Symmetries of Figures

HSG.CO.A.3 — Lesson 3 of 5: Congruence Unit

In this lesson:

• What makes a transformation a "symmetry"
• Rotational and reflective symmetries of figures
• Why some figures have more symmetry than others

Learning Objectives

By the end of this lesson, you should be able to:

1. Define symmetry as a rigid motion that maps a figure onto itself
2. Identify all rotational symmetries of rectangles, parallelograms, trapezoids, and regular polygons
3. Identify all lines of reflective symmetry for those same figures
4. Describe symmetries precisely — angle of rotation, line of reflection
5. Explain why figures with more equal sides and angles have more symmetries

From Rigid Motions to Symmetry

A 90° rotation of this square: every vertex moves — yet the square looks identical.

Symmetry: A Self-Mapping Rigid Motion

A symmetry of figure is a rigid motion such that:

• Every point of maps to a point of
• Every point of is the image of some point of
• The figure occupies the same set of points — even though individual points may move

Two Types of Symmetry for Finite Figures

Only rotations and reflections can be symmetries of finite figures:

• Rotational symmetry: rotate by (where ) — figure maps to itself
• Reflective symmetry: reflect across a line — figure maps to itself

Translation is NOT a symmetry of any finite figure — it moves the whole figure to a new location

The Identity: Every Figure's Free Symmetry

Every figure has at least one symmetry: the identity

• Identity = 0° rotation = "do nothing"
• It trivially satisfies (every point stays put)
• We include it when listing all symmetries (makes the math cleaner)
• Important: 360° rotation = 0° rotation = identity — don't count it twice

The interesting symmetries are the non-trivial ones (beyond 0°)

Quick Check: Can Translation Be a Symmetry?

True or False:

"A translation can be a symmetry of a rectangle."

Think about this before advancing — what does a translation do to a figure?

Translation Cannot Be a Figure Symmetry

A translation moves every point to a new location — the rectangle no longer overlaps itself.

✓ Translations can only be symmetries of infinite figures (like an infinite wallpaper pattern)
✓ For finite figures: only rotations and reflections can be symmetries

Rotational Symmetry: Finding All Valid Angles

A figure has rotational symmetry if a rotation by (where ) maps it to itself.

• The center is usually the geometric center of the figure
• The order = number of rotations that work (including identity)
• To find: test which angles send each vertex to another vertex position

Regular Hexagon: Six Rotational Symmetries

• Rotate by 60°: vertex 1 → position of vertex 2, vertex 2 → vertex 3, ...
• Test 120°, 180°, 240°, 300° — all work
• 6 rotational symmetries: 0°, 60°, 120°, 180°, 240°, 300°

Rectangle: Testing Rotation Angles

• Test 90°: long sides swap with short sides — does not work ✗
• Test 180°: each vertex maps to the opposite vertex — works ✓
• 2 rotational symmetries: 0° and 180° (order 2)

Parallelogram and Trapezoid Rotational Symmetry

Parallelogram (non-rectangle):

• 180° works: opposite vertices swap — each vertex maps to the opposite position
• 90° fails: adjacent sides have unequal lengths and non-right angles
• 2 rotational symmetries: 0° and 180°

Isosceles trapezoid:

• 180° fails: the two parallel sides have different lengths — top and bottom swap incorrectly
• Only the identity works: 1 rotational symmetry (order 1)

Pattern for Regular Polygon Symmetries

For a regular -gon, the rotational symmetry angles are:

• Regular polygon has rotational symmetries (order )
• More sides → more symmetry → smaller angular step
• Example: equilateral triangle (): 120°, 240°, 360°=0°

Why 60° works for the hexagon:

Check-In: Find Octagon Symmetries

A regular octagon has 8 sides.

How many rotational symmetries does it have?
At what angles?

Work through the pattern before advancing. Hint:

Octagon Has Eight Rotational Symmetries

8 rotational symmetries at:

• Smallest angle: — not a "standard" angle, but valid
• Pattern: every 45° step is a symmetry
• Order 8: the octagon looks the same from 8 orientations

Reflective Symmetry: Lines That Fold Figures

A figure has reflective symmetry if reflecting across line maps it to itself. Line is a line of symmetry.

