A rectangle is defined to be an "equiangular parallelogram".
So a rectangle is any four-sided polygon, having two pairs of parallel opposite sides, and four angles which are equal in measure. With this definition, we must still "prove" that each angle measures 90 degrees.
The second rectangle conjecture is more interesting, and says that the diagonals each have the same length.
Conjecture (Rectangle Conjecture I ): The measure of each angle in a rectangle is 90 degrees.
Proof: This follows directly from the Quadrilateral Sum Conjecture which says that the sum of angles in any convex quadrilateral is equal to 360 degrees. Since a rectangle has four angles of equal measure, the measure of each must be 360/4, or 90 degrees.
Conjecture (Rectangle Conjecture II ): The diagonals of a rectangle are equal in length.
Proof: This follows from the Side/Angle/Side congruence for triangles. Triangle ABD is congruent to triangle DCA, and therefore, BD is congruent to CA.