A diamond is actually an informal term for a rhombus, which has four equal sides(congruent is not used to refer to equal sides), with the opposite sides being parallel to one another.
So a "diamond" has four equal sides and two pairs of parallel sides.
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A square is a four-sided polygon with two congruent parallel sides.
A four-sided polygon with opposite sides parallel and all sides congruent is a square.
Rhombus
Four right angles; opposite sides are parallel; opposite sides are congruent; diagonals are congruent.
A square and a rhombus