A Square
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No, both shapes have all four sides equal.
A rhombus is the only shape besides a square that can have equal sides.
An infinite number. A polygon with any number of sides can be equilateral (ie have sides of equal length).
No. Parallelograms have only 4 sides with 2 sets of parallel sides: they include squares, rectangles, and rhombuses (rhombi, diamond shapes). Because opposite angles are equal, each pair of parallel sides is equal in length. (For squares and "equilateral rhombi" all four sides are equal in length.)
They can have some (or all) equal angles. A rectangle, for example, has all four angles equal but, because its sides are not the same measure, it is not a regular shape.