Lesson 16

More Symmetry

Let’s describe more symmetries of shapes.

16.1: Which One Doesn't Belong: Symmetry

Which one doesn’t belong?


Circle with center A.


Triangle A B C. Single tick marks on sides A C and C B.


Quadrilateral A B C D. Single arrows on opposite sides D A and C B. Double arrows on opposite sides D C and A B.


6-sided irreguar shape B C D E F G that has all right angles.


16.2: Self Rotation

Determine all the angles of rotation that create symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:

  • the name of your shape
  • the definition of your shape
  • drawings of each rotation that creates symmetry
  • a description in words of each rotation that creates symmetry, including the center, angle, and direction of rotation
  • one non-example (a description and drawing of a rotation that does not result in symmetry)

Finite figures, like the shapes we have looked at in class, cannot have translation symmetry. But with a pattern that continues on forever, it is possible. Patterns like this one that have translation symmetry in only one direction are called frieze patterns.

Frieze pattern, octagon, centered with half octagons on both sides.

  1. What are the lines of symmetry for this pattern?
  2. What angles of rotation produce symmetry for this pattern?
  3. What translations produce symmetry for this pattern if we imagine it extending horizontally forever?

16.3: Parallelogram Symmetry

Clare says, "Last class I thought the parallelogram would have reflection symmetry. I tried using a diagonal as the line of symmetry but it didn’t work. So now I’m doubting that it has rotation symmetry."

Lin says, "I thought that too at first, but now I think that a parallelogram does have rotation symmetry. Here, look at this."

How could Lin describe to Clare the symmetry she sees?


A parallelogram inside two concentric circles.


A shape has rotation symmetry if there is a rotation between 0 and 360 degrees that takes the shape to itself. A regular hexagon has many angles that work to create rotation symmetry. Here is one of them. What other angles would create a rotation where the image is the same as the original figure?

Regular hexagon. At its center, a 60 degree angle is marked, and the angle sides extend to two vertices of the hexagon, forming a triangle. An arc of a circle also connects the two vertices.

Can you think of a shape that has translation symmetry?

There aren’t any polygons with translation symmetry, but an infinite shape like a line can be translated such that the translation takes the line to itself.

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 \(m\).

    Reflect $A$ across 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.

  • rotation symmetry

    A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as \(0^\circ\) and \(360^\circ\) that leave every point on the figure where it is.)

  • 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 \(t\).