Checkpoint 8 Maths · Geometry and Measure · 8Gg.01 · 8Gg.09

Quadrilaterals and Regular Polygons

Learn the properties of every special quadrilateral, how they fit into a hierarchy, and why a regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

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In the curriculum

Identify and describe the hierarchy of quadrilaterals: know the properties of each special quadrilateral and explain which shapes are special types of other shapes (for example, every square is a rectangle). (8Gg.01)
Understand that for a regular polygon, the number of sides = the number of lines of symmetry = the order of rotational symmetry. (8Gg.09)
This means: use sides, angles, parallel lines, diagonals and symmetry to name and sort quadrilaterals, and use the 'n sides → n lines → order n' rule for regular polygons.

1. Key words

Word / phrase	Meaning / use
quadrilateral	شكل رباعي — a 2D shape with 4 straight sides
parallel	متوازي — lines that never meet; they stay the same distance apart (marked with arrows ›)
perpendicular	عمودي — lines that meet at a right angle (90°)
adjacent sides	أضلاع متجاورة — sides that are next to each other (they share a vertex)
opposite sides	أضلاع متقابلة — sides that face each other (they do not touch)
diagonal	قطر — a straight line joining two opposite vertices
bisect	ينصف — cut into two equal parts
hierarchy	تسلسل هرمي — an order from general (at the top) to special (at the bottom)
regular polygon	مضلع منتظم — all sides equal AND all angles equal
line of symmetry	محور تماثل — a mirror line: one half is the reflection of the other
order of rotational symmetry	رتبة التماثل الدوراني — how many times a shape fits onto itself in one full turn

2. The special quadrilaterals

Diagrams use marks to show properties. Dashes (|, ||) on sides mean those sides are equal. Arrows (›, ››) mean those sides are parallel. A small square in a corner means a right angle.

Square, rectangle and rhombus. Same marks = equal sides; same arrows = parallel sides.

Parallelogram, trapezium and kite.

Shape	Sides	Angles	Diagonals
square	4 equal sides; 2 pairs of parallel sides	4 right angles	equal; bisect each other; cross at 90°; bisect the angles
rectangle	opposite sides equal; 2 pairs of parallel sides	4 right angles	equal; bisect each other
rhombus	4 equal sides; 2 pairs of parallel sides	opposite angles equal	bisect each other at 90°; bisect the angles
parallelogram	opposite sides equal; 2 pairs of parallel sides	opposite angles equal	bisect each other
trapezium	at least 1 pair of parallel sides	the two angles at the ends of each non-parallel side add to 180°	no special property
kite	2 pairs of adjacent equal sides	1 pair of opposite angles equal (between the unequal sides)	cross at 90°; one diagonal bisects the other

An isosceles trapezium is a trapezium whose two non-parallel sides are equal. It has 1 line of symmetry and equal diagonals.

Diagonals are the quickest test

Diagonals equal and bisect each other → rectangle (or square).
Diagonals bisect each other at 90° → rhombus (or square).
Diagonals cross at 90° but only one is cut in half → kite.
Diagonals bisect each other only → parallelogram.

3. The hierarchy of quadrilaterals

A hierarchy puts the most general shape at the top and the most special shape at the bottom. A shape lower down has all the properties of every shape it is joined to above it, plus extra ones. So it is a special type of each of them.

The hierarchy of quadrilaterals. Follow the arrows upwards: a square is a rectangle and a rhombus; a rhombus is a parallelogram and a kite; a parallelogram is a trapezium.

A square is a special rectangle (a rectangle with 4 equal sides).
A square is a special rhombus (a rhombus with right angles).
A rectangle and a rhombus are both special parallelograms.
A rhombus is a special kite (both pairs of adjacent sides are equal, so all 4 sides are equal).
A parallelogram is a special trapezium (it has two pairs of parallel sides, so it certainly has at least one).
Every one of them is a quadrilateral.

Reading the hierarchy the other way does NOT work. 'Every square is a rectangle' is true, but 'every rectangle is a square' is false: most rectangles have two long and two short sides. To show that a statement like this is false, give one counter-example (one shape that breaks it).

