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# Reflection Symmetry

## Properties of a shape with one or more lines of symmetry.

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Practice Reflection Symmetry
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Reflection Symmetry

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

#### Guidance

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.

Example A

Does a square have reflection symmetry?

Solution: 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.

Example B

How many lines of symmetry does a square have?

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

Example C

Does a trapezoid have reflection symmetry?

Solution: 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.

Concept Problem Revisited

When you reflect the regular pentagon below across line f\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 f\begin{align*}f\end{align*} is a line of symmetry because it causes the pentagon to have reflection symmetry.

#### Vocabulary

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.

#### Guided Practice

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.

1.

2.

3.

1. Yes, it has one horizontal line of symmetry

2. No, it does not have reflection symmetry.

3. Yes, it has one vertical line of symmetry.

#### Practice

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 n\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.

### Vocabulary Language: English

Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Isosceles Triangle

Isosceles Triangle

An isosceles triangle is a triangle in which exactly two sides are the same length.
Line of Symmetry

Line of Symmetry

A line of symmetry is a line that can be drawn to divide a figure into equal halves.
Line Symmetry

Line Symmetry

A figure has line symmetry or reflection symmetry when it can be divided into equal halves that match.
reflection symmetry

reflection symmetry

A figure has reflection symmetry if it can be reflected across a line and look exactly the same as it did before the reflection.
Symmetry

Symmetry

A figure has symmetry if it can be transformed and still look the same.