The family of quadrilaterals

What is it about?

A quadrilateral $ABCD$ has four corners, the sides $a = [AB]$ , $b = [BC]$ , $c = [CD]$ , $d = [DA]$ , the interior angles $\alpha, \beta, \gamma, \delta$ and the two diagonals $e = [AC]$ and $f = [BD]$ . The corners are labelled anticlockwise. In every quadrilateral:

\alpha + \beta + \gamma + \delta = 360^\circ

Some quadrilaterals have special properties and their own name. Here – as in many textbooks – they are defined by their symmetry or by their sides and angles:

Type	Definition	Symmetry
trapezium (US: trapezoid)	(at least) one pair of opposite sides is parallel	–
isosceles trapezium	trapezium with a line of symmetry through the midpoints of the parallel sides	1 line
kite	a diagonal is a line of symmetry	1 line
parallelogram	both pairs of opposite sides are parallel	point symmetry
rhombus	all four sides are equal	2 lines (the diagonals), point symmetry
rectangle	all four angles are right angles	2 lines, point symmetry
square	rhombus and rectangle at the same time	4 lines, point symmetry

In the house of quadrilaterals the square is at the very top. An arrow from X to Y means: X is a special case of Y, so every X is also a Y. Every square is a rectangle and a rhombus – but not every rectangle is a square. The higher a quadrilateral is in the house, the more properties it has.

The marks in the picture: arrows on the sides show parallel sides, ticks show equal sides (same number of ticks = same length), a quarter circle with a dot shows a right angle and a dash-dot line a line of symmetry. $Z$ is the centre of symmetry. One grid square is $0.5\,\text{cm}$ wide.

Try it

Drag a corner of the general quadrilateral until two opposite sides are parallel. Which type lights up in the house? Move the whole quadrilateral by grabbing its inside: does the type change?
Look at the parallelogram with diagonals. What do you notice about the diagonals? Where is the centre of symmetry $Z$ ?
Move C and D by the same amount until you get a rhombus. Which properties are added? Which types light up in the house now?
Load the kite and drag C downwards until you get an arrowhead. Is it still a kite?
Tap “Rectangle” in the house: the card gives the definition and says what your quadrilateral is still missing. “Show an example” transforms the quadrilateral.
Set up a tilted square. How many lines of symmetry does it have? Even standing on a corner, it is still a square.
Build shapes in the mode build a shape and classify shapes in the mode classify.

Tasks

Task 1: True or false?

(a) Every square is a rhombus. (b) Every rhombus is a square. (c) Every rectangle is a trapezium. (d) Every parallelogram is an isosceles trapezium. (e) Every rhombus is a kite.

Show solution

(a) True – a square has four equal sides. (b) False – a rhombus usually has no right angles. (c) True – in a rectangle even both pairs of opposite sides are parallel. (d) False – a slanted parallelogram has equal legs, but no line of symmetry. (e) True – in a rhombus both diagonals are lines of symmetry.

Task 2: Classify

Classify the quadrilaterals as precisely as possible: shape 1, shape 2, shape 3.

Show solution

Shape 1 is a (tilted) rectangle: four right angles, but not all sides are equal. Shape 2 is a parallelogram: opposite sides parallel and equal, no right angles, no line of symmetry. Shape 3 is a kite: $|AB| = |AD|$ and $|CB| = |CD|$ , the diagonal $e$ is a line of symmetry.

Task 3: The missing corner

Load task Drag D so that $ABCD$ becomes a parallelogram. How many squares to the right of A and above A is D then? Can you also place D so that you get a rectangle?

Show solution

From B to C you go 2 squares right and 6 up. For $d \parallel b$ and $|AD| = |BC|$ you have to go the same way from A: D lies 2 squares right of A and 6 squares above A. Show solution A rectangle is not possible with these points A, B, C, because the angle at B is not a right angle.

Task 4: What follows from the diagonals?

The diagonals of a quadrilateral bisect each other and are perpendicular. Which types of quadrilateral are possible?

Show solution

If the diagonals bisect each other, the quadrilateral is a parallelogram. If they are also perpendicular, all four sides are equal: it is a rhombus – possibly even a square.

Task 5: Give reasons

Use the definition to explain: every rectangle is an isosceles trapezium.

Show solution

In a rectangle $a$ and $c$ are parallel, so it is a trapezium. In addition, the common perpendicular bisector of $a$ and $c$ is a line of symmetry – exactly what the definition of the isosceles trapezium requires. Show rectangle

Ideas for the classroom

Use: about grade 5 (recognising and describing special quadrilaterals) and grade 7 (quadrilaterals with line and point symmetry, ordering them in the house of quadrilaterals). Suitable for introducing the topic on the whiteboard, for pair work (one builds, the other classifies) and for practice in the two task modes.
Definitions: The simulation uses the inclusive, symmetry-based definitions (a square is also a rectangle, a parallelogram is also a trapezium). An isosceles trapezium is an axially symmetric trapezium here, so slanted parallelograms do not belong to it. Non-convex kites (“arrowheads”) count as kites because they are symmetric about a diagonal – some textbooks only consider convex ones. Please check against the textbook you use.
Model: The corners lie on grid points, so parallel, perpendicular, equal and symmetric are detected exactly (without rounding). Crossed quadrilaterals and quadrilaterals with three corners on one line cannot be dragged. If you enter them in the number fields, a notice appears instead of a classification.
Controls: Drag corners, grab the inside to move the shape, tap the boxes in the house for the definition and “what is missing?”. Switching off “Show the classification in the house” turns exploring into a puzzle.
Common stumbling blocks:
- “A square is not a rectangle.” – The names do not exclude each other; a square meets all conditions of a rectangle.
- “A square standing on a corner is a rhombus and not a square.” – The position does not matter.
- “The diagonal of a parallelogram is a line of symmetry.” – If you reflect in it, the shape does not map onto itself.
- “In a rectangle the diagonals are perpendicular.” – That is only true for the square.