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A rhombus is always equilateral. It need not be equiangular, but if it is, then most people look at it and call it a "square".
Yes
Square
The rhombus has adjacent angles being the same, and all sides are equal. However, it is not necessarily equiangular.
A regular polygon must be equiangular as well as equilateral. A rhombus is an example of a polygon that is equilateral but not equiangular.