Quadrilateral JKLM is given as a parallelogram. By a theorem, opposite sides JK and LM are congruent (1). By the definition of parallelogram, opposite sides KJ and ML are parallel. By the theorem on alternate interior angles, angles KJL and MLJ are congruent (2). Segments JN and PL are given as congruent (3). Using the three statements of congruence labeled above (1), (2), and (3), we now prove that triangles JKN and LMP are congruent by SAS. Sides of the triangles KN and PM are congruent by CPCTC. Sides of quadrilateral KNMP are given as parallel. Therefore, quadrilateral KNMP is a parallelogram by the theorem: If two sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
