Parallelogram Parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The
Cyclic quadrilateral vertices , , , and sides , , , and , the lengths of the diagonals and can be expressed in terms of the sides as so showing Ptolemy's theorem According to "Ptolemy's second theorem", using the same notations as above. For the sum of the diagonals we have the inequality Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality. Moreover, In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.