No, they are not.
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a trapezoid has a pair of symmetry lines but a square diamond or parallelogram all sides are parallel
There are no lines of symmetry; However, the lines opposite are parallel to each other
A general trapezoid is a 4 sided quadrilateral with a pair opposite parallel lines of different lengths and no lines of symmetry But an isosceles trapezoid has the same properties as above except that it has 1 line of symmetry
The shape has to be a parallelogram since it has no lines of symmetry and all sides are equal and parallel
A quardrilateral with 1 pair of parallel lines and have no equal sides is a trapezium.