Lesson 7 ~25 min Unit 3 · Geometry +85 XP

Introducing Quadrilaterals

A quadrilateral is a closed shape with four straight sides, four vertices and four interior angles. There are six "special" quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium, kite — and they fit into a family tree based on which features they share.

Today's hook: Squares, rectangles, diamonds, kites — they're all quadrilaterals. But some are more special than others. Why?

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Think First

warm-up

Look around the room. Find five things shaped like quadrilaterals (a window, a book, a screen...). For each one, decide which "special" quadrilateral fits best: square, rectangle, parallelogram, rhombus, trapezium or kite. Was any one of them more than one type at once?

Record your answer in your workbook.

What Is a Quadrilateral?

+5 XP

A quadrilateral is a closed plane figure made up of four straight sides, four vertices (corners) and four interior angles. The four interior angles ALWAYS add to $360^{\circ}$. Six special quadrilaterals get their own names because of a defining feature: square, rectangle, parallelogram, rhombus, trapezium, kite.

Every quadrilateral has 4 sides, 4 vertices, 4 angles. The angle sum is always $360^{\circ}$ — you can prove this by drawing one diagonal and splitting the shape into two triangles ($180^{\circ} + 180^{\circ}$). Each special quadrilateral has one extra rule: a pair of parallel sides, four equal sides, four right angles, and so on.

Interior angle sum of any quadrilateral $= 360^{\circ}$

Closed and straight

All four sides must be straight, and the shape must close back to its starting vertex.

Angles sum to 360°

A useful check whenever you find angles in any quadrilateral.

Six special types

Square, rectangle, parallelogram, rhombus, trapezium, kite — each defined by its own feature.

What to write in your book

A quadrilateral has 4 straight sides, 4 vertices, and 4 interior angles.
The four interior angles of any quadrilateral sum to $360^{\circ}$ (proof: split with a diagonal into two triangles, $180^{\circ} + 180^{\circ}$).
Six special quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium, kite.

Quick check — what is the sum of the interior angles of any quadrilateral?

What You'll Master

objectives

Know

Definition of a quadrilateral (4 sides, 4 vertices, 4 angles)
Angle sum $= 360^{\circ}$
Defining property of each special quadrilateral
The names: square, rectangle, parallelogram, rhombus, trapezium, kite

Understand

Why the angle sum equals $360^{\circ}$ (two triangles)
How the special quadrilaterals are related (a hierarchy)
Why every square is also a rectangle AND a rhombus AND a parallelogram

Can Do

Identify the special quadrilateral from a description or diagram
Name every category a given quadrilateral belongs to
Apply the $360^{\circ}$ angle sum to find a missing angle

Words You Need

vocabulary

SquareFour equal sides AND four right angles.

RectangleFour right angles (opposite sides equal, but not all four).

ParallelogramTwo pairs of parallel sides.

RhombusFour equal sides (a "diamond" shape).

TrapeziumExactly ONE pair of parallel sides (NSW definition).

KiteTwo pairs of adjacent (touching) equal sides.

DiagonalA line segment joining two non-adjacent vertices of a quadrilateral.

HierarchyAn ordering: every square is a rectangle; every rectangle is a parallelogram.

Spot the Trap

heads-up

Wrong: "A square is just a square — it's not a rectangle." Actually, a square HAS four right angles, so by definition it IS a rectangle — just a very special one.

Right: A square is a rectangle, a rhombus, AND a parallelogram, all at once.

Wrong: "A trapezium has at least one pair of parallel sides." In NSW, the definition is EXACTLY one pair of parallel sides — otherwise it would be a parallelogram.

Right: A trapezium has EXACTLY one pair of parallel sides — the other pair is not parallel.

The Quadrilateral Family Tree

+5 XP

The six special quadrilaterals form a hierarchy. Some are more special than others, meaning they have ALL the features of a "parent" type PLUS something extra. A square is the most special — it sits at the top of the tree and inherits properties from rectangle, rhombus and parallelogram.

Starting from the most general: quadrilateral → trapezium (one pair parallel) OR parallelogram (two pairs parallel). A parallelogram with 4 right angles is a rectangle; a parallelogram with 4 equal sides is a rhombus. A shape that's BOTH a rectangle AND a rhombus (4 right angles AND 4 equal sides) is a square. The kite sits separately — defined by adjacent equal pairs of sides, not by parallel sides.

Square = rectangle + rhombus (most special)

Square = rectangle AND rhombus

It has 4 right angles (rectangle) AND 4 equal sides (rhombus).

Rectangle ≠ square (always)

A rectangle doesn't HAVE to have equal sides — but if it does, it's a square.

Kite is its own branch

A kite doesn't need parallel sides — it just needs 2 pairs of ADJACENT equal sides.

