four sides, four right angles, 2 pairs of parallel lines
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They are both two dimensional. They only have right angles at line intersections. Opposing sides are parallel. Opposing sides are the same size.
A square may be classified as a rectangle, a parallelogram, a rhombus, a polygon, and a quadrilateral.
1x5 2x4 3x3 a square, but a square IS a special type of rectangle, Also 4x2 and 5x1 which are the first two rotated through 90 degrees. Whether these should be considered the same or different depends on the context of the question.
Shapes can be classified in many ways. These ways can be general or specific. Of the terms you mentioned, the most general is the quadrilateral. Parallelograms, squares, rectangles and rhombuses are all quadrilaterals (they have 4 sides) but quadrilaterals are not necessarily any of the others. The next classification is the parallelogram, which adds the qualification that opposite sides be parallel. Squares, rectangles and rhombuses are parallelograms, but a parallelogram does not have to be a square, rectangle or rhombus.
A rectangle is technically a parallelogram and it belongs to the class of polygons known as quadrilaterals