Rectangle, rhombus, square, kite, trapezium as special cases
The parallelogram family tree, and two quadrilaterals that sit outside it entirely.
Every parallelogram this course has drawn so far has been a general one: opposite sides equal, opposite angles equal, nothing more specific claimed. Add one extra condition and it earns a sharper name.
The same kind of parallelogram as the last two lessons, A, B, C and D. A live reading names it, based on the angle at A and on whether AB and AD happen to be equal.
A parallelogram, A, B, C and D, where A, B and D are draggable and C always follows to keep the shape a genuine parallelogram. A live reading names it a general parallelogram, a rectangle, a rhombus, or a square, depending on whether the angle at A is a right angle and whether AB and AD happen to be equal. This needs JavaScript switched on.
Each extra condition earns its own name without changing anything else about the shape underneath — it is still every property the last two lessons proved, plus one more fact. A parallelogram with a right angle is a rectangle. A parallelogram with all four sides equal is a rhombus. A parallelogram that is both — a right angle and all four sides equal — is a square.
Two more names belong to this family of shapes without belonging to the parallelogram itself. A kite has two pairs of adjacent sides equal, rather than two pairs of opposite sides — it is not required to have any parallel sides at all. A trapezium has exactly one pair of parallel sides, not two. Neither is a parallelogram in general, so neither can be reached by dragging the figure above into a special case: they are a different branch of the same family tree.
Forcing a rectangle, and forcing a rhombus
In any parallelogram, AD is parallel to BC, and AB crosses both as a transversal, so the angle at A and the angle at B are co-interior angles: they sum to 180 degrees. If the angle at A is 90, the angle at B must be 90 too. Opposite angles are already equal (the last lesson but one), so the angle at C matches A and the angle at D matches B — all four are 90. One right angle, once the shape is already known to be a parallelogram, is enough to force the rest, and that is what makes it a rectangle.
Opposite sides of a parallelogram are already equal, AB to DC and AD to BC. Adding “AB equals AD” links the two pairs together — AB, DC, AD and BC are now all equal to each other, not just equal in their own pair. One equality between adjacent sides is enough to force all four sides equal, which is what makes it a rhombus. A rectangle and a rhombus are not two unrelated shapes with their own separate rules — they are the one parallelogram rule, already proved, with a single extra fact spreading across the whole shape. A square needs no argument of its own at all: it is just both extra facts holding at once.
The next lesson turns this around: instead of being told which extra condition makes which name, the reader builds the classification for themselves — which shapes force which others, and which do not.
A parallelogram has one angle of 90 degrees. What can be said about the other three?
All three are also 90 degrees — it is a rectangle, and every angle in a rectangle is a right angle, by the same argument this lesson just made.
A parallelogram has AB equal to 6 units and AD equal to 6 units. What kind of parallelogram must it be?
A rhombus at least — all four sides are forced equal. It is only a square if, in addition, one of its angles is a right angle, which this information alone does not say.
Is a square a rhombus?
Yes. A square meets the rhombus condition (all four sides equal) and the rectangle condition (a right angle) at once, so every square is also a rhombus and also a rectangle — a square is not a fourth, separate thing sitting beside the other two.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.