yes
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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.
Six - three from diagonal to diagonal, three from midpoint of a side to midpoint to a side.
No because diagonal lines do not count. Even though they might fit perfectly inside the rectangle, but those don't count. Only lines going horizontally or vertically count as lines of symmetry. :)
How many diagonal lines does a 1million sided shape have
A parallelogram need have no lines of symmetry.