Chapter 6 of 8

All 8 chapters
  1. 1Straight lines and parallel lines, readunderstood
  2. 2Polygons, readunderstood
  3. 3Right angles and turns, readunderstood
  4. 4Measuring angles in degrees, readunderstood
  5. 5Triangles and four sided shapes, readunderstood
  6. 6Lines of symmetry, readunderstood
  7. 7Solid shapes and nets, readunderstood
  8. 8Grids and directions, readunderstood

M2.5.6 Chapter 6

Lines of symmetry

When do the two halves of a folded shape match?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find lines of symmetry by folding paper
  • Complete a drawing so both sides of a mirror line match

Fold a leaf down its middle. Often the two halves almost match. A shape is symmetrical when its two folded halves match exactly.

Fold a paper rectangle in different ways. Some folds work. Some do not.

Draw it first

Fold a rectangle three ways

  1. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.

    1 Draw a rectangle that is not a square.

  2. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.fold 1: halves match

    2 Fold it down the middle. The halves match. The fold is a line of symmetry.

  3. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.2 lines of symmetry

    3 Fold it the other way. The halves match again. So it has 2 lines.

  4. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.diagonal: no match

    4 Fold from one corner to the opposite corner. This fold is a diagonal. The corners do not meet. It is not a line of symmetry.

A line of symmetry is a fold. The two halves match exactly.

Six shapes with their lines of symmetry drawn as dashed lines. The labels are square, 4 lines; rectangle, 2 lines; rhombus, 2 lines; equilateral, 3 lines; isosceles, 1 line; and scalene, 0 lines.square4 linesrectangle2 linesrhombus2 linesequilateral3 linesisosceles1 linescalene0 lines
Dashed lines are lines of symmetry. A scalene triangle has none.

A regular hexagon has 6 lines of symmetry. A circle has more than you can count.

Look at the rhombus. Its two diagonals are lines of symmetry. A rectangle is different.

A line of symmetry also works as a mirror line. What is on one side is a mirror picture of the other.

See it three ways

With things

Fold a sheet in half and open it. Draw half a house on one side. Count the steps from each corner to the fold. Count the same steps on the other side.

As a picture

A grid of dots with a dashed mirror line down the middle. The left half of a house is drawn with solid lines. Its mirror twin on the right is drawn with dashed lines. The label is mirror line.mirror line

In symbols

A corner 2 steps left of the line has a twin. The twin is 2 steps right, at the same height.

Try it

Tear a symmetrical shape

You need: a sheet of paper, a pencil.

  1. Fold the paper in half. Press the fold flat.
  2. Draw half of a leaf or a tree. Its middle must touch the fold.
  3. Tear along your line through both layers. Tear, do not cut.
  4. Open the shape. Look at the fold.

What you will see: The two halves match. The fold is a line of symmetry. You made a symmetrical shape.

If you cannot try it, this is what happens.
A six sided shape with a dashed vertical line down its middle. The two halves match. The label is fold line.fold line

A wrong idea many people have

The diagonal of a rectangle is a line of symmetry.

Why it feels right: The diagonal cuts the rectangle into two triangles. They are the same size.

The test that breaks it: Fold a paper rectangle corner to corner. One corner sticks out past the edge. The corners do not meet.

What is true: Halves that are the same size are not enough. They must match when you fold. A rectangle that is not a square has 2 lines.

Key words

symmetrical
a shape with at least one line of symmetry.
line of symmetry
a fold line where the two halves match exactly.
mirror line
a line of symmetry, used to draw a mirror picture.

Back to your guess

The answer is B: Only a line that makes two halves that match when folded.

A line works only when the two halves match when you fold. Many lines through the middle fail the test. The fold decides.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How many lines of symmetry does a square have?

    Tap your answer.

    1. Not this one. You found the two folds through the middle. A square also has two diagonal folds. Fold paper to check. Try another.

    2. Yes, that is right. A square folds 4 ways. Down the middle, across the middle, and along each diagonal.

    3. Not this one. You counted both ends of each line. One line is one fold. Count each fold once. Try another.

    See the answer

    4

    A square folds 4 ways. Down the middle, across the middle, and along each diagonal.

    • If you chose A: You found the two folds through the middle. A square also has two diagonal folds. Fold paper to check.
    • If you chose C: You counted both ends of each line. One line is one fold. Count each fold once.

