A rhombus or square (if all four angles are also equal).
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Well, honey, that's not a parallelogram, that's a rectangle. A parallelogram can have opposite sides equal in length and opposite angles equal in measure, but not necessarily right angles. So, in this case, you're describing a rectangle, not a parallelogram. But hey, at least you're on the right track!
A rhombus is a parallelogram with 4 equal sides. Many parallelograms do not have 4 equal sides. All parallelograms have 2 pairs of equal opposite sides.
No. A trapezoid is a 4-sided polygon with one pair of parallel sides. A parallelogram with 4 equal sides is a rhombus. (A parallelogram with 4 equal sides and 4 right angles is a square, which is a special case of a rhombus.)
The shape you are looking for is a quadrilateral. If you have a square, all angles MUST be right angles. A parallelogram can also have 4 equal sides in length and parallel sides, but a parallelogram doesn't have right angles.
Both shapes have 4 sides, two pairs equal in length. In a parallelogram the equal length sides are opposite each other. In a kite the equal length sides are adjacent.