Mathematics Year 7: Classifying quadrilaterals in a hierarchy, and explaining why
Mathematics, Year 7 (Portugal): Trapezoid, parallelogram, rectangle, rhombus, square and kite as families inside families, and how to say why a figure belongs.
Each name is a set of properties
Trapezoid: at least one pair of parallel sides. Parallelogram: two pairs. Rectangle: four right angles. Rhombus: four equal sides. Square: four equal sides and four right angles. Kite: a convex quadrilateral with two pairs of equal consecutive sides. A figure that has the properties of a name has that name. So a square is a rectangle and a rhombus; a rectangle and a rhombus are parallelograms; a parallelogram is a trapezoid; and a rhombus is a kite. It is a hierarchy: families inside families.
Why a hierarchy saves work
What is true for a family holds for every family inside it: the diagonals of a parallelogram bisect each other, so those of the rectangle, the rhombus and the square do too. Each smaller family adds a property: the rectangle adds right angles to the parallelogram, the rhombus adds equal sides, and the square adds both. That is why the square has all the diagonal properties you saw: those of a rectangle (equal) and those of a rhombus (perpendicular).
A case: which names fit, and how to say it
A quadrilateral has four sides of 5 cm and angles, in order, of 60°, 120°, 60° and 120°. Check: 60 + 120 + 60 + 120 = 360°, as it must be, because (4 − 2) × 180° = 360°. Four equal sides: a rhombus. Every rhombus is also a parallelogram, a trapezoid and a kite, so all those names fit. It is not a rectangle or a square: no right angles. To say it: «It is a rhombus because it has four sides of 5 cm; it is not a square because its angles measure 60° and 120°, not 90°.»
The trap: thinking the names exclude each other
The slip: «a square is not a rectangle, because all its sides are equal». A rectangle only asks for four right angles; the square has them, so it is a special rectangle. A more specific name does not cancel the general one, just as being from Lisbon does not stop you being Portuguese. Watch out for the trapezoid: some books ask for exactly one pair of parallel sides, which leaves the parallelogram out. Here we use «at least one pair», the definition that makes the hierarchy work; check which one your book uses.
Where you see this
A screen, a window, a sheet of paper: rectangles. A square tile is at once a square, a rectangle, a rhombus and a parallelogram, and you pick the name that says what you need. And explaining a shape to a friend in a message, without a drawing, forces you to use properties: «four equal sides and perpendicular diagonals». The clearer your words, the closer the figure your friend imagines gets to yours.
Test what you learned
Which chain goes from the most special to the most general quadrilateral without a mistake?
- a) square → rectangle → parallelogram → trapezoid → quadrilateral
- b) square → parallelogram → rectangle → trapezoid → quadrilateral
- c) square → rectangle → kite → trapezoid → quadrilateral
- d) square → trapezoid → parallelogram → rectangle → quadrilateral
Correct answer: a) square → rectangle → parallelogram → trapezoid → quadrilateral — Right: each name is a special case of the next. A square has four right angles (rectangle), two pairs of parallel sides (parallelogram) and at least one pair (trapezoid).
A quadrilateral has four sides of 5 cm and angles, in order, of 60°, 120°, 60° and 120°. Which names fit it?
- a) Square and rectangle, because all its sides are equal
- b) Only rhombus: it is not a parallelogram or a trapezoid
- c) Rectangle and parallelogram, because opposite angles are equal
- d) Rhombus, parallelogram, trapezoid and kite; not a rectangle or a square
Correct answer: d) Rhombus, parallelogram, trapezoid and kite; not a rectangle or a square — Right: four equal sides make a rhombus, and a rhombus is also a parallelogram, a trapezoid and a kite. There are no right angles, so no rectangle or square.
Tomás says: «A square is not a rectangle, because all its sides are equal.» Where does he go wrong?
- a) He does not go wrong: a rectangle must have sides of different lengths
- b) The square is a trapezoid, and that is why it is a rectangle
- c) A rectangle only asks for four right angles, and the square has them: it is a special rectangle
- d) The square is a rectangle because its diagonals are perpendicular
Correct answer: c) A rectangle only asks for four right angles, and the square has them: it is a special rectangle — Right: a more specific name (square) does not cancel a more general one (rectangle). Every square is a rectangle; not every rectangle is a square.
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