Depends on the quadrilateral. Eg in a cyclic Quad, opposite angles are supplementary. Note that it is impossible that there would be only one pair of supp. angles as the internal angles total 360o
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All supplementary angles do not form a linear pair. The opposite angles of any quadrilateral inscribed in a circle (a cyclic quadrilateral) are supplementary but they are not a linear pair. However, all linear pair are supplementary.
A parallelogram.
Not necessarily. A linear pair of angles must be supplementary but supplementary angles need not form a linear pair. For example, the opposite angles of a cyclic quadrilateral are supplementary but they are (by definition) not next to one another.
Sometimes but not always depending on what type the 4 sided quadrilateral is.
Supplementary angles are any angles in which their degrees add to a sum of 180o. In the related links you will find an example of Supplementary angles.