Depends how it's cut
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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.
A polygon with 2n sides (n>1) can have opposite sides congruent.
A square's opposite sides are parallel and all the sides are congruent.
Yes. A parallelogram has congruent opposite sides as well as congruent opposite angles.
A rhombus.