Can a non-degenerate polygon with all sides equal have unequal angles?

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I have always been hearing that a regular polygon is a polygon with equal sides and equal angles, but I never considered the fact that it may be possible for a polygon with all sides equal but unequal angles. Is this even possible? A lucid example would be greatly appreciated.

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Yes, of course. Consider a rhombus, or a home plate type shape (an equilateral pentagon with unequal angles), or take six toothpicks and attach them end-to-end in a hexagon and observe that you can change the angles freely.