AMSCO'S Geometry. New York - Rye High School

The Square 401EXAMPLE 2In square PQRS, SQ is a diagonal.SRFind mPSQ.SolutionA square is a rhombus, and the diagonal of a rhombusbisects its angles. Therefore, the diagonal SQbisects PSR. Since a square is a rectangle, PSRis a right angle and mPSR 90. Therefore,1mPSQ 2 (90) 45.PQExercisesWriting About Mathematics1. Ava said that the diagonals of a square separate the square into four congruent isoscelesright triangles. Do you agree with Ava? Justify your answer.2. Raphael said that a square could be defined as a quadrilateral that is both equiangular andequilateral. Do you agree with Raphael? Justify your answer.Developing SkillsIn 3–6, the diagonals of square ABCD intersect at M.3. Prove that AC is the perpendicular bisector of BD.CD4. If AC 3x 2 and BD 7x 10, find AC, BD, AM, BM.5. If AB a b, BC 2a, and CD 3b 5, find AB, BC, CD,and DA.M6. If mAMD x 2y and mABC 2x y, find the values ofx and y.BAIn 7–11, tell whether each conclusion follows from the given premises. If not, draw a counterexample.7. If a quadrilateral is a square, then all sides are congruent.If all sides of a quadrilateral are congruent then it is a rhombus.Conclusion: If a quadrilateral is a square, then it is a rhombus.8. A diagonal of a parallelogram separates the parallelogram into two congruent triangles.A square is a parallelogram.Conclusion: A diagonal of a square separates the square into two congruent triangles.9. If a quadrilateral is a square, then all angles are right angles.If a quadrilateral is a square, then it is a rhombus.Conclusion: In a rhombus, all angles are right angles.