# Lesson 15

Symmetry

### Lesson Narrative

In this lesson and the next, students determine the cases where applying a certain rigid motion to a shape doesn’t change it. This is the idea of symmetry. A shape is said to have symmetry if there is a rigid transformation that takes the shape to itself. Students first study reflection symmetry using lines of symmetry, and then they study rotation symmetry in a subsequent lesson. Translation symmetry isn’t mentioned explicitly, but students were exposed to the idea that a line has translation symmetry in a previous lesson. Students apply their understanding of rigid transformations to identify shapes where there is a line of symmetry which reflects the shape onto itself, satisfying the definition of reflection symmetry. The fact that reflecting a segment across its perpendicular bisector exchanges its endpoints will be useful in the next unit when students study triangle congruence.

In one activity, each group is assigned a different shape to consider. Students make use of structure when they discuss which lines of symmetry apply to a type of shape generally, rather than limiting their thinking to a given example (MP7).

### Learning Goals

Teacher Facing

• Describe (orally and in writing) the reflections that take a figure onto itself.

### Student Facing

• Let’s describe some symmetries of shapes.

### Required Preparation

Print and cut up slips from the blackline master. The blackline master for this lesson contains 8 different shapes. Each group of 2–4 students will be investigating a shape. Prepare enough copies of the blackline master so that each student in each group gets a copy of the shape their group will investigate. (Note: Students will repeat this process for rotation symmetry in the next lesson; it may be easier to prepare twice as many shapes once rather than repeat the process.)

### Student Facing

• I can describe the reflections that take a figure onto itself.

Building On

### Glossary Entries

• assertion

A statement that you think is true but have not yet proved.

• congruent

One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

• directed line segment

A line segment with an arrow at one end specifying a direction.

• image

If a transformation takes $$A$$ to $$A'$$, then $$A$$ is the original and $$A'$$ is the image.

• line of symmetry

A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

• reflection

A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line.

In the figure, $$A'$$ is the image of $$A$$ under the reflection across the line $$m$$.

• reflection symmetry

A figure has reflection symmetry if there is a reflection that takes the figure to itself.

• rigid transformation

A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

• rotation

A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

$$P'$$ is the image of $$P$$ after a counterclockwise rotation of  $$t^\circ$$ using the point $$O$$ as the center.

​​​​​Quadrilateral $$ABCD$$ is rotated 120 degrees counterclockwise using the point $$D$$ as the center.

• symmetry

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

• theorem

A statement that has been proved mathematically.

• translation

A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

In the figure, $$A'$$ is the image of $$A$$ under the translation given by the directed line segment $$t$$.