Show that if daigonals of parralelogram bisect each at right angles ,

1.	Show that if daigonals of parralelogram bisect each at right angles , then it is a rhombus
Answer» Given: Diagonals of quadrilateral intersect each other at right angles.To Prove: Quadrilateral is a rhombus.Proof : In {tex}\\triangle{/tex}AOB and {tex}\\triangle{/tex}AOD,AO = AO . . . [Common]OB = OD . . . [Given]{tex}\\angle{/tex}AOB = {tex}\\angle{/tex}AOD . . .[Each 90o]{tex}\\therefore{/tex}\xa0{tex}\\angle{/tex}AOB\xa0{tex}\\cong{/tex}\xa0{tex}\\triangle{/tex}AOD . . . [By SAS property]{tex}\\therefore{/tex}\xa0AB = AD . . . [c.p.c.t.] . . . . (1)Similarly, we can prove thatAB = BC . . . . (2)BC = CD . . . . (3)CD = AD . . . . (4)From (1), (2), (3) and (4)AB = BC = CD = DASince opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that ABCD is a rhombus.

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