Symmetries of Quadrilaterals

Learning Focus

Classify quadrilaterals by their lines of symmetry and rotational symmetry.

Big idea

A figure has line symmetry if a reflection maps it onto itself and rotational symmetry if a rotation of less than $360^\circ$ does.

• Square: $4$ lines, rotations by $90^\circ$.
• Rectangle and rhombus (non-square): $2$ lines, rotation by $180^\circ$.
• Isosceles trapezoid: $1$ line.
• General parallelogram: $0$ lines, but $180^\circ$ rotational symmetry.

Worked example

A square has $4$ lines of symmetry (two diagonals, two midlines). Its smallest rotation onto itself is $\frac{360^\circ}{4}=90^\circ$.

A parallelogram that is not a rectangle or rhombus has no reflection symmetry, but a $180^\circ$ turn about its center maps it to itself.

Takeaways

• Count lines of symmetry by trying reflections.
• Rotational symmetry angle is $360^\circ/n$.
• Special quadrilaterals have more symmetry.