They may, or may not, be congruent. Congruent means that they have the same measure.
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quadrilaterals are consecutive and opposite angles always congruent?
yes, all angles of a rectangle are 90 degrees and therefore congruent
Not unless the parallelogram is a rectangle. In every parallelogram, consecutive angles are supplementary.
All rectangles, including squares.
There's lots of useful things you can discover when parallel lines are cut by a transversal, most of them having to do with angle relationships. Corresponding angles are congruent, alternate interior angles are congruent, same side or consecutive interior angles are supplementary, alternate exterior angles are congruent, and vertical angles are congruent.