Add a right angle → rectangle; equalize the sides → rhombus; do both → square.
The parallelogram is a starting clay. Press all four of its corners to right angles and it stiffens into a rectangle; stretch all four sides to one length and it tightens into a rhombus; do both at once and it locks into the most special quadrilateral of all, the square. Each new condition keeps every parallelogram property you already know — opposite sides equal, opposite angles equal, diagonals bisecting each other — and quietly adds a fresh one. And the freshest news always shows up in the diagonals: in a rectangle they grow equal, in a rhombus they cross at a right angle, and in a square they do both. By the end you'll read a whole family in a single glance.
A rectangle is a parallelogram with one right angle. That single 90° corner is contagious. Consecutive angles of a parallelogram are supplementary (they add to 180°), so if ∠A = 90°, then ∠B = 180° − 90° = 90° too — and opposite angles are equal, so ∠C = ∠D = 90° as well. One right angle forces all four.
Since a rectangle is still a parallelogram, it keeps everything: opposite sides parallel and equal, diagonals bisecting each other. But it earns one brand-new property — the diagonals are equal: AC = BD. Look at triangles △ABC and △BAD: they share side AB, have BC = AD (opposite sides), and a right angle at B and at A. By SAS they are congruent, so their third sides match: AC = BD.
A parallelogram is a rectangle if either: it has one right angle, or its diagonals are equal. (The second test is the converse of the property we just proved.)
Switch between the four shapes and watch what the diagonals do. The marks never lie — count the ticks.
Here is a small wonder that falls straight out of the rectangle. Take a right triangle △ABC with the right angle at C and the hypotenuse AB. Spin a copy of it a half-turn about the midpoint of AB and the two triangles snap together into a rectangle — the hypotenuse becomes one of its diagonals. We already know a rectangle's diagonals are equal and bisect each other, so the half-diagonal from C to the midpoint M equals half of every diagonal.
That gives a clean, useful fact:
In a right triangle, the median to the hypotenuse is half the hypotenuse: if M is the midpoint of the hypotenuse AB, then CM = AM = MB = ½ AB. So M is the same distance from all three vertices.
Read that last line again: the right-angle vertex C stays a fixed distance ½ AB from the center M no matter where it sits — which means C rides a circle. That is the very first whisper of a big Stage-18 idea: the right angle always rides a semicircle on its hypotenuse.
Slide the right-angle vertex C along the arc. The hypotenuse never changes, and CM always equals half of it.
Run down the other branch. A rhombus is a parallelogram with four equal sides — a "pushed-over square," a diamond on a playing card, the cells of a chain-link fence. It keeps every parallelogram property, then adds two of its own, and both live in the diagonals.
The diagonals are perpendicular. Like any parallelogram, the diagonals bisect each other at the common midpoint O. In △ABO and △ADO we have AB = AD (all four sides equal), the shared side AO, and BO = DO (the diagonals bisect each other) — so SSS makes them congruent. The two angles at O are equal and add to 180°, so each is 90°: AC ⊥ BD. (The diagonals split the rhombus into four congruent right triangles.)
The diagonals bisect the angles. From that same congruence, ∠BAO = ∠DAO — diagonal AC splits ∠A exactly in half, and likewise every diagonal bisects the two angles it reaches.
A parallelogram is a rhombus if either: it has four equal sides (one pair of adjacent sides equal is enough — the opposite ones already match), or its diagonals are perpendicular.
The rectangle's diagonals are equal but not perpendicular; the rhombus's diagonals are perpendicular but not equal. Mixing these up is the single most common slip in this whole lesson.
A square is the meeting point of both branches: it is a rectangle and a rhombus at once — four equal sides and four right angles. It therefore inherits everything. Its diagonals are equal (from the rectangle), perpendicular (from the rhombus), they bisect each other (from the parallelogram), and they bisect the angles — slicing each 90° corner into two 45° halves (from the rhombus).
You can reach a square from either side of the family: a rectangle with two equal adjacent sides is a square, and a rhombus with one right angle is a square. Either route adds the one condition the shape was missing.
Stand back and the whole picture is a tree of single steps (the diagram at the top of this lesson). Start with any quadrilateral; make both pairs of opposite sides parallel and you have a parallelogram; from there one branch adds right angles (→ rectangle) and the other adds equal sides (→ rhombus); the two branches rejoin at the square.
