Maths · 5 Geometry

5.4 Symmetry

Before you start

The names of the special triangles (equilateral, isosceles, scalene) and quadrilaterals (square, rectangle, rhombus, parallelogram, kite, trapezium).
That a full turn is $360 °$ .
Plotting points on a coordinate grid, and that $x = 4$ is a vertical line and $y = 1$ is a horizontal line.
What a prism, pyramid, cylinder and cone look like, and that a prism has the same cross-section all along its length.

Syllabus objectives covered (2)

Recognise line symmetry and order of rotational symmetry in two dimensions.
Recognise symmetry properties of prisms, cylinders, pyramids and cones.

Part 1 of 7

Line symmetry: the fold-and-match test

A line of symmetry (also called a mirror line) splits a shape into two halves that are mirror images of each other. If you folded the shape along that line, one half would land exactly on top of the other, with nothing sticking out.
The precise version of the test: every point on one side of the line has a partner on the other side, the same distance from the line, and the line joining the two partners crosses the mirror line at right angles.
Cutting a shape into two equal halves is not enough. The halves must be mirror images across that line. This is exactly where the diagonal of a rectangle fails.
Take a rectangle $A B C D$ with $A B = 8$ cm and $A D = 5$ cm, and imagine folding it along the diagonal $A C$ . The fold would have to send side $A B$ onto side $A D$ , because they are the two sides that meet at $A$ . But one is 8 cm and the other is 5 cm, so corner $B$ lands somewhere that is not $D$ . The two triangles are congruent, yet the fold does not match them up.
graph diagram — not drawn yet
Rectangle ABCD, 8 units wide and 5 units tall, with its two lines of symmetry drawn dashed (one vertical through the midpoints of AB and DC, one horizontal through the midpoints of AD and BC) and the diagonal AC drawn dotted, plus a faint outline showing where triangle ABC would land if folded along AC, overhanging the rectangle.
The dashed lines are lines of symmetry. Folding along the diagonal makes the triangle overhang, so the diagonal is not one.
A shape can have no lines of symmetry, one, two, or more. A circle has infinitely many: every diameter is a line of symmetry.
To find them all, test each likely line in turn: lines through a vertex and the opposite vertex, lines through a vertex and the midpoint of the opposite side, and lines through the midpoints of opposite sides. Keep only the ones that pass the fold test.

Part 2 of 7

Rotational symmetry and its order

A shape has rotational symmetry if you can turn it about a fixed point, the centre of rotation, through less than a full turn and it fits exactly onto its own outline.
The order of rotational symmetry is the number of positions in one full turn of $360 °$ in which the shape looks exactly as it did at the start. The starting position (which is also the $360 °$ position) counts once.
A square fits onto itself after turns of $90 °$ , $180 °$ , $270 °$ and $360 °$ : four positions, so its order is 4.
Every shape fits onto itself after a full turn, so the smallest possible order is 1. A shape of order 1 is said to have no rotational symmetry. The order is never 0.
The order and the smallest angle of turn that works are linked, because the equal turns have to add up to one full turn.
$order = \frac{360 °}{smallest angle of turn}$
So a shape of order 5 first fits after $360 ° \div 5 = 72 °$ , and a shape that first fits after $30 °$ has order $360 \div 30 = 12$ .
For order 2 the turn is a half-turn, and the centre is the midpoint between each point and the point it lands on. For a parallelogram, that is where the diagonals cross.
A useful check that links the two kinds of symmetry: a flat shape with at least one line of symmetry has exactly as many lines as its order of rotational symmetry. A shape can have rotational symmetry with no lines at all (a parallelogram: 0 lines, order 2), but 2 lines with order 4 is impossible. If your two answers break this, test the shape again.

