Quadrilaterals, four-sided polygons, form a genuine hierarchy rather than a flat list of unrelated separate shapes, every square is a rectangle, every rectangle is a parallelogram, but the reverse relationship doesn't hold in either direction, not every parallelogram is a rectangle, and not every rectangle is a square. That one-way relationship is defined entirely by which specific properties, equal side lengths, right angles, parallel opposite sides, each successive shape in the hierarchy is required to have on top of the ones before it.
Each shape in the hierarchy adds a specific requirement on top of the one before it
A parallelogram requires only that its opposite sides be parallel, the most general category in this particular hierarchy. A rectangle is a parallelogram that additionally requires right angles at every corner. A rhombus is a parallelogram that additionally requires all four sides be equal in length. A square requires both of those additional properties at once, right angles and equal sides together, which is exactly why a square counts as a rectangle, a rhombus and a parallelogram simultaneously, it's the single shape satisfying every requirement in the whole hierarchy at once. The general polygon angle-sum formula, the interior angles of any polygon with n sides sum to n minus two, times 180 degrees, extends this same systematic logic to shapes with any number of sides at all.
Proving a shape is a square means proving every broader category's properties too
Because the hierarchy is nested, proving a specific shape actually is a square means proving it satisfies every more general category's requirements along the way, that it's a parallelogram, that it's additionally a rectangle, and that it's additionally a rhombus, you genuinely can't skip straight to claiming it's a square without establishing each of those broader properties first. That layered proof structure is exactly what geometry problems asking a student to classify or formally prove a given shape's type are actually testing.
What we're still unsure about
That quadrilaterals form a nested hierarchy defined by cumulative properties is well established, uncontroversial geometry. What's genuinely a persistent, documented classroom misconception is treating rectangle and square as mutually exclusive categories, the way they'd casually be used in everyday English, rather than square correctly being understood as a special, more demanding case of rectangle, and mathematics educators continue to actively debate exactly how much of that specific, persistent misconception comes from ordinary spoken language's looser usage of the two words, versus how much comes from how the shapes are actually taught and pictured in the classroom itself.
This sits inside Polygons & Quadrilaterals, one of eight topics in Geometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.