# Symmetry, Reflections and Turns

> Find lines of symmetry and rotational symmetry, reflect shapes in mirror lines, follow a shape through flips and turns, and predict what folded and cut paper looks like.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/symmetry/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The paper snowflake
Isla folds a square of paper in half, then in half again. She snips a small triangle from the corner where the folds meet. When she opens it, there is a diamond shaped hole right in the middle.

"How did one cut make that?" asks her little brother. Isla explains that each fold is a mirror line. A cut on the folded paper is copied on the other side of every fold. Two folds make four copies, and the four copies of the triangle join up into a diamond.

**In short:** A line of symmetry splits a shape into two halves that are mirror images. A shape has rotational symmetry if it looks the same after part of a turn. A reflection flips a shape over a mirror line, and every fold in paper acts like a mirror line.

## Before you start

- The names of triangles and quadrilaterals (see Triangles and Quadrilaterals).
- Quarter, half and full turns (see Types of Angles).

## Lines of symmetry

If you fold a shape along a **line of symmetry**, the two halves match exactly.

| Shape | Lines of symmetry |
|---|---|
| Square | 4 |
| Rectangle (not a square) | 2 |
| Rhombus (not a square) | 2 |
| Kite | 1 |
| Isosceles trapezium | 1 |
| Parallelogram (not a rectangle) | 0 |
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Regular hexagon | 6 |
| Circle | Endless |

### Exam tip: Regular shapes
A regular shape (all sides and angles equal) has as many lines of symmetry as it has sides.

### Watch out: The parallelogram trap
A parallelogram looks balanced, but folding it along either diagonal or across the middle never makes the halves match. It has no lines of symmetry.

## Rotational symmetry

A shape has **rotational symmetry** if it fits on itself before a full turn. The **order** is how many times it fits in one full turn.

- A square fits 4 times: order 4.
- A rectangle and a parallelogram fit twice: order 2.
- The letter Z fits twice, but has no line of symmetry.

## Reflections

A reflection flips a shape over a **mirror line**. Each point of the image is the **same distance** from the mirror line as the point it came from, on the other side.

### Worked example: On a grid
A point is 3 squares to the left of a vertical mirror line. Its reflection is 3 squares to the right, on the same row.

### Key idea: Two flips make a turn
Reflect a shape in a vertical line, then in a horizontal line. The result is the same as a **half turn**. Reflecting in the same line twice puts the shape back where it started.

### Worked example: Following the corners
A square has W at the top left, X at the top right, Y at the bottom right and Z at the bottom left.

- Reflect in the vertical line: W and X swap, and Z and Y swap. Top row: X, W.
- Reflect in the horizontal line: the rows swap. Top row: Y, Z.

That is a half turn, with Y now at the top left.

## Folding and cutting paper

### Key idea: Every fold is a mirror line
1. Unfold one fold at a time, in the reverse order.
2. Each time, reflect every cut across that fold.
3. A cut on a fold opens out into a shape that is symmetric about that fold.

### Worked example: Cutting on the fold
Fold a sheet in half and cut a semicircle from the folded edge. Open it: the semicircle and its reflection make a whole circle.

## Completing a symmetric pattern

To make a grid pattern with 4 lines of symmetry, it must match across the vertical line, the horizontal line and both diagonals. Reflect each shaded square in every line, and shade all the images.

### Challenge: Selective level: shapes from clues
A quadrilateral has no parallel sides, two sides of 6 cm next to each other, two sides of 9 cm next to each other, and one right angle. Two pairs of equal sides next to each other make a kite. A kite has exactly 1 line of symmetry, through the corners where the equal pairs meet.

A robot faces north and makes 7 quarter turns clockwise. Every 4 quarter turns is a full turn, so 7 is the same as 3 quarter turns clockwise. It faces west.

**Mind map**

```text
Symmetry
  Lines of symmetry
    Fold and halves match
    Regular shape: one per side
    Parallelogram has none
  Rotational
    Order: fits per full turn
  Reflections
    Same distance from mirror
    Two flips make a half turn
  Paper folding
    Each fold is a mirror
    Unfold in reverse order
```

## Quick summary

- A line of symmetry splits a shape into two mirror halves.
- A square has 4 lines, a rectangle 2, a kite 1 and a parallelogram 0.
- A regular shape has as many lines of symmetry as sides.
- Rotational order is how many times a shape fits on itself in a full turn.
- Reflecting in a vertical then a horizontal line is the same as a half turn.
- When paper is folded, each cut is reflected across every fold.

## Quiz

Try these questions from the chapter quiz.

1. How many lines of symmetry does a square have?
   - A. 2
   - B. 3
   - C. 4
   - D. 8
   - Answer: C. 4. Two through the middles of the sides, and two through the corners.

2. How many lines of symmetry does a rectangle that is not a square have?
   - A. 1
   - B. 2
   - C. 4
   - D. 0
   - Answer: B. 2. One across the middle each way. The diagonals are not lines of symmetry.

3. Which capital letter has a horizontal line of symmetry?
   - A. A
   - B. B
   - C. N
   - D. Z
   - Answer: B. B. The top and bottom halves of B match. A has a vertical line only.

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