Part 3 of 7

Triangles and quadrilaterals: symmetry explains their properties

Each fact below is for the general shape. A square is a special rectangle and a special rhombus with extra symmetry, so 'a rectangle has 2 lines' means a rectangle that is not a square. The same goes for 'isosceles' meaning not equilateral.
Equilateral triangle: 3 lines, each through a vertex and the midpoint of the opposite side; order 3. The symmetry is the reason all three angles are equal, so each is $60 °$ .
Isosceles triangle: 1 line, through the vertex between the two equal sides and the midpoint of the third side; order 1. The fold swaps the two base angles, so they must be equal. The same fold shows that the line cuts the base in half at right angles and cuts the top angle in half.
Scalene triangle: no sides are equal, so no fold can match them. 0 lines, order 1.
Square: 4 lines (both diagonals and both lines through midpoints of opposite sides); order 4. Rectangle: 2 lines, through the midpoints of opposite sides, not the diagonals; order 2.
Rhombus: 2 lines, and they are its diagonals; order 2. Because each diagonal is a mirror line, the diagonals cross at right angles, and each diagonal cuts the angles at its ends in half.
Parallelogram: 0 lines; order 2, about the point where the diagonals cross. The half-turn swaps opposite sides and opposite angles, which is why they are equal, and it sends each half of a diagonal onto the other half, which is why the diagonals cut each other in half.
Kite: 1 line, the diagonal through the two vertices where the pairs of equal sides meet; order 1. The fold swaps the other two vertices, so that pair of opposite angles is equal, and the diagonals cross at right angles.
Isosceles trapezium: 1 line, through the midpoints of the two parallel sides; order 1. The fold swaps the two angles at each parallel side, so they are equal in pairs, and it swaps the diagonals, so they are equal in length. A trapezium that is not isosceles has 0 lines and order 1.
graph diagram — not drawn yet
Six quadrilaterals in two rows, each labelled with its name: square, rectangle, rhombus, parallelogram, kite and isosceles trapezium. On each, every line of symmetry is drawn dashed, and a dot marks the centre of rotation on the four with order 2 or more. Under each, a label gives 'lines: n, order: m' with the values 4 and 4, 2 and 2, 2 and 2, 0 and 2, 1 and 1, 1 and 1.
Rectangle lines go through midpoints of sides; rhombus lines are the diagonals; the parallelogram has a centre but no lines.

Part 4 of 7

Regular polygons and the circle

A regular polygon has all sides equal and all angles equal. A regular polygon with $n$ sides has $n$ lines of symmetry and rotational symmetry of order $n$ .
When $n$ is odd, every line of symmetry runs from a vertex to the midpoint of the opposite side. A regular pentagon has 5 such lines.
When $n$ is even, half the lines join pairs of opposite vertices and half join midpoints of opposite sides. A regular hexagon has $3 + 3 = 6$ .
graph diagram — not drawn yet
A regular hexagon with its six lines of symmetry: three drawn as solid lines joining opposite vertices and three drawn as dashed lines joining midpoints of opposite sides, all crossing at the centre, with the angle of 60 degrees marked between two neighbouring vertices at the centre.
Three lines through vertices and three through midpoints of sides: 6 lines, not 12.
The trap with even $n$ : a line through two opposite vertices is one line. Count lines, not the vertices they touch, or a hexagon comes out with 12.
The smallest angle of turn is $360 ° \div n$ : $60 °$ for a hexagon, $45 °$ for an octagon, $36 °$ for a decagon.
A circle fits onto itself after a turn of any angle about its centre, and every diameter is a line of symmetry. It has infinitely many lines of symmetry and rotational symmetry of infinite order.

