Yes
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A parallelogram, other than a rhombus or rectangle.
Trapezoid
Oh, dude, a quadrilateral can only have a maximum of 4 lines of symmetry because it has four sides. It's like trying to fit five people in a four-seater car - not gonna work, my friend. So, to answer your question, no, a quadrilateral cannot have more than 4 lines of symmetry.
A parallelogram.
A regular hexagon has more than 4 lines of symmetry. Even number sided polygons that are regular and have more than 4 sides have more than 4 lines of symmetry. Circles have more than four lines of symmetry. Squares also have 4 lines of symmetry.