Yes
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a rhombus
Parallelogram, trapezoid, rhombus and a kite are some of them Note that a square and a rectangle have diagonals of equal lengths
Quadilateral.
Yes.
A Rhombus or a paralellogram or what is known as a "diamond-shape", like on a playing card. These would fit the criteria given - They are all names for a quadrilateral with four equal sides with non-equal angles.