The arrows only run one way. Every square is a rectangle and a rhombus; every rectangle and rhombus is a parallelogram — but reading backward fails: a parallelogram need not be a rectangle, and a rectangle need not be a square. The narrower the shape, the more properties it carries.
"Is a" only points down the tree: square ⊂ rectangle ⊂ parallelogram, and square ⊂ rhombus ⊂ parallelogram. A square is the rare shape that belongs to every box.
| Shape | 4 equal sides? | 4 right angles? | diagonals equal? | diagonals ⊥? | diagonals bisect angles? |
|---|---|---|---|---|---|
| Parallelogram | ✗ | ✗ | ✗ | ✗ | ✗ |
| Rectangle | ✗ | ✓ | ✓ | ✗ | ✗ |
| Rhombus | ✓ | ✗ | ✗ | ✓ | ✓ |
| Square | ✓ | ✓ | ✓ | ✓ | ✓ |
Every row keeps the row above it and earns one more ✓. (All four also bisect each other — a property they share.)
• A rectangle is a parallelogram with a right angle (⇒ all four right angles); its diagonals are equal.
• A rhombus is a parallelogram with four equal sides; its diagonals are perpendicular and bisect the angles.
• A square is both — four equal sides and four right angles; its diagonals are equal, perpendicular, bisect each other, and bisect the angles (into 45°).
• In a right triangle, the median to the hypotenuse equals half the hypotenuse.
• square ⊂ {rectangle, rhombus} ⊂ parallelogram — never the reverse.
A parallelogram has one angle measuring 90°. What kind of parallelogram is it, and what are its other three angles?
It is a rectangle. Consecutive angles are supplementary, so the neighbors are 180° − 90° = 90°; opposite angles are equal, so the fourth is 90° as well. All four angles are 90°.
A rectangle's two diagonals are each 10 units long, meeting at O. What is special about them, and how long is each of the four pieces?
A rectangle's diagonals are equal (and they bisect each other, like any parallelogram). So both diagonals are 10, and each half is 10 ÷ 2 = 5: AO = BO = CO = DO = 5.
A rhombus has side length 6. Find its perimeter. Are its diagonals perpendicular?
All four sides are equal, so the perimeter is 4 × 6 = 24. And yes — a rhombus's diagonals are always perpendicular.
A rhombus has diagonals of length 6 and 8. Since the diagonals are perpendicular and bisect each other, find the side length of the rhombus.
The diagonals split it into four right triangles with legs 6 ÷ 2 = 3 and 8 ÷ 2 = 4. By the Pythagorean theorem the side is √(3² + 4²) = √25 = 5.
True or false: every square is a rectangle, and every square is a rhombus. Explain.
Both true. A square has four right angles (so it qualifies as a rectangle) and four equal sides (so it qualifies as a rhombus). The reverse fails: most rectangles and most rhombi are not squares.
A right triangle has hypotenuse 13. How long is the median drawn from the right angle to the hypotenuse?
The median to the hypotenuse is half the hypotenuse: 13 ÷ 2 = 6.5.
Six questions to lock it in. Tap the answer you think is right.
This lesson grows the parallelogram family by the add-one-condition method, which is exactly how the Common Core asks students to organize quadrilaterals. The big idea is hierarchy: a square is not a separate creature from a rectangle — it is a rectangle (and a rhombus) with one extra constraint. Encourage your student to read the family tree downward ("a square is a rectangle") and to notice that each step inherits everything above it and earns one new diagonal property.
The misconception to watch is the diagonal swap: students routinely claim a rectangle's diagonals are perpendicular, or a rhombus's are equal. They are not — it's the reverse. A rectangle's diagonals are equal (it's a "stretched" frame); a rhombus's are perpendicular (it's a "pushed" diamond). The "Don't swap them" note and the morph widget exist precisely to drill this. A second slip is treating "rectangle" and "square" as unrelated rather than square ⊂ rectangle.
Common Core: HS G-CO.C.11 (prove theorems about parallelograms, including the special cases) and G-SRT.B.5 (use congruence to solve problems and prove relationships), building on 8.G. The median-to-the-hypotenuse fact also previews the inscribed right angle of Stage 18 (G-C).