Any polygon that's equilateral must be equiangular.
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Yes. Any polygon with more than 3 sides can be equiangular without being equilateral or equilateral without being equiangular.
No, an equilateral triangle has to be equiangular, but an equiangular triangle does NOT have to be equilateral
A rectangle is not always equilateral, although it is always equiangular. A rhombus is not always equiangular, but is always equilateral. A square is both equilateral and equiangular.
Every equilateral triangle is equiangular, and every equiangular triangle is equilateral.
Yes, by definition. If it not equilateral AND equiangular, then it is not regular.