Any polygon with more than 4 sides.
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A quadrilateral has two and only two diagonals. That's the maximum and also the minimum. If it has more or fewer than two diagonals, it's not a quadrilateral.
Most shapes will have more than two sides. If the sides are straight lines, the shapes MUST have more than two sides.
Hexagon
A decagon and any polygon above a pentagon has more than two lines of symmetry
Yes. For example, dodecagons, squares and triangles.