The rhombus has adjacent angles being the same, and all sides are equal. However, it is not necessarily equiangular.
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A rectangle
A rectangle comes to mind. A rectangle is equiangular (all angles the same) and is also a quadrilateral (four-sided), but is not equilateral (all sides of the same length).
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, all the angles of an equilateral triangle are always 60 degrees.
A regular polygon must be equiangular as well as equilateral. A rhombus is an example of a polygon that is equilateral but not equiangular.