Geometry Symmetry of Polygons and Rigid Motions Worksheet (PDF)

Geometry math worksheet: Symmetry of Polygons and Rigid Motions – Maths PDF

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This geometry symmetry worksheet for Geometry covers symmetry of polygons and rigid motions (Common Core G-CO.A.3, G-CO.B.6), with a free PDF to print.

These 20 problems practice line symmetry, rotational symmetry and rigid motions of polygons (Common Core G-CO.A.3, G-CO.B.6). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with signs, wheels, tiles and coordinates.

Geometry · Common Core State Standards: G-CO.A.3, G-CO.B.6

Geometry symmetry worksheet: exercises 1 to 10

Exercise 1 – Square lines

How many lines of symmetry does a square have?

Exercise 2 – Table top

A table top is a rectangle 90 cm by 60 cm. How many lines of symmetry does the table top have, and where do they lie?

Exercise 3 – Regular hexagon

Regular hexagon.

How many lines of symmetry does a regular hexagon have, and what is its smallest angle of rotation?

Exercise 4 – Pentagon rotation

Find the smallest positive rotation that maps a regular pentagon onto itself.

Exercise 5 – Octagon rotation

A stop sign is a regular octagon. Find its number of lines of symmetry and its smallest rotation angle.

Exercise 6 – Equilateral triangle

Equilateral triangle.

List the lines of symmetry of an equilateral triangle and the rotations (less than 360°) that map it onto itself.

Exercise 7 – Parallelogram

A parallelogram with center O.

Parallelogram ABCD has no right angles and unequal adjacent sides. Does it have line symmetry? Does it have rotational symmetry? Describe.

Exercise 8 – Isosceles trapezoid

Isosceles trapezoid.

How many lines of symmetry does an isosceles trapezoid have? Does it have rotational symmetry?

Exercise 9 – Rhombus

A rhombus is not a square. State its number of lines of symmetry and its smallest rotation.

Exercise 10 – Twelve-sided polygon

List every rotation less than 360° that maps a regular 12-gon onto itself. How many rotations are there, counting 360°?

Exercises 11 and up

Exercise 11 – Reflect a rectangle

Rectangle with the line x = 3.

Rectangle ABCD has A(0, 0), B(6, 0), C(6, 4) and D(0, 4). Reflect it across the line x = 3. Where does each vertex go? Is this a symmetry?

Exercise 12 – Rotate a rectangle

The center of the same rectangle is (3, 2). A 180° rotation about the center has rule (x, y) → (6 − x, 4 − y). Find the image of each vertex.

Exercise 13 – Find n from the angle

A regular polygon maps onto itself under a rotation of 40° but not under any smaller positive rotation. Find the number of sides.

Exercise 14 – Distances are preserved

Triangle ABC.

Triangle ABC has AB = 7, BC = 5 and CA = 5. It is reflected and then rotated to give triangle A′B′C′. Find the perimeter of A′B′C′ and the measure of an angle in A′B′C′ that corresponds to a 53° angle in ABC.

Exercise 15 – Square symmetries

A square has vertices \((2, 2), (-2, 2), (-2, -2), (2, -2)\). List all rotations about the origin (less than 360°) and all reflections that map it onto itself. How many symmetries are there including the identity?

Exercise 16 – Capital letters

Which of the letters A, B, F, H, M, R, T have a vertical line of symmetry?

Exercise 17 – Pinwheel

Pinwheel with 6 blades.

A pinwheel at a school fair has 6 identical blades evenly spaced. Jamal spins it by 150°. Does the pinwheel look the same? Which rotation angles below 180° give an identical look?

Exercise 18 – Wheel spokes

A bike wheel has 10 evenly spaced spokes. Find the smallest rotation that maps the spokes onto themselves and list the rotations below 180°.

Exercise 19 – Road signs

A yield sign is an equilateral triangle and a stop sign is a regular octagon. Which has more lines of symmetry, and how many more? Which has the larger smallest rotation, and by how much?

Exercise 20 – Rhombus tile

A rhombus tile has diagonals of 10 inches and 24 inches. A reflection across the longer diagonal is a symmetry of the tile. Find the side length and the perimeter.

Key facts behind this geometry symmetry worksheet

Key facts

A regular n-gon has n lines of symmetry and rotational symmetry every 360/n degrees.
A rectangle that is not a square has 2 lines of symmetry and a 180 degree rotation.
A parallelogram that is not a rectangle or rhombus has no line of symmetry but has a 180 degree rotation.
Rigid motions preserve distance and angle measure.
To test a symmetry, check that every vertex maps to a vertex.

How to use this geometry symmetry worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the geometry symmetry worksheet as a PDF with the button below and keep it with your notes.

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