One of the properties of parallelograms is that any pair of consecutive angles are supplementary. This is a straightforward application of the properties of parallel lines intersected by a transversal line.
Problem
ABCD is a parallelogram. Show that the pairs of consecutive angles are supplementary.
Strategy
The definition of a parallelogram is that both pairs of opposing sides are parallel. So it is a straightforward application of the properties of parallel lines to show that the consecutive angles are supplementary, as we have already proven this for parallel lines here: Consecutive Interior Angles Theorem
A parallelogram consists of two pairs of parallel lines, intersected by transversal lines (the other sides).
Proof
(1) AB||CD //Given, definition of a parallelogram
(2) m∠ABC + m∠DCB = 180° // consecutive interior angels between 2 parallel lines
(3) m∠BAD + m∠CDA = 180° // consecutive interior angels between 2 parallel lines