What to write in your book

Family tree: quadrilateral → (trapezium OR parallelogram OR kite) → rectangle / rhombus → square.
A square is BOTH a rectangle and a rhombus (4 right angles AND 4 equal sides).
A kite sits on its own branch — defined by 2 pairs of ADJACENT equal sides, not parallel sides.

True or false?

Every square is also a rectangle, a rhombus, and a parallelogram.

Naming from Descriptions

+5 XP

To identify a quadrilateral from a list of features, look for the most specific name. If a shape has 4 right angles AND 4 equal sides, it's not just a "rectangle" — it's a square. Always go as far down the family tree as the facts allow.

Decision flow:
• 4 right angles + 4 equal sides → square
• 4 right angles only → rectangle
• 4 equal sides only → rhombus
• 2 pairs of parallel sides → parallelogram
• Exactly 1 pair of parallel sides → trapezium
• 2 pairs of adjacent equal sides → kite

Specific > general: a square is a "square" before it's a "rectangle".

Look at all properties

Not just sides — angles, parallel-marks, tick-marks all give clues.

Multiple names allowed

A square is correctly called a square, but ALSO a rectangle, rhombus, parallelogram.

"Exactly one" matters

A trapezium has EXACTLY one pair of parallel sides — not "at least one".

What to write in your book

Always choose the MOST specific name: 4 equal sides + 4 right angles → square (not just rectangle).
4 right angles only → rectangle; 4 equal sides only → rhombus.
NSW trapezium definition: EXACTLY one pair of parallel sides (not "at least one").

Fill in the blank.

Three angles of a quadrilateral are $90^{\circ}, 100^{\circ}$ and $80^{\circ}$. The fourth angle is °.

Watch Me Solve It · 3 examples

Watch Me Solve It · Name the quadrilateral

+15 XP per step

PROBLEM

A quadrilateral has all four sides equal in length, but its angles are NOT right angles. What is its most specific name?

1
Check the conditions
4 equal sides ✓ | No right angles ✗
2
Walk the family tree
Equal sides + right angles → square. No right angles, so NOT a square.
3
Identify
A quadrilateral with 4 equal sides (but not 4 right angles) is a rhombus.
It's also a parallelogram, but "rhombus" is more specific.

AnswerRhombus.

Watch Me Solve It · Missing angle in a quadrilateral

+15 XP per step

PROBLEM

Three angles of a quadrilateral are $80^{\circ}, 100^{\circ}$ and $120^{\circ}$. Find the fourth angle.

1
Recall the angle sum
Sum of interior angles of a quadrilateral $= 360^{\circ}$ (∠ sum of quad)
2
Add the three known angles
$80 + 100 + 120 = 300^{\circ}$
3
Subtract
Fourth angle $= 360 - 300 = 60^{\circ}$
Check: $80 + 100 + 120 + 60 = 360^{\circ}$ ✓

AnswerFourth angle $= 60^{\circ}$.

Watch Me Solve It · "List every name"

+15 XP per step

PROBLEM

A quadrilateral has 4 equal sides AND 4 right angles. List every category from the family tree it belongs to.

1
Identify the most specific name
4 equal sides + 4 right angles → square
2
Trace up the family tree
Square → rectangle (4 right angles) → parallelogram (2 pairs parallel)
3
Also via the other branch
Square → rhombus (4 equal sides) → parallelogram
So the shape is correctly called: square, rectangle, rhombus, parallelogram, and quadrilateral.

AnswerSquare, rectangle, rhombus, parallelogram, quadrilateral.

Common Pitfalls

heads-up

Confusing "rhombus" with "kite"

A rhombus has FOUR equal sides; a kite has only TWO PAIRS of adjacent equal sides (not all four the same length).

Fix: Rhombus = 4 equal. Kite = 2 short + 2 long (each pair adjacent).

Calling a square "just a square"

A square inherits properties from rectangle, rhombus AND parallelogram. Failing to recognise this can lose marks in proofs.

Fix: Whenever you use rectangle or rhombus properties on a square, that's valid.

Using "at least one pair parallel" for trapezium

The NSW Stage 4 definition is EXACTLY one pair. Some textbooks use the inclusive version, but for HSC/NSW use exactly one.

Fix: If both pairs are parallel, it's a parallelogram, not a trapezium.

Copy Into Your Books

Definitions

Square: 4 equal sides + 4 right angles
Rectangle: 4 right angles
Parallelogram: 2 pairs parallel
Rhombus: 4 equal sides

More definitions

Trapezium: exactly 1 pair parallel
Kite: 2 pairs adjacent equal sides
Quadrilateral: any 4-sided closed plane figure

Angle sum

$a + b + c + d = 360^{\circ}$
Reason: (∠ sum of quad)
Proof: split into 2 triangles

Hierarchy

Square = rectangle + rhombus
Rectangle, Rhombus → parallelogram
Always pick the MOST specific name

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Naming & Angle Sum

4 problems

Four quick drills on naming and the $360^{\circ}$ angle sum. Solve, then reveal.

