yes
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All regular polygons do.
almost all even number of sided polygons, regular has to be
There would be too many to name. All regular polygons have a line of symmetry, but there are still polygons that don't.
Yes.
Rectangles, circles, and all regular polygons with even numbers of sides have.A pedant adds:Not strictly true because a line of symmetry is an infinite line, it is not just the segment that is within the shape. And an infinite line cannot be bisected.