# Quadrilaterals

> Know the square, rectangle, rhombus, parallelogram, trapezium and kite, and use the 360 degree rule.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/quadrilaterals/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The kite workshop
Before the school kite festival, Mr Okafor shows the class how to make a kite. "Two short sticks of equal length at the top, two long ones at the bottom," he says. The shape that appears is a **kite**: two pairs of equal sides next to each other.

Then he shows them the box the paper came in. Its side is a **rectangle**. The tiles on the floor are **squares**, and the slanted bike rack makes a **parallelogram**.

All of these are **quadrilaterals**: shapes with 4 straight sides. Each has its own special features, and test questions love to use them.

**In short:** A quadrilateral is a shape with 4 sides, and its angles add to $360^\circ$. Squares, rectangles, parallelograms, rhombuses, trapeziums and kites are special quadrilaterals, told apart by their equal sides, parallel sides and right angles.

## Before you start

- **Parallel** lines go the same way and never meet, like train tracks. Diagrams mark them with matching arrows.
- The angles in a triangle add to $180^\circ$ (see Triangles).
- Angles on a straight line add to $180^\circ$.

## Angle sum of a quadrilateral

### Key idea: Always $360^\circ$
The four angles inside any quadrilateral add up to $360^\circ$.

Why? A line from one corner to the opposite corner cuts it into **two triangles**, and $2 \times 180^\circ = 360^\circ$.

### Worked example: Missing angle
A quadrilateral has angles $120^\circ$, $90^\circ$ and $80^\circ$. The fourth is $360 - 120 - 90 - 80 = 70^\circ$.

## The quadrilateral family

*Diagram: Six quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium and kite*
*The six special quadrilaterals.*

| Shape | Sides | Angles | Parallel sides |
|---|---|---|---|
| **Square** | all 4 equal | all $90^\circ$ | 2 pairs |
| **Rectangle** | opposite sides equal | all $90^\circ$ | 2 pairs |
| **Parallelogram** | opposite sides equal | opposite angles equal | 2 pairs |
| **Rhombus** | all 4 equal | opposite angles equal | 2 pairs |
| **Trapezium** | can be anything | can be anything | exactly 1 pair |
| **Kite** | 2 pairs of equal sides next to each other | 1 pair of opposite angles equal | none |

### Exam tip: Family tree
A square is a special rectangle, because all its sides are equal. It is also a special rhombus, because all its angles are $90^\circ$. Rectangles and rhombuses are special parallelograms. So every square is also a parallelogram!

## Using the parallel sides

In a parallelogram, two angles next to each other along a side add to $180^\circ$. (They are **co-interior** angles between parallel lines, explained in the next chapter.)

### Worked example: Parallelogram
One angle of a parallelogram is $100^\circ$.

The opposite angle is also $100^\circ$. Each neighbour is $180 - 100 = 80^\circ$.

Check: $100 + 80 + 100 + 80 = 360$.

### Worked example: Trapezium
A trapezium has its parallel sides at the top and bottom. Its top two angles are $90^\circ$ and $120^\circ$. Each bottom angle pairs with the top angle above it to make $180^\circ$. So the bottom angles are $90^\circ$ and $60^\circ$.

### Worked example: Kite
A kite has angles of $x$, $x$, $y$ and $z$, with the two $x$ angles opposite each other. If $y = 100^\circ$ and $z = 60^\circ$, then $2x = 360 - 160 = 200$ and $x = 100^\circ$.

## Exterior angles

If one side is stretched past a corner, the outside angle and the inside angle make a straight line. A quadrilateral with angles $130^\circ$, $95^\circ$ and $80^\circ$ has a fourth angle of $55^\circ$, and the exterior angle there is $180 - 55 = 125^\circ$.

### Challenge: Selective level: algebra in shapes
The angles of a quadrilateral are $x^\circ$, $2x^\circ$, $3x^\circ$ and $4x^\circ$. Then $10x = 360$ and $x = 36$. The angles are $36^\circ$, $72^\circ$, $108^\circ$ and $144^\circ$.

A rhombus has one angle three times another. Neighbouring angles add to $180^\circ$, so $x + 3x = 180$ and the angles are $45^\circ$ and $135^\circ$.

An isosceles trapezium (its two slanting sides are equal) has two top angles of $125^\circ$. Each bottom angle is $180 - 125 = 55^\circ$.

**Mind map**

```text
Quadrilaterals
  Angle sum
    360°
    Two triangles
  Two pairs of parallel sides
    Square
    Rectangle
    Rhombus
    Parallelogram
  One pair
    Trapezium
  None
    Kite
  Shortcuts
    Opposite angles equal in parallelograms
    Neighbours on a parallel side add to 180°
```

## Quick summary

- A quadrilateral has 4 sides and its angles add to $360^\circ$.
- Square: 4 equal sides, 4 right angles. Rectangle: opposite sides equal, 4 right angles.
- Parallelogram: opposite sides parallel and equal, opposite angles equal.
- Rhombus: 4 equal sides, opposite angles equal.
- Trapezium: exactly 1 pair of parallel sides.
- Kite: 2 pairs of equal sides next to each other, 1 pair of equal opposite angles.
- Neighbouring angles along a parallel side add to $180^\circ$.

## Quiz

Try these questions from the chapter quiz.

1. A quadrilateral has angles $120^\circ$, $90^\circ$ and $80^\circ$. What is the fourth angle?
   - A. $60^\circ$
   - B. $70^\circ$
   - C. $80^\circ$
   - D. $110^\circ$
   - Answer: B. $70^\circ$. $360 - 290 = 70$.

2. Which shape has exactly one pair of parallel sides?
   - A. rectangle
   - B. rhombus
   - C. trapezium
   - D. kite
   - Answer: C. trapezium. A trapezium has exactly one pair of parallel sides.

3. Which shape has four equal sides but its angles are not all $90^\circ$?
   - A. square
   - B. rhombus
   - C. rectangle
   - D. kite
   - Answer: B. rhombus. A rhombus has 4 equal sides. If the angles were all $90^\circ$ it would be a square.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/quadrilaterals/#quiz