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The side of a triangle or parallelogram used to help find area is called the base. Area is base times height in a parallelogram. In a triangle, it is 1/2 times base times height.
No, a parallelogram is not a trapezoid.
A quadrilateral, in general, is not a parallelogram. If it is a parallelogram then you will have some additional information about its sides and angles. If you do not have such information it is not possible to prove that it is a parallelogram. Draw a diagonal which will divide the quadrilateral into two triangles and use the additional information that you have to show that the triangles are congruent. This can then be used to show equality of sides or of angles: the latter can then be used to show that sides are parallel. Note that the choice of which diagonal may influence how (if at all) you proceed.
Base x Height
Square = parallelogram and a square trapezoid = trapezoid Parallelogram = Parallelogram