Cambridge Primary Checkpoint · Mathematics · Stage 6 · Geometry and Measure

Quadrilaterals: Properties and Classification

Name, describe, sort and sketch quadrilaterals using their sides, angles, parallel sides, lines of symmetry and diagonals.

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Identify and name quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium, kite. [6Gg.01]
Describe them using their sides, angles, parallel sides, lines of symmetry and diagonals. [6Gg.01]
Classify (sort) quadrilaterals by their properties — for example in Venn and Carroll diagrams — and understand that some shapes are special types of others. [6Gg.01]
Sketch quadrilaterals with given properties, on plain paper or on a grid. [6Gg.01]

A quadrilateral is any flat (2D) shape with four straight sides and four vertices (corners). Some quadrilaterals are so useful that they have their own names. To get full marks you must know the properties of each one so well that you can name a shape from a description, sort shapes into groups, and sketch a shape that fits a list of clues.

1. Key words

Word / phrase	Meaning / use
quadrilateral	شكل رباعي — a 2D shape with 4 straight sides
vertex (plural vertices)	رأس — a corner where two sides meet
parallel	متوازي — always the same distance apart; they never meet (shown with matching arrows)
perpendicular	عمودي — meeting at a right angle (90°)
right angle	زاوية قائمة — exactly 90° (shown with a small square)
acute / obtuse angle	زاوية حادة / منفرجة — less than 90° / between 90° and 180°
diagonal	قطر — a straight line joining two opposite vertices
bisect	ينصّف — cut into two equal parts
line of symmetry	محور تماثل — a mirror line: the two halves match exactly
adjacent	متجاور — next to each other
opposite	متقابل — across from each other
classify	يصنّف — sort into groups using properties

How to read the marks on a diagram: small dashes (ticks) show sides of equal length — sides with the same number of ticks are equal. Matching arrows show parallel sides. A small square in a corner shows a right angle.

2. Meet the quadrilaterals

The named quadrilaterals. Ticks = equal sides, arrows = parallel sides, small squares = right angles.

Shape	Sides	Angles	Parallel sides
square	4 equal sides	4 right angles	2 pairs
rectangle	opposite sides equal	4 right angles	2 pairs
rhombus	4 equal sides	opposite angles equal (2 acute, 2 obtuse)	2 pairs
parallelogram	opposite sides equal	opposite angles equal (2 acute, 2 obtuse)	2 pairs
trapezium	no rule	no rule (some have 2 right angles)	1 pair
isosceles trapezium	the two non-parallel sides are equal	2 pairs of equal angles (next to each parallel side)	1 pair
kite	2 pairs of equal adjacent sides	1 pair of opposite angles equal	none
arrowhead (delta)	2 pairs of equal adjacent sides	one angle is reflex (more than 180°)	none

The arrowhead is a "kite pushed in": it has the same equal sides as a kite but one corner points inwards. Any other four-sided shape with no special properties is simply called an irregular quadrilateral (or just a quadrilateral).

3. Lines of symmetry

Imagine folding the shape along a line. If the two halves fit exactly on top of each other, the fold is a line of symmetry. Use tracing paper in the test if it helps — it is allowed.

Dashed red lines are lines of symmetry. A parallelogram has none: folding along a diagonal or through the middle does not make the halves match.

Watch out: the diagonals of a rectangle are NOT lines of symmetry, and the diagonals of a rhombus ARE. Test it by folding a paper rectangle corner to corner — the halves do not match. A general trapezium and an irregular quadrilateral have no lines of symmetry. A kite and an arrowhead each have one.

Some shapes also look the same after a turn (rotational symmetry — you study this fully in the symmetry lesson): a square looks the same 4 times in a full turn (order 4); a rectangle, rhombus and parallelogram 2 times (order 2); kites and trapeziums only once (order 1).

4. Diagonals

Draw both diagonals and ask three questions: Are they equal in length? Do they cut each other in half (bisect each other)? Do they meet at a right angle (are they perpendicular)?

Diagonals drawn as dashed blue lines. A small red square shows that the diagonals meet at 90°.

Shape	Diagonals equal?	Bisect each other?	Perpendicular?
square	yes	yes	yes
rectangle	yes	yes	no
rhombus	no	yes	yes
parallelogram	no	yes	no
kite	no	no — only one is cut in half	yes
isosceles trapezium	yes	no	no

Notice the square says yes to everything: it has all the properties of a rectangle AND all the properties of a rhombus.

5. Classifying: the quadrilateral family

A shape can belong to more than one group. A square has 2 pairs of parallel sides, so it IS a parallelogram. It has 4 right angles, so it IS a rectangle. It has 4 equal sides, so it IS a rhombus. We say a square is a special rectangle and a special rhombus.

Each arrow means "…has a special type called…". Follow the arrows backwards: a square is a rectangle, a rhombus, a parallelogram, a kite and a quadrilateral.

Every square is a rectangle — but not every rectangle is a square (its sides need not all be equal).
Every square is a rhombus — but not every rhombus is a square (it need not have right angles).
Every rectangle and every rhombus is a parallelogram.
A rhombus is also a special kite (its two pairs of adjacent sides are equal).
In this course a trapezium has exactly one pair of parallel sides, so a parallelogram is not counted as a trapezium.