Part 5 of 7

Completing a shape so it has the symmetry asked for

A grid question may give you part of a shape and a mirror line or a centre, and ask you to add the rest. Work one vertex at a time, then join them up.
**Reflection in a vertical line $x = a$ :** a point $h$ units to one side of the line goes to $h$ units on the other side, at the same height. In coordinates, $x$ becomes $2 a - x$ and $y$ stays the same.
$(x, y) \mapsto (2 a - x, y)$
With the mirror line $x = 4$ , the point $(1, 3)$ is 3 units to the left, so its partner is 3 units to the right: $(7, 3)$ . Check: $2 \times 4 - 1 = 7$ .
**Reflection in a horizontal line $y = b$ ** works the same way vertically: $(x, y) \mapsto (x, 2 b - y)$ .
**Rotational symmetry of order 2 about a point $(h, k)$ :** each vertex goes straight through the centre and the same distance out the other side. Write the step from the centre to the vertex, then make the opposite step from the centre.
$(x, y) \mapsto (2 h - x, 2 k - y)$
About the centre $(4, 3)$ , the point $(6, 4)$ is 2 right and 1 up. Go 2 left and 1 down from the centre instead: $(2, 2)$ . Check: $8 - 6 = 2$ and $6 - 4 = 2$ .
Order 4 needs quarter-turns. A step of ' $p$ right, $q$ up' from the centre becomes ' $q$ left, $p$ up' after a quarter-turn anticlockwise; do that three times to get the three new copies.
When you have finished, test the whole shape with the fold or the turn. Adding an extra square that has no partner breaks the symmetry you were asked for.

Part 6 of 7

Planes of symmetry of solids

In three dimensions the mirror is a flat sheet, a plane of symmetry. It cuts the solid into two halves that are mirror images of each other.
The prisms and pyramids here are right ones: a right prism has its length at right angles to its ends, and a right pyramid has its apex directly above the centre of its base.
Prisms. Each line of symmetry of the cross-section, swept along the full length of the prism, makes a plane of symmetry. On top of those, there is one more: the plane halfway along, parallel to the two ends. It works because the two ends are identical.
$planes of a prism = (lines of symmetry of the cross-section) + 1$
So a prism whose cross-section is an equilateral triangle has $3 + 1 = 4$ planes; a cuboid with three different edge lengths has $2 + 1 = 3$ ; a prism on a regular hexagon has $6 + 1 = 7$ .
graph diagram — not drawn yet
A right prism with an equilateral triangle cross-section drawn in perspective, shown four times side by side; in each copy one plane of symmetry is shaded: three copies show a plane running along the length through one edge and the middle of the opposite rectangular face, and the fourth shows the plane halfway along, parallel to the triangular ends.
Three planes come from the triangle's lines of symmetry; the fourth is the halfway plane.
One family can trip the count: a cuboid with two square faces. If you take a rectangular face as the cross-section you get $2 + 1 = 3$ , which is too few. Choose the square faces as the ends and the rule works: a $3 \times 3 \times 5$ cuboid is a square prism with $4 + 1 = 5$ planes. A cube has 9 planes, because every face is square: 3 parallel to pairs of opposite faces and 6 through pairs of opposite edges.
Cylinder. It is a prism with a circular cross-section. Every plane that contains the central axis is a plane of symmetry, so there are infinitely many of those, plus the one plane at right angles to the axis, halfway along.
Pyramids. The two ends are different (a base at the bottom, a single point at the top), so there is no halfway plane. Every plane of symmetry passes through the apex, and there is one for each line of symmetry of the base. A square-based pyramid has 4; a pyramid on a regular pentagon has 5; a pyramid on a rectangle that is not a square has 2.
The exception is a regular tetrahedron, a triangular pyramid whose four faces are all equilateral triangles. It has 6 planes, not 3, because any of its four faces can act as the base.
Cone. Like a pyramid on a circular base: infinitely many planes, every one passing through the apex and the centre of the base. There is no plane at right angles to the axis, because no such plane could swap the apex with the circular base.
graph diagram — not drawn yet
A cylinder and a cone side by side, each with its central axis drawn as a vertical dashed line and one vertical plane of symmetry containing the axis shaded; the cylinder also has a horizontal plane shaded halfway up, and the cone has no horizontal plane.
Both have infinitely many planes through the axis; only the cylinder has the halfway plane.

