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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,$\angle$COD =$90 \degree$

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 $\triangle$BAD and $\triangle$ABC,

AB=AB               (common)

BD=AC

$\triangle$BAD $\cong$ $\triangle$ABC   (By SSS)

$\angle BAD = \angle ABC$    (CPCT)

$\angle$BAD+$\angle$ABC =$180 \degree$           (Co-interior angles)

2. $\angle$ABC = $180 \degree$

$\angle$ABC =$90 \degree$

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

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