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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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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=D

     

 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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