A rectangle. It could be a square too but a square is just a special type of rectangle.
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Yes. A parallelogram is effectively a rectangle that doesn't have right angles.
A parallelogram with no right angles and four congruent sides is a Rhombus.
No, a quadrilateral with congruent diagonals but no right angles is not necessarily a parallelogram. In order for a quadrilateral to be classified as a parallelogram, it must have both pairs of opposite sides parallel. The property of having congruent diagonals does not guarantee that the sides are parallel, so the quadrilateral may not be a parallelogram.
A square has 4 congruent sides and 4 right angles, in addition to having all of the properties of a parallelogram. A kite is not a parallelogram. It has two pairs of consecutive congruent sides, and a pair of congruent opposite angles.
A square is the parallelogram that has all sides congruent. A square also has four angles that are all right angles.