CC-10
Symmetry
Common Core State Standards
MACC.912.G-CO.1.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
MP 7
You can use what you know about reflections and rotations to identify types of
symmetry. A figure has symmetry if there is a rigid motion that maps the figure
onto itself.
A figure has line symmetry, or reflectional symmetry, if there is a reflection for which
the figure is its own image. The line of reflection is called the line of symmetry.
A figure has a rotational symmetry, if its image, after a rotation of less than 360°, is
exactly the same as the original figure. A figure has point symmetry if a 180° rotation
about a center of rotation maps the figure onto itself.
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1. Use a straightedge to copy the rhombus at the right.
a. How many lines of reflection, or lines of symmetry, does the rhombus have?
b. Draw all of the lines of symmetry.
2. Do all parallelograms have reflectional symmetry? Explain your reasoning.
3. The isosceles trapezoid at the right has only 1 pair of parallel sides. How many
lines of symmetry does the trapezoid have?
4. Do all isosceles trapezoids have reflectional symmetry? Do all trapezoids have
reflectional symmetry? Explain.
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5. Use a straightedge to copy the regular hexagon at the right.
a. How many lines of symmetry does a regular hexagon have?
b. Draw all of the lines of symmetry.
6. What are the center and angle(s) of the rotations that map the regular hexagon
onto itself?
7. Do all regular polygons have rotational symmetry? Explain your reasoning.
8. Do all regular polygons have point symmetry? Explain.
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Copy and cut out the shapes below. Shade 12 of each square to represent the orange
sections. Arrange the shapes to make a design that has both reflectional symmetry and
rotational symmetry.
9. Draw the design you made.
10. How many lines of symmetry does your design have? Sketch each line of
symmetry.
11. Why are the colors of the tiles important to the symmetry?
12. Does your design have more than one of angle of rotation that maps it onto itself?
If so, what are they?
13. Can you change the center of rotation and still map the figure onto itself? Explain.
Exercises
Tell what type(s) of symmetry each figure has. Sketch the figure and the line(s) of
symmetry, and give the angle(s) of rotation when appropriate.
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17. Vocabulary If a figure has point symmetry, must it also have rotational
symmetry? Explain.
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18. Writing
A quadrilateral with vertices (1, 5) and (–2, –3) has point symmetry about
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the origin.
a. Show that the quadrilateral is a parallelogram.
b. How can you use point symmetry to find the other vertices?
19. Error Analysis Your friend thinks that the regular pentagon in the diagram has
10 lines of symmetry. Explain and correct your friend’s error.
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