Yes. Think of a kite shape. Symmetry along its length but no parallel sides.
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The TRAPEZIUM has one pair of opposite sides that are parallel.In America, a trapezium is a convex quadrilateral with no parallel sides. In the United Kingdom and Australia, a trapezium is defined as a convex quadrilateral with one pair of parallel sides (a trapezoid in America).A trapezium always has two parallel sides. Some trapeziums have 1 line of symmetry, these are called isoceles trapeziums
draw a square then tilt one line from each pair of parallel sides away from its pair then adjust the lengths of the top and bottom sides to make it match
A parallelogram is a quadrilateral in which each of two sets of opposite sides consists of two parallel lines (or for math purists I should say, parallel line segments).
You cannot have one parallel line. There must be another line that it is parallel to. Also, 1 symmetry is meaningless - you can have one axis of symmetry or rotational symmetry of order one. The answer will depend on which one it is.
The diagonals of a rectangle aren't lines of symmetry unless it's square.