Math resources Geometry Symmetry

Lines of Symmetry

# Lines of symmetry

Here you will learn about lines of symmetry, including symmetry properties within polygons, angle properties, and symmetry of different line graphs.

Students first learn about line symmetry in grade 4 with their work with 2D shapes in geometry.

## What are lines of symmetry?

Lines of symmetry are straight lines that divide a shape into two equal parts, where one part is an exact reflection or mirror image of the other.

In geometry, many 2D shapes have line symmetry.

For example,

Equilateral
triangle
SquareRectangleRegular
hexagon

3 lines of
symmetry

4 lines of
symmetry

2 lines of
symmetry

6 lines of
symmetry

Line symmetry can be seen in nature.

For example,

Line symmetry can be seen in art.

For example,

### Lines of symmetry in regular shapes

Regular polygons are polygons that have equal side lengths and equal angle measures.

For example,

Equilateral Triangle

3 lines of symmetry

Notice how each line of symmetry can be a fold line.

Square

4 lines of symmetry

Notice how each line of symmetry can be a fold line.

Regular pentagon

5 lines of symmetry

Notice how each line of symmetry can be a fold line.

Regular hexagon

6 lines of symmetry

Notice how each line of symmetry can be a fold line.

Notice how the number of lines of symmetry in these regular polygons is equal to the number of sides and angles the polygons have.

All regular polygons share the property that the number of sides is equal to the number of lines of symmetry.

### Lines of symmetry in irregular shapes

Irregular polygons are polygons that do not have all equal angles and all equal side lengths. However, they can have line symmetry.

For example, these irregular polygons have line symmetry.

Let’s look at different shapes (specifically quadrilaterals) and their number of lines of symmetry.

of Symmetry
Square
4
Rectangle
2
Parallelogram
0
Rhombus
2
Kite
1
Trapezoid
0
Isosceles Trapezoid
1
Irregular
0

### Lines of symmetry in circles

The diameter of a circle splits the circle into two equal sized half circles, or semicircles. Think of the diameter as the line of reflection.

A circle can be folded in-half many ways, meaning that there are an infinite number of lines of symmetry.

## Common Core State Standards

How does this apply to 4th grade math?

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

## How to identify lines of symmetry

In order to draw/identify lines of symmetry:

1. Locate the center of the 2D shape.
2. Draw a vertical line through the center and check for line symmetry.
3. Draw a horizontal line through the marked center and check for line symmetry.
4. Draw a line from each vertex through the center and check for symmetry.
5. State the number of lines of symmetry.

## Lines of symmetry examples

### Example 1: regular shapes

How many lines of symmetry does the equilateral triangle have?

1. Locate the center of the 2D shape.

2Draw a vertical line through the center and check for line symmetry.

The vertical line through the center is a line of symmetry dividing the triangle into two equal halves that are mirror images of each other. Notice that it can be a fold line.

3Draw a horizontal line through the marked center and check for line symmetry.

4Draw a line from each vertex through the center and check for line symmetry.

5State the number of lines of symmetry.

The equilateral triangle has 3 lines of symmetry. An equilateral triangle has 3 equal sides and 3 equal angles, so it will have 3 lines of symmetry.

### Example 2: the regular pentagon

How many lines of symmetry does the regular pentagon have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

Draw a horizontal line through the marked center and check for line symmetry.

Draw a line from each vertex through the center and check for line symmetry.

State the number of lines of symmetry.

### Example 3: irregular shapes

How many lines of symmetry does the trapezoid have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

Draw a horizontal line through the marked center and check for line symmetry.

Draw a line from each vertex through the center and check for line symmetry.

State the number of lines of symmetry.

### Example 4: irregular shapes

How many lines of symmetry does the irregular hexagon have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

Draw a horizontal line through the marked center and check for line symmetry.

Draw a line from each vertex through the center and check for line symmetry.

State the number of lines of symmetry.

### Example 5: irregular shapes

How many lines of symmetry does the parallelogram have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

Draw a horizontal line through the marked center and check for line symmetry.

Draw a line from each vertex through the center and check for line symmetry.

State the number of lines of symmetry.