The fold test: fold the figure along the candidate line — do both halves match perfectly?

• Every point of must have its reflected image also on
• Dividing into congruent pieces is not sufficient — halves must be mirror images

Rectangle: Two Lines of Symmetry

• Horizontal midline: top maps to bottom — ✓ works
• Vertical midline: left maps to right — ✓ works
• Diagonal: does NOT work ✗ — critical misconception case

Rectangle has exactly 2 lines of symmetry

Why the Rectangle Diagonal Fails

Consider rectangle: , , , . Diagonal from to .

Reflect across diagonal :

• Reflected image of is not
• The reflected lands outside the rectangle

Key insight: A line of symmetry must map the entire figure onto itself — not just divide it into congruent pieces.

Parallelogram: Zero Lines of Symmetry

A non-rectangular parallelogram has no lines of symmetry:

• Midlines: adjacent angles are unequal → neither midline works
• Diagonals: angles are not right angles → neither diagonal works
• Any other line: unequal adjacent angles always prevent a match

0 lines of symmetry — even though it has 180° rotational symmetry

This proves rotational and reflective symmetry are independent

Trapezoid: One or Zero Lines of Symmetry

Isosceles trapezoid:

• Perpendicular bisector of the bases is a line of symmetry ✓
• No other line works — only that one midline balances the equal legs
• 1 line of symmetry

General (non-isosceles) trapezoid:

• The two legs have different lengths → no fold line balances them
• 0 lines of symmetry — no symmetry of any kind (except identity)

Regular Hexagon: Six Lines of Symmetry

• 3 lines through opposite vertex pairs
• 3 lines through midpoints of opposite sides
• 6 lines of symmetry total — matching the 6 rotational symmetries

Check-In: Find Pentagon Symmetry Lines

A regular pentagon has 5 sides.

How many lines of symmetry does it have?
Describe each one.

Think about where the fold lines must go — through vertices, or through midpoints?

Pentagon Has Five Lines of Symmetry

5 lines of symmetry — one per vertex:

• Each line passes through one vertex and the midpoint of the opposite side
• No "opposite vertex pairs" since 5 is odd — no two vertices are directly across
• Regular -gon always has exactly lines — matching the rotational symmetries

Complete Symmetry Profiles for All Figures

Key Patterns That Determine Symmetry Count

More equal sides and angles → more symmetries:

• Square (4 equal sides, 4 equal angles) → 4 + 4
• Rectangle (2 pairs equal sides, 4 equal angles) → 2 + 2
• Parallelogram (2 pairs equal sides, 2 pairs equal angles) → 2 + 0

The two symmetry types are independent:

• Parallelogram: 2 rotational, 0 reflective → rotation without reflection
• Isosceles trapezoid: 0 non-trivial rotational, 1 reflective → reflection without rotation

Rotational and Reflective Symmetry Are Independent

Can a figure have lines of symmetry but no rotational symmetry?

→ Yes — the isosceles trapezoid: 1 line of symmetry, 0 non-trivial rotations

Can a figure have rotational symmetry but no lines of symmetry?

→ Yes — the non-rectangular parallelogram: 180° rotational symmetry, 0 lines

These counterexamples prove the two types are truly independent properties

Key Takeaways and Misconception Warnings

✓ A symmetry is a rigid motion where — figure maps onto itself
✓ Identity (0° = 360°) is always a symmetry — don't count it twice
✓ Regular -gon: exactly rotational symmetries and lines of symmetry
✓ Rectangle: 2 + 2 · Parallelogram: 2 rotational + 0 reflective · Isosceles trapezoid: 0 + 1

Rectangle diagonal ≠ line of symmetry — congruent triangles ≠ self-mapping
Parallelogram has 0 lines of symmetry — despite 180° rotational symmetry
Rotational and reflective symmetry are independent — one does not imply the other

Connections to Future Geometry Topics

This lesson connects to:

• HSG.CO.A.4 (next): formal definitions of rotations and reflections — made rigorous
• HSG.CO.B.7: triangle congruence — a triangle's line of symmetry proves base angles are equal
• HSG.CO.C.11: parallelogram theorems — 180° rotational symmetry explains why opposite sides and angles are equal

Symmetry is also at the heart of crystallography, molecular chemistry, and symmetry groups