AMSCO'S Geometry. New York - Rye High School

396 QuadrilateralsSPProof Since all sides of a rhombus are congruent, we know that PQ > RQ. Thus,PQR is isosceles and its base angles are equal in measure.Let mPRQ mRPQ x.RQ60°x x 60 1802x 60 1802x 120x 60Therefore, mPRQ 60, mRPQ 60, and mPQR 60. TrianglePQR is equilateral since an equiangular triangle is equilateral.Since opposite angles of a rhombus have equal measures, mRSP 60.By similar reasoning, RSP is equilateral.EXAMPLE 2Given: ABCD is a parallelogram.AB 2x 1, DC 3x 11, AD x 13.Prove: ABCD is a rhombus.DCABProof(1) Since ABCD is a parallelogram,opposite sides are equal in length:(2) Substitute x 12 to find thelengths of sides AB and AD:(3) Since ABCD is a parallelogramwith two congruent consecutivesides, ABCD is a rhombus.DC AB3x 11 2x 1x 12AB 2x 1 AD x 13 2(12) 1 12 13 25 25ExercisesWriting About Mathematics1. Rochelle said that the diagonals of a rhombus separate the rhombus into four congruentright triangles. Do you agree with Rochelle? Explain why or why not.