, decagon 10). It is regular when all its sides and all its angles are equal, and concave when a corner points in.

Source: content/maths/angles/quadrilaterals/lesson.md