AMSCO'S Geometry. New York - Rye High School

The Parallelogram 381ADBCQuadrilateral ABCD is a parallelogram because AB CD and BC DA.The symbol for parallelogram ABCD is ~ABCD.Note the use of arrowheads, pointing in the same direction, to show sidesthat are parallel in the figure.Theorem 10.1A diagonal divides a parallelogram into two congruent triangles.GivenParallelogram ABCD with diagonal ACDCProveProofABC CDASince opposite sides of a parallelogram are parallel, alternate interior anglescan be proved congruent using the diagonal as the transversal.ABStatements1. ABCD is a parallelogram. 1. Given.Reasons2. AB CD and BC DA2. A parallelogram is a quadrilateralin which two pairs of oppositesides are parallel.3. BAC DCA and 3. If two parallel lines are cut by aBCA DACtransversal, alternate interiorangles are congruent.4. AC > AC4. Reflexive property of congruence.5. ABC CDA 5. ASA.We have proved that the diagonal AC divides parallelogram ABCD intotwo congruent triangles. An identical proof could be used to show that BDdivides the parallelogram into two congruent triangles, ABD CDB.The following corollaries result from this theorem.Corollary 10.1aOpposite sides of a parallelogram are congruent.Corollary 10.1bOpposite angles of a parallelogram are congruent.The proofs of these corollaries are left to the student. (See exercises 14and 15.)