Parallelograms: 2-fold
Square: 4-fold
n-fold symmetries refer to rotational symmetries. Consequently, any symmetries about axes that these and other quadrilaterals may have are not relevant to this question.
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Not all 4 sided quadrilaterals have lines of symmetry.
Not all 4 sided quadrilaterals have lines of symmetry although some of them do have lines of symmetry.
No.
Rhombus
A square
Not all 4 sided quadrilaterals have lines of symmetry.
square
Not all 4 sided quadrilaterals have lines of symmetry although some of them do have lines of symmetry.
Parallelograms
Squares.
No.
A square
Types of quadrilaterals that have lines of symmetry: squares, rectangles, some rhombuses (standard diamond shape), boomerang shape, and trapezoids.
Rhombus
A square
yes
trapezoid for one