A parallelogram is a quadrilateral because it has 4 sides and all quadrilaterals have 4 sides such as a square, a rectangle, a rhombus ... etc
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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.
A shape that is a parallelogram but not a quadrilateral doesn't exist.
No. A parallelogram is a special kind of quadrilateral.
It's not a matter of 'finding' it. Either the quadrilateral is a parallelogram,or else it is not one.If the opposite sides of the quadrilateral are parallel, then it's a parallelogram.
A quadrilateral is a parallelogram if it has two pairs of (opposite) sides that are parallel.