A note about the word 'trapezium'

In the hierarchy we use the definition 'at least one pair of parallel sides', so a parallelogram counts as a special trapezium. Some books say 'exactly one pair'. In a hierarchy question, always place the parallelogram under the trapezium, as in the diagram above.

الفكرة الأساسية بالعربي

الشكل اللي تحت في الشجرة بياخد كل خواص الشكل اللي فوقه وزيادة. يعني المربع مستطيل (لأن زواياه قائمة) والمربع معين (لأن أضلاعه متساوية). لكن العكس مش صح: مش كل مستطيل مربع. ولو عايز تثبت إن جملة غلط، هات مثال واحد بس يكسرها (counter-example).

4. How to identify a quadrilateral, step by step

Count the pairs of parallel sides (look for arrows, or check gradients on a grid). 0 pairs → kite or a general quadrilateral; 1 pair → trapezium; 2 pairs → parallelogram family.
Check the sides: all equal? opposite equal? adjacent pairs equal?
Check the angles: right angles? opposite angles equal?
Check the diagonals if they are given: equal? cross at 90°? bisect each other?
Choose the most special name that fits all the facts, then check upwards in the hierarchy: the shape is also every shape above it.

On a coordinate grid, compare the 'steps' along each side (across and up). Two sides with the same step (for example 1 right, 3 up) are parallel and equal. Count squares to compare horizontal and vertical lengths. Do not measure a slanted length by counting squares along it.

5. Symmetry of regular polygons

A regular polygon has all sides equal and all angles equal. For every regular polygon:

The rule (8Gg.09)

number of sides = number of lines of symmetry = order of rotational symmetry. A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n. The smallest turn that fits it back onto itself is 360° ÷ n.

Every regular polygon with n sides has n lines of symmetry (dashed) and rotational symmetry of order n.

Where are the lines? If n is odd (triangle, pentagon, heptagon…), each line goes from a vertex to the midpoint of the opposite side. If n is even (square, hexagon, octagon…), half of the lines join opposite vertices and half join the midpoints of opposite sides.

Rotational symmetry of a regular pentagon: 5 turns of 72° make a full turn.

Regular polygon	Sides	Lines of symmetry	Order of rotational symmetry	Smallest turn
equilateral triangle	3	3	3	120°
square	4	4	4	90°
regular pentagon	5	5	5	72°
regular hexagon	6	6	6	60°
regular octagon	8	8	8	45°
regular decagon	10	10	10	36°
regular dodecagon	12	12	12	30°

The rule only works for regular polygons. Compare the quadrilaterals:

Quadrilateral	Lines of symmetry	Order of rotational symmetry
square (regular)	4	4
rectangle	2	2
rhombus	2	2
parallelogram	0	2
kite	1	1
isosceles trapezium	1	1
trapezium (general)	0	1

Order 1 means the shape only fits onto itself after a full turn, so it has no rotational symmetry. A shape with all sides equal is not always regular: the hexagon below has 6 equal sides, but its angles are not all equal (130° and 100°), so it has only 2 lines of symmetry and order 2.

Six equal sides, but NOT a regular hexagon: the angles are different.

قاعدة المضلع المنتظم

المضلع المنتظم = كل الأضلاع متساوية وكل الزوايا متساوية. لو عنده n ضلع، يبقى عنده n محور تماثل، ورتبة التماثل الدوراني بتاعته n، وأصغر لفة بترجّعه زي ما كان = 360 ÷ n. القاعدة دي للمنتظم بس — المستطيل مثلاً 4 أضلاع لكن محورين بس.

6. Worked examples

Worked example 1 [2 marks] A quadrilateral has four equal sides. Its angles are 70°, 110°, 70° and 110°. Draw a ring around every name that is correct for this shape: square rhombus rectangle parallelogram kite

Four equal sides but no right angles → it is a rhombus, not a square.
No right angles → it is not a rectangle.
Go up the hierarchy from rhombus: a rhombus is a parallelogram and a kite.
Answer: rhombus, parallelogram, kite. (2 marks for all three and no wrong ones; 1 mark for two correct and no wrong ones.)

Common mistake: ringing only 'rhombus'. The question says 'every name', so think about the hierarchy.

