a right triangle,
a pentagon w/ a right angle,
alot of shapes with right angles.
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A polygon need not have ANY perpendicular or parallel lines. For example, consider an equilateral triangle. It can happen that two sides of a polygon, extended if necessary, meet at a point where they form a 90 degree angle. Those two lines are perpendicular. There may be pairs of lines such that, no matter how far you extend them in either direction, they will never meet. Such lines are parallel. A triangle cannot have parallel lines but it can have perpendicular lines. Any polygon of 4 or more sides can have sides that are perpendicular or parallel (or some of each).
A square and a rectangle will meet the given criteria.
For the perimeter, you just go round the shape, adding the lengths of each side as you go along. There may or may not be simple formulae of the area for a shape with unequal sides. If all you know is the [different] side lengths then only a triangle has a relatively simple formula for its area. If you also know that some of the sides are parallel, then you can find the area of a rectangle. Then, if you also know the perpendicular distance between the parallel sides, you can do a parallelogram. If, in addition you know which sides are parallel, a trapezium. And so on.
A regular pentagon does not have parallel or perpendicular lines. Some other possibilities include, but are not limited to:a circlean ovalan equilateral triangle or any triangle that is not a right triangle.(A right angle's lines are perpendicular. Those are what make it a 90 degree angle.)
A polygon with 4 or more sides can have a pair of parallel sides.