No, only in certain, limited circumstances. Eg where a quadrilateral is (can be) circumscribed within a circle.
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Wouldn't think so, think of a trapezium shaped like a square with a triangle on one side... The angles at one end of the figure are both 90o, and the angles at the other end will be supplementary, but not opposite angles.
no * * * * * But they can be.
No. Opposite angles are equal. Adjacent angles are supplementary.
Yes.
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.