Part 7 of 7

Axes of rotational symmetry of solids

An axis of rotational symmetry is a straight line through a solid such that turning the solid about it brings it back onto its own position before a full turn. The order about that axis is the number of positions in one full turn in which it looks the same.
Prisms. About the axis running along the length through the centres of the two ends, the order equals the order of rotational symmetry of the cross-section. An equilateral triangular prism has order 3; a regular hexagonal prism, order 6; a square prism, order 4.
A prism can also have axes at right angles to its length. A cuboid with three different edge lengths has three axes, each through the centres of a pair of opposite faces, and each of order 2. So a question names the axis it means: read which line is marked or described, and give the order about that one.
Pyramids. The axis runs from the apex to the centre of the base, and the order equals the order of rotational symmetry of the base. A square-based pyramid has order 4 about it; a pyramid on a regular hexagon, order 6.
Cylinders and cones. The central axis is an axis of rotational symmetry, and a turn of any angle about it works, so the order about it is infinite. A cylinder also has axes at right angles to its central axis through its centre, each of order 2; a cone has no other axis, because turning it over would put the apex at the bottom.
Keep planes and axes apart. A square-based pyramid has 4 planes of symmetry and order 4 about its axis, but they are different things: a plane is a mirror, an axis is a line you turn about. Answer the one the question asks for.

Worked examples

Example 1. Write down the number of lines of symmetry and the order of rotational symmetry of (a) a rhombus that is not a square, (b) an isosceles trapezium.

Show the working

(a) Test a diagonal $A C$ of rhombus $A B C D$ . All four sides are equal, so $A B = A D$ and $C B = C D$ . Folding along $A C$ sends $A B$ onto $A D$ and $C B$ onto $C D$ , so $B$ lands on $D$ . Both diagonals pass the test.
Test a line through the midpoints of $A B$ and $D C$ . That fold would swap $A$ and $B$ , so the angles at $A$ and $B$ would have to be equal. Next to each other in a rhombus they add to $180 °$ , so they are equal only if both are $90 °$ , which would make it a square. These lines fail.
Now turn it. A half-turn about the point where the diagonals cross sends $A$ to $C$ and $B$ to $D$ , so it fits. A quarter-turn would put diagonal $A C$ where $B D$ is, but the diagonals of a rhombus that is not a square have different lengths. So the order is 2.
Check: 2 lines and order 2 agree with the rule that a shape with lines of symmetry has as many lines as its order.
(b) The isosceles trapezium folds onto itself along the line through the midpoints of its two parallel sides. That is 1 line. Any other fold fails: the fold across the middle would swap the two parallel sides, which have different lengths, and a fold along a diagonal would send each parallel side onto a slanting side, which cannot equal both of them.
A half-turn would put the longer parallel side where the shorter one was, so it does not fit until a full turn. Order 1, and the check agrees: 1 line, order 1.

Answer: (a) 2 lines of symmetry (the diagonals), order 2. (b) 1 line of symmetry, order 1.

Example 2. The smallest angle through which a regular polygon can be turned about its centre to fit exactly onto itself is $40 °$ . Work out how many sides the polygon has, and describe its lines of symmetry.

Show the working

The equal turns must add up to $360 °$ , so divide to find the order.
$order = \frac{360 °}{40 °} = 9$
A regular polygon with $n$ sides has order $n$ , so the polygon has 9 sides.
It also has 9 lines of symmetry. Because 9 is odd, no line can join two vertices: each one runs from a vertex to the midpoint of the opposite side.

Answer: 9 sides; it has 9 lines of symmetry, each through one vertex and the midpoint of the opposite side.

Example 3. On a coordinate grid, triangle $T$ has vertices $(1, 2)$ , $(3, 2)$ and $(3, 5)$ . A second triangle is added so that the whole shape has rotational symmetry of order 2 about the point $(4, 3)$ . Find the vertices of the second triangle.

