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Yes, it is possible for a shape to have at least one pair of parallel lines and no lines of symmetry. For example, an irregular trapezoid has one pair of parallel sides but lacks any lines of symmetry due to its uneven angles and side lengths. This demonstrates that parallel lines and symmetry are independent properties in geometric shapes.
The diagonals of rectangles are rotational lines of symmetry but not reflective. To be reflective lines, folding along the line has to give the same shape on each side.
Six - three from diagonal to diagonal, three from midpoint of a side to midpoint to a side.
It depends on the nature of the irregularities. If you have an octagon where each alternate side is equal then you can have 8 lines of symmetry.
3 Lines of symmetry