1 A quadrilateral has 4 right angles. Its sides are NOT all equal. Name it.

4 right angles only.Rectangle
2 Three angles of a quadrilateral are $90^{\circ}, 90^{\circ}, 90^{\circ}$. Find the fourth.

$360 - 270 = 90^{\circ}$.$90^{\circ}$ (it's a rectangle)
3 A quadrilateral has 2 pairs of adjacent equal sides but no parallel sides. Name it.

2 pairs of adjacent equal sides.Kite
4 Three angles of a quadrilateral are $70^{\circ}, 130^{\circ}, 95^{\circ}$. Find the fourth.

$360 - 70 - 130 - 95 = 65^{\circ}$.$65^{\circ}$

Complete in your workbook.

Quick Check · 5 questions

A quadrilateral has 4 equal sides but its angles are NOT right angles. The most specific name is:

+10 XP

Three angles of a quadrilateral are $85^{\circ}, 95^{\circ}$ and $110^{\circ}$. Find the fourth.

+10 XP

The defining feature of a trapezium is:

+10 XP

Every square is also a:

+10 XP

The interior angles of any quadrilateral always add to:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Recall Easy 3 MARKS

Q6. For each description, name the most specific quadrilateral.
(a) Four right angles, opposite sides equal but adjacent sides different.
(b) Four equal sides, no right angles.
(c) Two pairs of adjacent equal sides; one pair of opposite angles equal.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. A quadrilateral has angles $(2x + 10)^{\circ}, (3x - 20)^{\circ}, (x + 30)^{\circ}$ and $(4x + 40)^{\circ}$.
(a) Set up an equation.
(b) Solve for $x$.
(c) State the four angles and verify they sum to $360^{\circ}$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. Decide whether each statement is TRUE or FALSE. Give a one-sentence reason.
(a) Every rectangle is a parallelogram.
(b) Every rhombus is a square.
(c) Every square is a trapezium (using NSW "exactly one pair parallel" definition).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Rhombus (4 equal sides without right angles).

2. D — $70^{\circ}$. $360 - 85 - 95 - 110$.

3. A — Exactly one pair of parallel sides.

4. C — Rectangle, rhombus AND parallelogram.

5. D — $360^{\circ}$ (two triangles).

Show Your Working Model Answers

Q6 (3 marks): (a) Rectangle [1]. (b) Rhombus [1]. (c) Kite [1].

Q7 (3 marks): (a) $(2x + 10) + (3x - 20) + (x + 30) + (4x + 40) = 360$ (∠ sum of quad) [1]. (b) $10x + 60 = 360 \Rightarrow 10x = 300 \Rightarrow x = 30$ [1]. (c) Angles: $70^{\circ}, 70^{\circ}, 60^{\circ}, 160^{\circ}$. Check: $70 + 70 + 60 + 160 = 360^{\circ}$ ✓ [1].

Q8 (3 marks): (a) TRUE — a rectangle has two pairs of parallel sides, so it's a parallelogram [1]. (b) FALSE — a rhombus has 4 equal sides but doesn't have to have right angles; only when it also has 4 right angles is it a square [1]. (c) FALSE under NSW "exactly one pair" definition — a square has TWO pairs of parallel sides, so it's NOT a trapezium [1].

Stretch Challenge · +25 XP, +10 coins

The Mystery Quadrilateral

A quadrilateral $ABCD$ has the following clues: $AB = BC$ and $AD = DC$ (two pairs of adjacent equal sides). The diagonal $BD$ is the axis of symmetry, and $\angle ABD = \angle CBD$. The diagonal $AC$ is perpendicular to $BD$. The angles at $A$ and $C$ are equal but obtuse. NO sides are parallel. (a) Name the quadrilateral. (b) Could it ever be a rhombus? Why or why not? (c) If $\angle ABC = 80^{\circ}$ and $\angle ADC = 100^{\circ}$, find the angles at $A$ and $C$.

Reveal solution

(a) The shape is a kite — two pairs of adjacent equal sides with one diagonal as the axis of symmetry. (b) It cannot be a rhombus, because a rhombus has all FOUR sides equal AND has parallel sides; here we're told the pairs are unequal in length and no sides are parallel. (c) Since the angle sum is $360^{\circ}$ and $\angle ABC + \angle ADC = 80 + 100 = 180^{\circ}$, the angles at $A$ and $C$ together sum to $360 - 180 = 180^{\circ}$, and they're equal (axis of symmetry along $BD$), so each $= 90^{\circ}$.

Quick Review

Quadrilateral

4 sides, 4 vertices, 4 angles summing to $360^{\circ}$.

Square

4 equal sides AND 4 right angles.

Rectangle

4 right angles.

Parallelogram

2 pairs of parallel sides.

Rhombus / Kite

Rhombus = 4 equal. Kite = 2 pairs adjacent equal.

Trapezium

Exactly 1 pair parallel (NSW).

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