Show the working

Order 2 means a half-turn. Each vertex goes straight through the centre $(4, 3)$ to the same distance on the other side, so write the step from the centre to the vertex, then make the opposite step.
$(1, 2)$ is 3 left and 1 down from $(4, 3)$ . Go 3 right and 1 up instead: $(7, 4)$ .
$(3, 2)$ is 1 left and 1 down. Go 1 right and 1 up: $(5, 4)$ .
$(3, 5)$ is 1 left and 2 up. Go 1 right and 2 down: $(5, 1)$ .
Check with the coordinate rule $(x, y) \mapsto (2 h - x, 2 k - y) = (8 - x, 6 - y)$ : $(1, 2) \mapsto (7, 4)$ , $(3, 2) \mapsto (5, 4)$ , $(3, 5) \mapsto (5, 1)$ . All three agree.

Answer: The second triangle has vertices $(7, 4)$ , $(5, 4)$ and $(5, 1)$ .

Example 4. Solid $P$ is a right prism whose cross-section is a regular hexagon. Solid $Q$ is a right pyramid whose base is a regular hexagon. Compare the symmetry of $P$ and $Q$ : the number of planes of symmetry, and the order of rotational symmetry about the axis through the centre of the hexagonal face or faces.

Show the working

Start with the hexagon itself: a regular hexagon has 6 lines of symmetry and order 6.
For $P$ , each of the 6 lines swept along the length gives a plane, and the halfway plane parallel to the two identical ends adds one more.
$6 + 1 = 7$
For $Q$ , each of the 6 lines gives a plane through the apex. There is no halfway plane, because the top is a point and the bottom is a hexagon. So 6 planes.
About the axis through the centre of the hexagon, each solid turns exactly as its hexagon does, so both have order 6.

Answer: $P$ has 7 planes of symmetry and $Q$ has 6; both have rotational symmetry of order 6 about the axis through the centre of the hexagon.

Where people go wrong

Mistake: Saying the diagonals of a rectangle are lines of symmetry, giving it 4 lines.

Instead: A rectangle that is not a square has 2 lines, through the midpoints of opposite sides.

Folding along a diagonal would send a long side onto a short side. The two triangles are congruent, but they are not mirror images across the diagonal.

Mistake: Writing order 0 for a shape with no rotational symmetry.

Instead: Write order 1.

Every shape fits onto itself after a full turn, and that position counts once. Order 1 is what 'no rotational symmetry' means.

Mistake: Giving a parallelogram two lines of symmetry.

Instead: A parallelogram that is not a rectangle or rhombus has 0 lines and order 2.

A fold along a diagonal or a midline would have to match unequal sides or unequal angles. It is the half-turn about the centre that fits, not a fold.

Mistake: Counting 12 lines of symmetry on a regular hexagon by counting one line from every vertex and every midpoint.

Instead: A regular hexagon has 6 lines.

Each line through a vertex also passes through the opposite vertex, and each line through a midpoint passes through the opposite midpoint, so counting the points counts every line twice.

Mistake: Giving a pyramid or cone a plane of symmetry halfway up, or forgetting the halfway plane of a prism.

Instead: Prism: lines of symmetry of the cross-section plus 1. Pyramid: lines of symmetry of the base only. Cone: infinitely many, all through the apex.

The halfway plane swaps the two ends, so it exists only when the two ends are identical, as they are on a prism or cylinder.

