A square is a rectangle. Every single one.
Say it backwards — a rectangle is a square — and now it's false. Most rectangles aren't square.
Being a special case only runs one direction. The narrow name can always borrow the wide name. The wide name can never borrow back.
This is exactly last session's trap, moved from weather sentences into shapes. If a shape is a square, then it is a rectangle. is a conditional statement, same as Ifit rains then the ground is wet. — and it has the same converse trap waiting inside it.
Each one below adds exactly one new requirement on top of the last.
Opposite sides parallel — and, as a result, equal in length. That single requirement is all this level asks for.
No claim about angles. No claim about all four sides matching each other.
A parallelogram, plus one more rule: all four angles are 90°.
Side lengths still aren't required to match each other.
A parallelogram, plus a different one more rule: all four sides are equal length.
Angles still aren't required to be right angles.
A rectangle and a rhombus, both requirements met at once.
Right angles from one side. Equal sides from the other. Nothing left to add.
Rectangle is Latin rectus, "straight/right," plus angulus, "corner" — literally "right-angled."
Square comes through French from Latin exquadrare, "to make four-cornered" — the same quadr- root as quadrilateral, just verbed.
Four pairs below. For each one: are the two sentences saying the same thing, or not? Answer by feel — nothing here is scored.
Ifa shape is a square then it is a rectangle. Ifa shape is a rectangle then it is a square.
Ifa shape is a rhombus then it is a parallelogram. Ifa shape is a parallelogram then it is a rhombus.
Ifa shape is a square then it is a rhombus. Ifa shape is a rhombus then it is a square.
Iftwo angles are vertical then they are congruent. Iftwo angles are congruent then they are vertical.
One picture, one family. Four buttons, each one lighting up everything that name is allowed to point at — never redrawn, only shaded differently.
Parallel is Greek para, "beside," plus allelon, "one another" — sides beside one another, never meeting.
-gram is Greek gramma, "something drawn" — a parallelogram is literally "a thing drawn with sides beside one another."
The whole jade ellipse is parallelograms. Every
quadrilateral in it has opposite sides parallel — that's the only promise this level makes.
The dot alone in the jade, outside both circles, is a normal slanted parallelogram: no right angles,
unequal adjacent sides, still fully qualified.
Every button lights up everything that name is allowed to point at, not just a typical example.
Pressing Rectangle shades the whole gold circle — including the sliver in the middle, because a square really is a rectangle. Rhombus does the same on the blue circle.
The dashed shape in the crossing zone is square — the only region that satisfies both circles' rules at once.
Quadrilateral is Latin quattuor, "four," plus latus / lateris, "side" — literally "four-sided."
Same root gives you lateral and bilateral.
Rhombus is Greek — rhombos, a spinning top that traces a lozenge as it wobbles. The shape got the toy's name.
A rule: Ifa quadrilateral is a rhombus then all four of its sides are equal. Four quadrilaterals are in front of you. You can check one thing about each. Which do you actually have to check to know if any of them break the rule? Tap the ones you'd check.
Only two matter: is a rhombus (do its sides actually turn out equal?) and does not have all sides equal (is it secretly claiming to be a rhombus anyway?). The other two can't break the rule no matter what you find — same shape of answer as the driving rule last session.
A quadrilateral has four equal sides. Is it definitely a square?
A quadrilateral is not a rectangle. Does that mean its opposite sides can't be equal?
Starting statement: Ifa quadrilateral is a square then it is a rectangle. Same pieces-and-slots as last session. Build each turn.
Which turn is guaranteed true here, same as last time? The contrapositive — "if not a rectangle, then not a square" — because anything failing to be a rectangle can't possibly be the narrower thing a square is.
The biconditional, back again: "A shape is a square if and only if it is both a rectangle and a rhombus." Both halves have to be true for the whole thing to be true — which is exactly the crossing zone on the map above. A biconditional built from two memberships is an intersection.
Three worked examples. The first is fully worked. The second has its middle step pulled — you supply it. The third is bare — you're on your own, with a nudge if you want one.
Three symbols in the proofs below:
∠ — angle. ≅ — congruent, "the same size and shape." ∥ — parallel, "never meet, no matter how far extended."
