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Diagonally opposite angles of a rectangle or square is one example.
No. All linear pair angles are supplementary, but supplementary angles do not have to be a linear pair.
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.
Angles that are congruent and supplementary must be right angles.
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.