Check yourself

Write down the number of lines of symmetry and the order of rotational symmetry of a kite that is not a rhombus.
Answer
1 line (the diagonal through the vertices where the pairs of equal sides meet); order 1.
A regular octagon: how many lines of symmetry, what order of rotational symmetry, and what is the smallest angle of turn that fits it onto itself?
Answer
8 lines, order 8, and $360 ° \div 8 = 45 °$ .
A shape fits onto itself after a turn of $72 °$ , and no smaller turn works. Write down its order of rotational symmetry.
Answer
$360 \div 72 = 5$ , so order 5.
Reflect the point $(2, 5)$ in the line $x = 4$ , and the point $(5, - 1)$ in the line $y = 1$ .
Answer
$(2 \times 4 - 2, 5) = (6, 5)$ and $(5, 2 \times 1 - (- 1)) = (5, 3)$ .
A flat shape has rotational symmetry of order 4 and at least one line of symmetry. How many lines of symmetry does it have?
Answer
4. A shape with any lines of symmetry has as many lines as its order of rotational symmetry.
A cuboid measures 2 cm by 3 cm by 5 cm. How many planes of symmetry does it have, and what is its order of rotational symmetry about the axis through the centres of the two 2 cm by 3 cm faces?
Answer
3 planes (the rectangle's 2 lines plus the halfway plane); order 2 about that axis.
A right pyramid has a rectangular base that is not a square. Write down its number of planes of symmetry and its order of rotational symmetry about the line from the apex to the centre of the base.
Answer
2 planes (one for each line of symmetry of the rectangle, both through the apex); order 2.
Explain why a cylinder has a plane of symmetry at right angles to its axis but a cone does not.
Answer
The cylinder's two ends are identical circles, so the plane halfway along swaps them. A cone has a circle at one end and a single point, the apex, at the other, so no plane at right angles to the axis can swap them.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Quadrilateral $A B C D$ has vertices $A (1, 1)$ , $B (5, 1)$ , $C (6, 4)$ and $D (2, 4)$ . It is a parallelogram that is neither a rectangle nor a rhombus.

(a) Write down the number of lines of symmetry of $A B C D$ . 1 mark No calculator

Mark scheme

B1 0 cao

(b) Write down the order of rotational symmetry of $A B C D$ . 1 mark No calculator

Mark scheme

B1 2 cao

(c) Work out the coordinates of the centre of rotational symmetry of $A B C D$ . 2 marks No calculator

Mark scheme

M1 midpoint of $A C$ or of $B D$ attempted, e.g. $(\frac{1 + 6}{2}, \frac{1 + 4}{2})$ oe
A1 $(3.5, 2.5)$ oe

(d) Point $D$ is moved to a new point $E$ so that $A B C E$ is an isosceles trapezium with $E C$ parallel to $A B$ and exactly one line of symmetry. Write down the equation of the line of symmetry of $A B C E$ . 1 mark No calculator

Mark scheme

B1 $x = 3$ cao

(e) Find the coordinates of $E$ . 2 marks No calculator

Mark scheme

M1 reflects $C$ in their line of symmetry, e.g. $C$ is 3 units right of $x = 3$ so $E$ is 3 units left, or $2 \times 3 - 6$ seen, or uses $A E = B C$ with $E$ on $y = 4$ ; FT their vertical line from (d)
A1 $(0, 4)$ cao

Question 2.

(a) The smallest angle through which a regular polygon can be turned about its centre to fit exactly onto itself is $24 °$ . Calculate the number of sides of the polygon. 2 marks No calculator

Mark scheme

M1 $360 \div 24$ oe
A1 15 cao

(b) Write down the number of lines of symmetry of this polygon. 1 mark No calculator

Mark scheme

B1 15; FT their (a) if a whole number

(c) Rectangle $P Q R S$ has $P Q = 8$ cm and $Q R = 5$ cm. Explain why the diagonal $P R$ is not a line of symmetry of the rectangle. 2 marks No calculator

Mark scheme

B1 reflection in $P R$ would have to map $Q$ onto $S$ , or side $P Q$ onto side $P S$ , or angle $Q P R$ onto angle $S P R$ oe
B1 but $P Q = 8$ cm and $P S = 5$ cm are not equal (or angle $Q P R \neq$ angle $S P R$ , since the diagonal does not cut the $90 °$ angle in half), so the two halves do not match oe

Question 3.