Worked example 2 [3 marks] Tick (✓) Correct or Not correct for each statement. If it is not correct, give a counter-example. (a) Every square is a rhombus. (b) Every rectangle is a square. (c) Every rhombus is a parallelogram.

(a) Correct. A square has 4 equal sides and 2 pairs of parallel sides, so it has all the properties of a rhombus.
(b) Not correct. Counter-example: a rectangle 6 cm by 2 cm has right angles but its sides are not all equal, so it is not a square.
(c) Correct. A rhombus has two pairs of parallel sides, so it is a parallelogram.

Worked example 3 [2 marks] The points A(1, 1), B(5, 1), C(6, 4) and D(2, 4) are joined in order. Write down the special name of ABCD. Give a reason.

ABCD on a coordinate grid.

AB: 4 right, 0 up. DC: 4 right, 0 up. Same step → AB and DC are parallel and equal.
AD: 1 right, 3 up. BC: 1 right, 3 up. Same step → AD and BC are parallel and equal.
Two pairs of parallel sides → parallelogram family. AB = 4 but AD is a slanted side about 3.2 units long, so the sides are not all equal (not a rhombus). The angle at A is not 90° (not a rectangle).
Answer: a parallelogram, because both pairs of opposite sides are parallel (and equal) but there are no right angles and the sides are not all equal.

Worked example 4 [2 marks] What am I? 'I am a quadrilateral. My diagonals are equal in length, they bisect each other and they cross at right angles.'

Diagonals equal and bisecting each other → rectangle family (rectangle or square).
Diagonals crossing at 90° and bisecting each other → rhombus family (rhombus or square).
The only shape in both families is the square.
Answer: square.

Worked example 5 [3 marks] A regular polygon has rotational symmetry of order 12. (a) How many lines of symmetry does it have? (b) How many sides does it have? (c) What is the smallest angle you can turn it through so that it looks the same?

(a) Regular polygon: lines of symmetry = order of rotation = 12.
(b) Number of sides = 12 (a regular dodecagon).
(c) 360° ÷ 12 = 30°.

Worked example 6 [1 mark] A regular polygon looks exactly the same after a turn of 40° about its centre. How many sides does it have? Working: the number of turns that make a full turn = 360° ÷ 40° = 9. Answer: 9 sides (a regular nonagon).

Common mistake: dividing 360 by the number of sides when the question gives the angle, or giving the angle when the question asks for the number of sides. Write the rule first: smallest turn = 360° ÷ n.

Common mistakes

Saying 'a square is not a rectangle'. It is: it has 4 right angles and opposite sides equal and parallel.
Reading the hierarchy downwards: 'all rectangles are squares' or 'all parallelograms are rhombuses' are false.
Thinking a rectangle has 4 lines of symmetry. Its diagonals are NOT lines of symmetry: it has only 2.
Thinking a parallelogram has lines of symmetry. It has none, but it has rotational symmetry of order 2.
Using the n-lines rule for a shape that is not regular (rectangle, kite, a hexagon with equal sides but different angles).
Writing 'order 0' for no rotational symmetry. The lowest order is 1.

How Checkpoint asks this

"Draw a ring around the correct name(s)." Read whether it says one name or every name.
"Tick Correct or Not correct" for statements like 'every rhombus is a square'. If it is not correct, a counter-example is the strongest reason.
"Complete the table" of properties (sides, angles, diagonals, lines of symmetry, order of rotation).
Points on a coordinate grid: "What type of quadrilateral is ABCD?" or "Write the coordinates of D so that ABCD is a parallelogram."
Regular polygons: "A regular polygon has 9 lines of symmetry. Write down its order of rotational symmetry." One-mark facts: write the number, not a sentence.
Explain questions: use mathematical words such as parallel, perpendicular, adjacent, opposite, diagonal and bisect.

Practice

Remember

Learn the properties of the square, rectangle, rhombus, parallelogram, trapezium and kite: sides, angles, diagonals, symmetry.
A shape lower in the hierarchy has all the properties of the shapes above it: a square is a rectangle and a rhombus; a rhombus is a parallelogram and a kite.
The hierarchy does not work upside down. One counter-example proves a statement false.
A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n; the smallest turn is 360° ÷ n.
The rule is only for regular polygons (equal sides AND equal angles).

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