  2. A rectangle is 8 cm long and 4 cm wide. How many lines of symmetry does it have?

    See the answer

    2

    The sides are not equal, so it is not a square. It folds down the middle and across the middle. The diagonals do not work.

  3. Which shape has no line of symmetry?

    Three shapes. A is a square. B is an isosceles triangle. C is a scalene triangle. The labels are A, B and C.ABC
    Tap your answer.

    1. Not this one. A square folds 4 ways. Try the folds on paper. Try another.

    2. Not this one. An isosceles triangle folds down its middle. The halves match. Try another.

    3. Yes, that is right. C is a scalene triangle. Its sides all differ, so no fold makes the halves match.

    See the answer

    Shape C

    C is a scalene triangle. Its sides all differ, so no fold makes the halves match.

    • If you chose A: A square folds 4 ways. Try the folds on paper.
    • If you chose B: An isosceles triangle folds down its middle. The halves match.

  4. Corner X is 3 steps left of the mirror line. How many steps right is its twin?

    A grid of dots with a dashed mirror line. The left half of a shape is drawn with solid lines. The corner X is at the top left of the shape. The labels are mirror line and X.mirror lineX
    See the answer

    3 steps

    The twin is as far from the mirror line as X. It is on the other side. X is 3 steps away, so its twin is 3 steps away.

  5. Ana folds a paper rectangle corner to corner. The corners do not meet. What does this show?

    Tap your answer.

    1. Yes, that is right. The two triangles are the same size. They do not match when folded. A line of symmetry needs a match.

    2. Not this one. The diagonal fails. Folds down the middle and across the middle work. Try another.

    3. Not this one. The fold was fine. Halves of the same size do not always match when folded. Try another.

    See the answer

    The diagonal is not a line of symmetry.

    The two triangles are the same size. They do not match when folded. A line of symmetry needs a match.

    • If you chose B: The diagonal fails. Folds down the middle and across the middle work.
    • If you chose C: The fold was fine. Halves of the same size do not always match when folded.

  6. From an earlier chapter

    How many corners does this polygon have?

    A polygon with 7 straight sides and a dot on each corner.
    See the answer

    7

    A polygon has as many corners as sides. Count the dots. There are 7. It is a heptagon.

  7. Look at the rhombus. How many lines of symmetry does it have?

    A rhombus that leans. One small tick on each side marks that all four sides are equal. The label is rhombus.rhombus
    Tap your answer.

    1. Yes, that is right. A rhombus folds along each diagonal and the halves match. Folds across the middle of its sides do not match.

    2. Not this one. It leans, but a lean does not stop a fold. Fold a paper rhombus corner to corner. The halves match. Try another.

    3. Not this one. This rhombus has 4 equal sides, but only 2 folds match. A square has 4 lines. Try another.

    See the answer

    2

    A rhombus folds along each diagonal and the halves match. Folds across the middle of its sides do not match.

    • If you chose B: It leans, but a lean does not stop a fold. Fold a paper rhombus corner to corner. The halves match.
    • If you chose C: This rhombus has 4 equal sides, but only 2 folds match. A square has 4 lines.

  8. Corner X is 3 steps left of the mirror line. Where is its twin?

    A grid of dots with a dashed mirror line. The left half of a house is drawn with solid lines. A dot marks the corner X, at the left end of the roof step. The labels are mirror line and X.mirror lineX
    Tap your answer.

    1. Not this one. You counted 1 from the left edge of the grid. Count steps from the mirror line instead. Try another.

    2. Yes, that is right. The twin is as far from the mirror line as X. It is on the other side. It keeps the same height.

    3. Not this one. This mirror line stands up. It flips left and right only. The twin stays at the same height. Try another.

    See the answer

    3 steps right, at the same height as X

    The twin is as far from the mirror line as X. It is on the other side. It keeps the same height.

    • If you chose A: You counted 1 from the left edge of the grid. Count steps from the mirror line instead.
    • If you chose C: This mirror line stands up. It flips left and right only. The twin stays at the same height.

Keep it

Copy this into your exercise book.

A line of symmetry is a fold where the halves match. A square has 4. A rectangle has 2. A scalene triangle has 0. A diagonal of a rectangle fails.

Draw: A rectangle with its 2 lines of symmetry dashed. Add its diagonal and write "no".

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (773 KB). It opens on any phone.