a parallelogram
/ par-uh-LEL-uh-gram /
Take a rectangle drawn on stretchy paper and shove the top edge sideways, keeping the top and bottom parallel and the same length; you get a slanted, leaning four-sided figure — a parallelogram. A leaning bookshelf brace, the lozenge pattern on a fence, and the shape a square box casts as a tilted shadow are parallelograms.
Precisely, a parallelogram is a quadrilateral with both pairs of opposite sides parallel. From that single definition a whole bundle of properties follows as theorems (not extra assumptions): opposite sides are equal in length; opposite angles are equal; consecutive angles are supplementary (they add to 180 degrees, since they are co-interior angles on parallel lines); and the diagonals bisect each other (each diagonal cuts the other into two equal halves at their crossing point). Each of these is also a test working backwards — for instance, if a quadrilateral's diagonals bisect each other, it must be a parallelogram; likewise if one pair of sides is both parallel and equal.
The parallelogram is the hub of the quadrilateral family: rectangles, rhombi, and squares are all special parallelograms, and the trapezoid sits just below it (one pair of parallel sides instead of two). The honest point to stress is the definition-versus-theorem distinction: 'opposite sides equal', 'diagonals bisect each other', and the rest are consequences proved from 'both pairs of opposite sides parallel', not part of the definition. Confusing a derived property for the definition is a common error; any one of several equivalent properties could serve as the definition, but in a given course only one is taken as primary and the others are theorems.
In parallelogram ABCD, if angle A = 70 degrees then angle C = 70 degrees (opposite) and angles B and D each equal 110 degrees (supplementary to A). The diagonals AC and BD cross at a point M with AM = MC and BM = MD.
Opposite angles equal, consecutive angles supplementary, diagonals bisecting.
'Diagonals bisect each other' and 'opposite sides equal' are theorems proved from the definition (both pairs of opposite sides parallel), not part of it — a frequent definition-versus-theorem confusion.