What happens when you reflect the regular pentagon below across line \begin{align*}f\end{align*}? Why is the line of reflection in this case called a line of symmetry?

### Reflection Symmetry

A shape has **symmetry** if it is indistinguishable from its transformed image. A shape has **reflection** **symmetry** if there exists a line of reflection that carries the shape onto itself. This line of reflection is called a **line of symmetry**. In other words, if you can reflect a shape across a line and the shape looks like it never moved, it has reflection symmetry.

A rectangle is an example of a shape with reflection symmetry. A line of reflection through the midpoints of opposite sides will always be a line of symmetry.

A rectangle has **two lines of symmetry**. You can imagine folding the rectangle along each line of symmetry and each half of the rectangle would match up perfectly. Remember that a shape has to have *at least one* line of symmetry for it to be considered a shape with reflection symmetry.

#### Recognizing Reflection Symmetry

Does a square have reflection symmetry?

Yes, because there exists at least one line of reflection that carries the square onto itself. Like a rectangle, a line through the midpoints of opposite sides will be a line of symmetry.

#### Identifying Lines of Symmetry

How many lines of symmetry does a square have?

A square has 4 lines of symmetry. Lines through the midpoints of opposites sides and lines through opposite vertices are all lines of symmetry.

#### Reflection Symmetry in Trapezoids

Does a trapezoid have reflection symmetry?

A generic trapezoid will not have reflection symmetry. An isosceles trapezoid will have reflection symmetry because the line connecting the midpoints of the bases will be a line of symmetry.

### Examples

#### Example 1

Earlier, you were asked why is the line of reflection called a line of symmetry.

When you reflect the regular pentagon below across line \begin{align*}f\end{align*}, the pentagon will look exactly the same. Without labeled points, it will be impossible to tell the difference between the original pentagon and its image. This means that the pentagon has **reflection symmetry.** Line \begin{align*}f\end{align*} is a **line of symmetry** because it causes the pentagon to have reflection symmetry.

Does the capital letter have reflection symmetry? If so, state how many lines of symmetry it has and explain where these lines of symmetry are.

#### Example 2

Yes, it has one horizontal line of symmetry.

#### Example 3

No, it does not have reflection symmetry.

#### Example 4

Yes, it has one vertical line of symmetry.

### Review

1. What does it mean for a shape to have symmetry?

2. What does it mean for a shape to have reflection symmetry?

3. What do you think it would mean for a shape to have translation symmetry? Can you think of any shapes or objects with translation symmetry?

For each of the following shapes, state whether or not it has reflection symmetry. If it does, state how many lines of symmetry it has and describe where the lines of symmetry are.

4. Equilateral triangle

5. Isosceles triangle

6. Scalene triangle

7. Parallelogram

8. Rhombus

9. Regular pentagon

10. Regular hexagon

11. Regular 12-gon

12. Regular \begin{align*}n\end{align*}-gon

13. Circle

14. Kite

15. In order to have reflection symmetry, must a polygon have at least two sides that are the same length? Explain.

16. Give examples of objects with reflection symmetry in nature.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.6.