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Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Q: 5  Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

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M mansi

Given : ABCD is a quadrilateral with  AC=BD,AO=CO,BO=DO,\angleCOD =90 \degree

 To prove: ABCD is a square.

Proof: Since the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a rhombus.

           Thus, AB=BC=CD=DA

           

               In \triangleBAD and \triangleABC,   

                AD=BC              (proved above )

               AB=AB               (common)

               BD=AC

             \triangleBAD \cong \triangleABC   (By SSS)

           \angle BAD = \angle ABC    (CPCT)

            \angleBAD+\angleABC =180 \degree           (Co-interior angles)

                   2. \angleABC = 180 \degree

                         \angleABC =90 \degree 

Hence,  the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

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