A shape has a line of symmetry if it can be folded along that line so the two halves match exactly. A shape can have zero, one, or many lines of symmetry — a rectangle has $2$, an equilateral triangle has $3$, and a circle has infinitely many.

A shape has rotational symmetry if it looks identical after being rotated by some angle less than a full turn about its centre. The order of rotational symmetry is the number of times it matches itself during one complete $360°$ turn. A square has order $4$; a shape with no repeats other than the full turn back to start has order $1$ (i.e. no rotational symmetry).

• Equilateral triangle: $3$ lines of symmetry, rotational order $3$.
• Isosceles triangle: $1$ line of symmetry, rotational order $1$.
• Square: $4$ lines of symmetry, rotational order $4$.
• Rectangle (non-square) / rhombus: $2$ lines of symmetry, rotational order $2$.
• Parallelogram (non-rhombus): $0$ lines of symmetry, rotational order $2$.
• Regular polygon with $n$ sides: $n$ lines of symmetry, rotational order $n$.

Yes, and this is a common exam trap. A parallelogram (that isn't a rhombus or rectangle) has rotational symmetry of order $2$ but zero lines of symmetry — rotating it $180°$ maps it onto itself, but no straight fold line makes the two halves match. Conversely, an isosceles triangle has exactly one line of symmetry but only rotational order $1$ (meaning, technically, no rotational symmetry beyond the trivial full turn). Always test the two properties separately rather than assuming a shape with one automatically has the other.

3D solids have planes of symmetry (a flat cutting plane that splits the solid into two mirror-image halves) instead of lines, and axes of symmetry (a line about which the solid can rotate onto itself). A cylinder has infinitely many planes of symmetry through its central axis, plus one horizontal plane through its midpoint; a cone has infinitely many planes through its apex and centre, and one axis of symmetry running through the apex and the centre of its base. A cuboid (with three different edge lengths) has exactly three planes of symmetry, each running parallel to a pair of faces through the solid's centre.

State the number of lines of symmetry and the order of rotational symmetry of a regular hexagon.

1. A regular hexagon has $6$ equal sides and $6$ equal angles.
2. Every regular polygon with $n$ sides has $n$ lines of symmetry: here, $6$.
3. Every regular polygon with $n$ sides has rotational symmetry of order $n$: here, order $6$.

State the number of lines of symmetry and the order of rotational symmetry of a (non-rhombus) parallelogram.

1. Try folding along any line: the two halves never match exactly, since opposite sides are equal but the shape is "slanted" — so there are $0$ lines of symmetry.
2. Rotating $180°$ about the centre maps the shape exactly onto itself (each side swaps with its opposite, equal side) — so the order of rotational symmetry is $2$.