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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

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Hello sir i want to know the best solution of the question from exercise 8.1of math of Quadrilaterals chapter of class 9th give me the best and easy for solving this question how i solve it of question no. 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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1. Given that,

Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at right angle at O.

To prove that,

The Quadrilateral ABCD is a square.

Proof,

In Î”AOB and Î”COD,

AO = CO (Diagonals bisect each other)

âˆ AOB = âˆ COD (Vertically opposite)

OB = OD (Diagonals bisect each other)

, Î”AOB â‰… Î”COD [SAS congruency]

Thus,

AB = CD [CPCT] â€” (i)

also,

âˆ OAB = âˆ OCD (Alternate interior angles)

â‡’ AB || CD

Now,

In Î”AOD and Î”COD,

AO = CO (Diagonals bisect each other)

âˆ AOD = âˆ COD (Vertically opposite)

OD = OD (Common)

, Î”AOD â‰… Î”COD [SAS congruency]

Thus,

AD = CD [CPCT] â€” (ii)

also,

â‡’ AD = BC = CD = AB â€” (ii)

also,Â  âˆ ADC = âˆ BCDÂ  [CPCT]

and âˆ ADC+âˆ BCD = 180Â° (co-interior angles)