(a) A right prism has a cross-section that is an equilateral triangle. Write down the number of planes of symmetry of the prism. 1 mark No calculator

Mark scheme

B1 4 cao

(b) Write down the order of rotational symmetry of this prism about the axis through the centres of its two triangular faces. 1 mark No calculator

Mark scheme

B1 3 cao

(c) A right pyramid has a square base. Write down (i) the number of planes of symmetry of the pyramid, (ii) its order of rotational symmetry about the line from its apex to the centre of its base. 2 marks No calculator

Mark scheme

B1 (i) 4 cao
B1 (ii) 4 cao

(d) Compare the planes of symmetry of a cylinder with the planes of symmetry of a cone. 2 marks No calculator

Mark scheme

B1 both have infinitely many planes of symmetry, each containing the central axis oe
B1 the cylinder also has one plane at right angles to its axis, halfway along, and the cone does not oe

How this is examined

Counting the lines of symmetry and the order of rotational symmetry of a flat shape: A pair of 1-mark Write down parts about a shape drawn on a grid or named in words, such as a quadrilateral or regular polygon Two whole numbers, one for each part. The order is at least 1: write 1, not 0, for a shape that fits only after a full turn. A rectangle that is not a square has 2 lines, not 4; a parallelogram has 0 lines and order 2.
Adding to a shape so that it has a stated line or rotational symmetry: A Complete on a grid, with the mirror line or the centre of rotation marked, about 2 marks Every new vertex in the right place: the same distance from the mirror line on the opposite side, or straight through the centre to the same distance out. The finished shape must pass the fold or turn test as a whole, with no extra squares shaded.
Working out a regular polygon from its smallest angle of turn: A 2-mark Work out or Calculate, giving the angle and asking for the number of sides or lines of symmetry The division $360 \div angle$ written down is the method; the whole-number result is the answer. For a regular polygon that number is the order, the number of sides and the number of lines of symmetry.
Counting the planes of symmetry of a prism, pyramid, cylinder or cone: A 1-mark Write down for a drawn or named solid, or a 2-mark Compare or Describe between two solids Prism: the cross-section's lines of symmetry plus the halfway plane (equilateral triangular prism: 4). Right pyramid: the base's lines only (square-based: 4). Cylinder and cone: 'infinitely many, each containing the axis', plus, for the cylinder only, 'one at right angles to the axis, halfway along'.
Using symmetry to explain why a line is or is not a line of symmetry, or why two angles are equal: A 1 or 2-mark Explain about a named triangle or quadrilateral Say what the reflection would have to do ('it would map side $P Q$ onto side $P S$ ') and why that works or fails, with the evidence ('but $P Q = 8$ cm and $P S = 5$ cm'). 'The halves look different' names nothing and explains nothing.
The order of rotational symmetry of a solid about a named axis: A 1-mark Write down, with the axis drawn on the solid or described in words The order of the cross-section (prism) or of the base (pyramid), about the axis through its centre; 'infinite' for a cylinder or cone about its central axis. Give the order about the axis named, not some other axis.

Words to know

Line of symmetry: A line that splits a flat shape into two halves that are mirror images; folding along it makes the halves match exactly.
Rotational symmetry: A shape has it if it fits onto its own outline after a turn of less than $360 °$ about a point.
Order of rotational symmetry: The number of positions in one full turn in which a shape looks the same as at the start. Always at least 1.
Centre of rotation: The fixed point a shape turns about.
Regular polygon: A polygon with all sides equal and all angles equal. With $n$ sides it has $n$ lines of symmetry and order $n$ .
Plane of symmetry: A flat surface that cuts a solid into two halves that are mirror images of each other.
Axis of rotational symmetry: A line through a solid about which it can be turned, through less than a full turn, to fit onto its own position.
Right prism / right pyramid: A prism whose length is at right angles to its ends; a pyramid whose apex is directly above the centre of its base.

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