This is a problem that came up during my geometry class, and, though logical, neither me or my teacher could come up with why.
Consider quadrilateral ABCD, listing clockwise, with A in the top left, depending on how you look at it.
Angle DAB is a right angle. Angle CDA is a right angle. Segment AD is congruent to segment BC.
Due to the fact that two lines perpendicular to a third are parallel, Segment AB is || to Segment CD.
From my thinking, since the opposite sides are parallel, and the other pair is congruent, and there are two consecutive right angles, the only possible way Segment BC could connect points B and C while being congruent to segment AD is if it meets segments AB and CD at right angles.
Is there a theorem that proves this figure is a parallelogram, or is my thinking simply a postulate? Is it possible to disprove my thinking?