AMSCO'S Geometry. New York - Rye High School

The Rhombus 395Theorem 10.15If a quadrilateral is equilateral, then it is a rhombus.GivenQuadrilateral ABCD with AB > BC > CD > DADCProveABCD is a rhombus.Proof It is given that in ABCD, AB > BC > CD > DA.Since both pairs of opposite sides are congruent,AABCD is a parallelogram. Two consecutive sides of parallelogramABCD are congruent, so by definition, ABCD is a rhombus.BTheorem 10.16If the diagonals of a parallelogram are perpendicular to each other, the parallelogramis a rhombus.GivenParallelogram ABCD with AC ' BDDCProveStrategyABCD is a rhombus.The diagonals divide parallelogram ABCD into fourtriangles. Prove that two of the triangles that share acommon side are congruent. Then use the fact that correspondingparts of congruent triangles are congruentto show that parallelogram ABCD has two congruentconsecutive sides.AEBThe proof of this theorem is left to the student. (See exercise 17.)SUMMARY To prove that a quadrilateral is a rhombus, prove that any one ofthe following statements is true:1. The quadrilateral is a parallelogram with two congruent consecutive sides.2. The quadrilateral is equilateral.3. The quadrilateral is a parallelogram whose diagonals are perpendicular toeach other.EXAMPLE 1PQRS is a rhombus and mPQR 60. Prove thatdivides the rhombus into two equilateral triangles.PRPQ60°SR