A rectangle, for example.
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Yes, a regular polygon is always equiangular.
Regular polygon is equilateral and equiangular. Irregular polygon is non-equilateral and non-equiangular.
It is a regular polygon
Yes, by definition. If it not equilateral AND equiangular, then it is not regular.
The term regular polygon is almost always used to refer to a polygon that has equal interior angles and equal side lengths. In that light, the equiangular polygon and a regular polygon are the same thing.