Carroll diagram example. Sort by two properties at the same time. Each shape goes in exactly one cell.

	has 4 right angles	does not have 4 right angles
all sides equal	square	rhombus
not all sides equal	rectangle	parallelogram, kite, trapezium

6. Sketching quadrilaterals

Read every clue and decide which shape it is first (use the tables above).
On a square grid, use the grid lines for parallel and perpendicular sides. To make a side parallel to another, move the same number of squares across and up.
Use a ruler. Mark equal sides with ticks, parallel sides with arrows and right angles with a small square — this shows the examiner you know the properties.
Check every clue again against your sketch before moving on.

الفكرة الأساسية

المربع هو مستطيل خاص ومعين خاص في نفس الوقت — عنده كل خواصهم.
متوازي الأضلاع: كل ضلعين متقابلين متوازيين ومتساويين. شبه المنحرف: زوج واحد بس من الأضلاع المتوازية. الطائرة الورقية (kite): زوجين من الأضلاع المتجاورة المتساوية.
اسأل 3 أسئلة عن الأقطار: متساويين؟ بينصّفوا بعض؟ متعامدين (90°)؟

7. Worked examples

Worked example 1 (1 mark) I am a quadrilateral. I have two pairs of parallel sides. All my sides are equal. I have no right angles. What am I?

Two pairs of parallel sides → parallelogram, rectangle, rhombus or square.
All sides equal → rhombus or square.
No right angles → not a square.
Answer: rhombus.

Worked example 2 (2 marks) Write each shape in the correct place in the Carroll diagram: square, rectangle, rhombus, kite, trapezium, irregular quadrilateral.

	diagonals meet at 90°	diagonals do not meet at 90°
has parallel sides	square, rhombus	rectangle, trapezium
no parallel sides	kite	irregular quadrilateral

Working: check each shape for both properties. Kite: diagonals perpendicular ✓, no parallel sides. Rectangle: parallel sides ✓, diagonals equal but not at 90°. Marks: 2 marks for all six correct; 1 mark for four or five correct.

Worked example 3 (1 mark) P, Q and R are three vertices of a parallelogram PQRS. Find the position of S.

To go from Q to R you move 1 square right and 3 squares up. Do the same from P to reach S.

Opposite sides of a parallelogram are parallel and equal, so PS must be the same "move" as QR.
Q → R is 1 right, 3 up. So P → S is also 1 right, 3 up.
Answer: S is 1 square right and 3 squares up from P (at (2, 4) if P is (1, 1)). Check: SR is 4 squares long, the same as PQ. ✓

Worked example 4 (1 mark) Zain says: "A square is not a rectangle because its sides are all the same length." Is he right? Explain.

Answer: No. A rectangle is a quadrilateral with 4 right angles (and opposite sides equal). A square has 4 right angles and its opposite sides are equal, so it meets every rule for a rectangle. It is a special rectangle whose sides happen to be all equal.

Worked example 5 (1 mark) ABCD is a parallelogram. Angle A = 65° and angle B = 115°. Write the sizes of angle C and angle D.

In a parallelogram, opposite angles are equal.
Angle C is opposite angle A, so angle C = 65°.
Angle D is opposite angle B, so angle D = 115°.
Answer: C = 65°, D = 115°.

Common mistake: thinking all four angles of a parallelogram are equal. That is only true for a rectangle (all 90°).

Worked example 6 (1 mark) Sketch a quadrilateral with exactly two lines of symmetry whose diagonals are not equal. Name it.

Two lines of symmetry → rectangle or rhombus. Diagonals not equal → not a rectangle. Answer: a rhombus — sketch 4 equal sides (ticks) with no right angles and draw the two diagonals as the dashed lines of symmetry. Common mistake: drawing a rectangle — its diagonals are equal.

Common mistakes

Thinking a rhombus is just a "tilted square". A square turned on its corner is still a square. A rhombus has no right angles (unless it is a square).
Saying a rectangle has 4 lines of symmetry. It has 2 — its diagonals are not lines of symmetry.
Saying a parallelogram has lines of symmetry. A (non-rectangular) parallelogram has none.
Saying "a square is not a rectangle". It is — a special one.
Mixing up trapezium (1 pair of parallel sides) and parallelogram (2 pairs).
Sketching freehand without a ruler, so that "parallel" sides clearly are not parallel. Use the grid and a ruler.

How Checkpoint asks this

Name the shape from a list of clues — go through the clues one at a time and cross out shapes that fail.
Sort shapes into a Carroll or Venn diagram. Every shape must go somewhere; check each shape against BOTH properties.
Tick true or false: "All rhombuses are squares". Look for one counter-example to show a statement is false.
Draw or complete a shape on a square grid: copy the "moves" of one side to draw the side parallel to it.
Join dots or draw a line to split a shape into two named quadrilaterals.
"Explain" questions: name the property that decides it (e.g. "it has no right angles") — vague answers like "it looks different" get no mark.

Practice

Remember

Describe a quadrilateral using sides, angles, parallel sides, lines of symmetry and diagonals.
Square: everything equal, 4 right angles, 4 lines of symmetry, diagonals equal and perpendicular.
Rectangle 2 lines of symmetry; rhombus 2 (its diagonals); parallelogram 0; kite 1; isosceles trapezium 1.
A square is a special rectangle AND a special rhombus; rectangles and rhombuses are special parallelograms.
Parallelogram: 2 pairs of parallel sides. Trapezium: 1 pair. Kite: none, but 2 pairs of equal adjacent sides.
On a grid, a parallel side copies the same "moves" (squares across and up).

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