Yes, a regular hexagon has ALL sides the same length and ALL angles equal. If you flatten it somewhat (like the profile of a lens) then the sides may still have the same length but some angles will be larger than others if opposite angles are the same than there will be symmetry. (there are many other options to retain symmetry in a hexagon this is but one)
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an irregular hexagon... __/\__ \ / \ / \ /
an irregular hexagon? yes, any polygon with 6 sides
A hexagon has 6 lines of symmetry.
A regular hexagon has six lines of symmetry.
if it's a regular hexagon it has 6 lines of symmetry