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There are infinite lines of symmetry, as in the case of a circle, the diagonal is the line of symmetry. The diagonal can start at an infinite number of places, and thus there are endless possible lines of symmetry.
There are infinitely many such shapes. To start with, any regular will do.
There would be too many to name. All regular polygons have a line of symmetry, but there are still polygons that don't.
There could be many answers to this. If were thinking in terms of geometry: a triangle has between 1 and 3 lines of symmetry a square will always have 4 lines of symmetry. Irregular shapes might have a single line of symmetry. For instance, a building may have a single line of symmetry. a line between two points may only have one line of symmetry. because it is only 2 dimensions, it is impossible for it to have more than one line of symmetry.
Six - three from diagonal to diagonal, three from midpoint of a side to midpoint to a side.