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# The quadrilateral family

Angles and shapes · Classifying shapes

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 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.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> 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.

<!-- Speaker note: 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. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Is a square a rectangle?**

- No: a rectangle has two long sides and two short sides
- Only a big square
- Yes

<!-- Speaker note: What happens: Yes. A rectangle is a quadrilateral with four right angles, and a square has four right angles. It is a special rectangle, with its sides equal too. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a vote; most say no. Keep the vote visible until after Investigate. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fquadrilaterals%2Fstep-2&code=widget%3A+geometry%0Ashape%3A+polygon%0Asides%3A+4%0Acorners%3A+1%2C1%3B+8%2C2%3B+7%2C6%3B+2%2C5%0Aclassify%3A+on%0Asum%3A+off%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/quadrilaterals/?full=step-2#step-2)

<!-- Speaker note: Watch for the arrows (parallel sides), ticks (equal sides) and right-angle marks. -->

<!-- Speaker note: Teacher: ask what the board added to its name as the rectangle appeared. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* Yes. A rectangle is a quadrilateral with four right angles, and a square has four right angles. It is a special rectangle, with its sides equal too.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Properties first

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

Now match each name to its property.

$$\text{4 equal sides, 4 right angles} \Rightarrow \text{square}$$

1) $\text{4 right angles} \Rightarrow \text{rectangle}$ — Right angles but sides not all equal.
2) $\text{4 equal sides} \Rightarrow \text{rhombus}$ — Equal sides but no right angles.
3) $\text{2 pairs parallel} \Rightarrow \text{parallelogram}$ — Then parallelogram, trapezium (at least one pair), kite.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 4 equal sides, 4 right angles square and say "Check the most special names first." Line 1: write 4 right angles rectangle and say "Right angles but sides not all equal." Line 2: write 4 equal sides rhombus and say "Equal sides but no right angles." Line 3: write 2 pairs parallel parallelogram and say "Then parallelogram, trapezium (at least one pair), kite." -->

<!-- Speaker note: Teacher: build a family tree on the board, from quadrilateral down to square. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Properties first

**Match each quadrilateral to its defining property.**

| | |
|---|---|
| `trapezium` | A. four equal sides |
| `parallelogram` | B. two pairs of parallel sides |
| `rhombus` | C. two pairs of equal sides next to each other |
| `rectangle` | D. four right angles |
| `kite` | E. at least one pair of parallel sides |

* Answer: trapezium → E. at least one pair of parallel sides; parallelogram → B. two pairs of parallel sides; rhombus → A. four equal sides; rectangle → D. four right angles; kite → C. two pairs of equal sides next to each other

<!-- Speaker note: A square has four equal sides and four right angles, so it fits the rhombus, rectangle and parallelogram rows too. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

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

```
 1  A square has four right angles.
 2  A rectangle is a quadrilateral with four right angles.
 3  A square also has two pairs of parallel sides, like a rectangle.
 4  But a square is not a rectangle, because all its sides are equal.
```

* Answer: line 4

<!-- Speaker note: Line 4: having an extra property (equal sides) does not take away the rectangle's properties. A square has four right angles, so it is a rectangle: a special one. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Always true?

**Which is always true?**

- Every kite is a rhombus
- Every rhombus is a parallelogram
- Every rectangle is a square
- Every parallelogram is a rhombus

* Answer: Every rhombus is a parallelogram

<!-- Speaker note: A rhombus has two pairs of parallel sides, so it has the parallelogram's property: every rhombus is a parallelogram. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A rhombus:** Each task changes one thing.
- **A plain parallelogram:** 
- **A kite:** 
- **One right angle:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/quadrilaterals/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: All four sides the same length, but no right angles. / Try corners (1, 1), (6, 1), (9, 5) and (4, 5): each side is 5. / Two pairs of parallel sides, with no right angles and the sides not all equal. / Two short equal sides next to each other at the top, two long equal sides next to each other at the bottom. / Try (5, 1), (8, 5), (5, 7), (2, 5). For students who need more support: Use a property checklist (parallel pairs, equal sides, right angles) for every shape before naming it. For early finishers: Draw a Venn diagram of rectangles and rhombuses. What goes in the overlap? Where do parallelograms go? 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. -->

<!-- Speaker note: Teacher: for the edge case, ask what co-interior angles say about the next corner. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Most special name

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

- Rectangle
- Trapezium
- Parallelogram
- Rhombus

* Answer: Parallelogram

<!-- Speaker note: Two pairs of parallel sides, nothing more: a parallelogram. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## All the names

**Which names fit a square?**

- Square, rectangle, rhombus, parallelogram, trapezium and kite
- Square and rectangle only
- Square and rhombus only
- Only square

* Answer: Square, rectangle, rhombus, parallelogram, trapezium and kite

<!-- Speaker note: A square has all these properties, so it belongs to all these families. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Angles in a parallelogram

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

* Answer: 110

<!-- Speaker note: Neighbouring angles of a parallelogram are co-interior: 180 − 70 = 110°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: triangles

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

* Answer: 110

<!-- Speaker note: 180 − 35 − 35 = 110°. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a right-angled trapezium

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

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/quadrilaterals/?full=step-6#step-6)

<!-- Speaker note: What this step is for: 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. Hints, in order: Make the top and bottom parallel, then make one side straight up. / Keep the other side slanted so there is only one parallel pair. -->

<!-- Speaker note: Teacher: ask why a trapezium can't have exactly three right angles. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A shape has two pairs of parallel sides and four equal sides but no right angles. Give all its names.**

* Rhombus, parallelogram and kite.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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). -->