Prove: In parallelogram ABCD, opposite angles are congruent — specifically, ∠A ≅ ∠C.
1. ABCD is a parallelogram. Given.
2. AB ∥ DC, with AC a transversal
crossing both. Definition of parallelogram — opposite sides are parallel.
3. ∠BAC ≅ ∠DCA and ∠DAC ≅ ∠BCA. Alternate interior angles, parallel lines cut by a
transversal.
4. ∠A = ∠BAC + ∠DAC, and ∠C = ∠DCA + ∠BCA. Angle addition.
5. ∠A ≅ ∠C. Substitution from steps 3 and 4 — equal parts added to equal parts.
Prove: A rhombus's diagonals bisect its angles — in rhombus WXYZ, diagonal WY bisects ∠W.
1. WXYZ is a rhombus, so all four sides are congruent:
WX ≅ WZ. Definition of rhombus.
2. WY ≅ WY. Reflexive property — shared side.
3. XY ≅ ZY. Same reason as step 1 — all four sides equal.
4. — your turn: what do steps 1–3 let you conclude about triangles WXY and
WZY?
Prove: Every square has diagonals that are both congruent and perpendicular.
A square belongs to both circles on the map. What does each circle promise about diagonals on its own?
Congruent: a square is a rectangle, and a rectangle's diagonals are always congruent
(they're each the hypotenuse of two congruent right triangles formed by a side and the other
side).
Perpendicular: a square is a rhombus, and a rhombus's diagonals are always
perpendicular (they bisect each other and split the rhombus into four congruent right
triangles).
Both at once only happens because a square sits in the crossing zone and
collects both guarantees — neither circle alone would give you both.
Four proofs below. Same four-slot shape as last session's broken proofs — find the ones that go wrong, and why.
Diagonal is Greek dia, "through/across," plus gonia, "angle" — a line running through, from one corner to another.
Same gonia root as "polygon" and "trigonometry" — anywhere angles are being measured.
Hypothesis: WXYZ is a rhombus.
Claim: therefore its diagonals are congruent, because all rhombuses are squares.
"All rhombuses are squares" is exactly backwards — it's the converse of "all squares are rhombuses," used as if it were free. A rhombus only guarantees perpendicular diagonals, not congruent ones. A tall, narrow rhombus makes this obvious: its diagonals are nowhere close to equal length. The conclusion is false, and the reasoning is exactly why.
Hypothesis: PQRS is a rectangle.
Claim: all four of its angles are congruent — "because a quadrilateral's angles add to
360°, and there are 4 of them, so 360 ÷ 4 = 90 each."
The conclusion here is true — a rectangle's angles really are all 90°. But the reasoning is broken: nothing about "four angles summing to 360°" forces them to be equal to each other. A kite has angles that sum to 360° too, and they usually aren't all 90°. The real reason is the definition of rectangle — all four angles are right angles by requirement, not by division. A true conclusion reached by a broken argument is still a broken proof.
Hypothesis: a quadrilateral is not a square.
Claim: therefore, it is not a rectangle either.
This is the inverse of "if square, then rectangle" — negate both halves, keep the order. A 4-by-9 rectangle is a perfect counterexample: not a square, but still very much a rectangle. The inverse doesn't carry the same guarantee as the original — the exact trap "measure first" opened on, one level up the family tree.
Hypothesis: a quadrilateral's opposite sides are not equal.
Claim: therefore, it is not a parallelogram.
This is the contrapositive of "if parallelogram, then opposite sides equal" — negate both halves and flip the order. If opposite sides aren't equal, the shape has already failed the one requirement parallelogram makes, so it can't possibly qualify. The contrapositive always carries the same truth as the original — still the one turn you can rely on.
Fill in what each root means, then check yourself. Screenshot this box when you're done — keep it as your own record.
The carry-forward, restated one more way: square → rectangle → parallelogram → quadrilateral is a one-way street. Every arrow points outward toward more shapes, and none of them point back.
four side · spinning top · beside one another · straight angle — and the family tree from today (parallelogram ⊃ rectangle, rhombus ⊃ square) is the same shape of relationship as converse/inverse/contrapositive from last time: one direction is guaranteed, the reverse direction needs its own proof.