You know that each diagonal of a rhombus bisects two angles of the rhombus. Get a better sense of whether this is true for other parallelograms.
ABCD is currently a rhombus. Change [math]\alpha\alpha'[/math] (m<DAB) to make squares and other rhombuses. Pay attention to the measures of the angles cut by the diagonals. Then, change the lengths of the sides to make rectangles and non-specific parallelograms. Is it still true that each diagonal has to bisect two angles of the parallelogram? (For an obvious example, make a short (AD & BC small), long (AB & CD large) parallelogram.)