C4.5 Symmetry

Learning objectives

C4.5.1Recognise line symmetry and order of rotational symmetry in two• Recognise line symmetry and order of rotational symmetry in two dimensions. Includes properties of triangles, quadrilaterals and polygons directly related to their symmetries.

Symmetry means a transformation leaves a shape looking unchanged: reflection tests line symmetry, while turning about a centre tests rotational symmetry.

Type	Test	What to count
line symmetry	reflect or fold across a candidate line	every distinct mirror line
rotational symmetry	rotate through one full turn about the centre	the number of matching positions, including the starting position

A line of symmetry must pair every point, edge and shaded region with a mirror image the same perpendicular distance on the other side. Test vertical, horizontal and diagonal candidates, but draw only those that reflect the entire figure.

Turn the shape through equal angles around its centre and count each exact match before reaching $360^\circ$ . If the smallest matching turn is $\theta$ , the order is $360^\circ/\theta$ . A half-turn match gives order $2$ .

Shape	Lines of symmetry	Rotational order
equilateral triangle	3	3
non-equilateral isosceles triangle	1	1
scalene triangle	0	1
square	4	4
non-square rectangle	2	2
non-square rhombus	2	2
parallelogram	0	2
kite	1	1
regular $n$ -gon	$n$	$n$

Rotational order is never zero: every shape matches after a full $360^\circ$ turn, so a shape with no smaller match has order $1$ . Do not count a diagonal as a mirror line unless the whole shape—including markings and shading—reflects onto itself.

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