### Example 6: irregular shapes

How many lines of symmetry does the shape have?

Locate the center of the 2D shape.

Draw a vertical line through the center and check for line symmetry.

Draw a horizontal line through the marked center and check for line symmetry.

Draw a line from each vertex through the center and check for line symmetry.

State the number of lines of symmetry.

### Teaching tips for lines of symmetry

• Use hands on manipulatives to help students gain understanding of lines of symmetry.

• Infuse creative activities such as origami so that students can build understanding through fun learning tasks.

### Easy mistakes to make

• Not visualizing that a line of symmetry is like a fold line
For example, thinking that this is a line of symmetry. If you were to fold the parallelogram along the line of symmetry, the two halves would not be mirror images of each other.

• Thinking that all quadrilaterals have \bf{4} lines of symmetry
The only quadrilateral with 4 lines of symmetry is a square because it is a regular polygon. The other quadrilaterals are not regular polygons so do not have 4 lines of symmetry.

• The number of sides always \textbf{=} the number of lines of symmetry
Although this is true for regular shapes, it is not true for all shapes.

• Thinking all triangles have \bf{3} lines of symmetry
For example, this right scalene triangle does not have any lines of symmetry.

Equilateral triangles are the only triangles that have 3 lines of symmetry because it is a regular triangle.

### Practice lines of symmetry questions

1. Find the number of lines of symmetry for the regular hexagon below.

4

2

6

3

Regular polygons have the same number of lines of symmetry as the number of sides that they have.

A hexagon has 6 sides so it has 6 lines of symmetry.

Marking the center, there are 6 lines that can be drawn through the center that cut the regular hexagon into equal halves that are mirror images of each other.

2. Find the number of lines of symmetry for the rectangle below.

1

2

0

4

After marking the center, the vertical line through the center and the horizontal line through the center are the only lines of symmetry.

Any other line drawn through the center will not result in the shape being divided into two equal halves that are mirror images.

3. Find the number of lines of symmetry for the kite below.

0

4

2

1

After marking the center, the only line of symmetry is the horizontal line.

There is no other line that can be drawn through the center that will divide the shape into two equal parts that are mirror images of each other.

4. Find the number of lines of symmetry for the irregular hexagon below.

2

0

3

6

After marking the center, the lines of symmetry are the vertical line through the center and the horizontal line through the center.

No other lines through the center will divide the shape into two equal parts that are mirror images of each other.

5. How many lines of symmetry does the smiley have?

0

1

infinite

8

Circles have an infinite amount of lines of symmetry. However, you have to take the eyes and mouth into account when figuring out the lines of symmetry.

There is only one line of symmetry vertically down the face that will divide the smiley face into two equal mirror images of each other.

6. How many lines of symmetry does the photo have?

0

1

2

infinite

The photo has two lines of symmetry, one that is horizontal through the center and the other that is vertical through the center.

There are no other lines that can be drawn that divides the photo into two equal halves that are mirror images of each other.

## Lines of symmetry FAQs

What are the types of symmetry?

There are two types of symmetry. Line symmetry is drawing a line through the center of a figure where both halves are mirror images of each other. Rotational symmetry involves turning a shape and determining how many times the shape appears as its original in one complete 360^{\circ} turn.

What is the axis of symmetry?

In algebra, you will learn about quadratic functions whose graph is a parabola. The parabola has a line going through the center that divides it into identical halves, which is the axis of symmetry. It is also the line of symmetry of the parabola.

Do all triangles have line symmetry?

No, a scalene triangle does not have any lines of symmetry. Meaning there are no imaginary lines or mirror lines that can be drawn through the center of the triangle, where one half is equal and the mirror image of the other half. Isosceles triangles have only one line of symmetry. Equilateral triangles are regular polygons, so they have 3 lines of symmetry.

Are all regular polygons symmetrical?

Yes, regular polygons are symmetrical shapes because they have equal sides and equal angles. They have line symmetry and rotational symmetry.

What does congruent mean?

Congruent means identical in shape and size. When drawing a dotted line representing line symmetry, each half of the shape is congruent.

## The next lessons are

• Angles in polygons
• Congruence and similarity
• Transformations

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