The classification, built by the student rather than handed over
The whole quadrilateral family in one figure, with nothing pre-built to already be any of it.
Every parallelogram figure so far in this chapter started as a parallelogram: three points were free and the fourth followed, so that it could never be anything else. This lesson removes that scaffolding.
A quadrilateral, A, B, C and D — all four points free this time, nothing built to already be a particular shape. A live reading names it, whatever it currently is.
A quadrilateral, A, B, C and D, all four points draggable and nothing pre-built to already be any particular shape. A live reading names it a general quadrilateral, a trapezium, a kite, a parallelogram, a rectangle, a rhombus, or a square, depending entirely on where the four points are dragged. This needs JavaScript switched on.
Try to make the reading say “a parallelogram.” Then try to make it say “a kite,” without it also being a parallelogram. Notice what has to be true each time, before reading on. Parallelogram, trapezium, kite, rectangle, rhombus and square were all named in the last two lessons; what is new here is arriving at each one without being told the condition first.
Dragging into each shape rather than being told the condition
A parallelogram needs opposite sides both equal and parallel at once — not two separate conditions to juggle, but one: side AB, as a displacement, has to match side DC exactly. (Matching displacement automatically makes AB and DC the same length (equal) and pointing the same way (parallel) — one condition doing the job of two.) Drag C until AB and DC are the same displacement and the shape locks into a parallelogram, every time, with nothing else required.
A kite needs two pairs of adjacent sides equal, arranged so the shape is symmetric about one diagonal — drag two points to mirror each other across the line through the other two, and both adjacent pairs come out equal together, because a reflection preserves length automatically. A trapezium needs exactly one pair of opposite sides parallel: easier to reach than a parallelogram, since only one direction has to be matched rather than two.
Rectangle, rhombus and square are not new work at all: they are a parallelogram, already built, with one more condition checked on top — a right angle, equal adjacent sides, or both, exactly as the last lesson found.
A definition a student has had to satisfy by dragging something into place is understood differently from one that was only read. “A parallelogram has opposite sides equal and parallel” is a sentence to memorise, right up until the moment of actually needing AB to match DC. The classification was never a list to learn — it was always a set of conditions to go looking for, and this lesson is the first time nothing hands them over first.
This closes the quadrilateral chapter’s parallelogram thread. The next phase turns to area — starting from a rectangle, then showing that a triangle is exactly half of one.
Drag the figure until it reads “a trapezium.” What is different about how you got there, compared to reaching “a parallelogram”?
Only one pair of opposite sides needs to end up parallel for a trapezium, against both pairs at once for a parallelogram — noticeably easier to land on, since half as much has to line up.
Can a shape be a kite and a trapezium at the same time?
Not in general — a kite’s defining symmetry is about adjacent sides being equal in pairs, which does not by itself force any side to be parallel to another, and a trapezium’s defining fact is exactly that one pair of sides is parallel. A shape could coincidentally satisfy both, but neither condition implies the other.
Why does the rectangle/rhombus/square check only ever run after the parallelogram check?
Because “a right angle” or “equal adjacent sides” only mean rectangle or rhombus given that the shape is already a parallelogram — a general quadrilateral can have a right angle at A without being anything special at all. The extra condition sharpens a name the shape has already earned, rather than standing on its own.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.