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in Quadrilaterals by (65.2k points)

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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Data : ABCD is a quadrilateral and bisect each other at right angles, then it is a square. 

AO = OC and AC = BD 

BO = OD 

∠AOB = ∠BOC = ∠COD + ∠DOA = 90°. 

To Prove: ABCD is a square. 

Proof: In ∆AOB and ∆COD, 

AO = OC BO = OD (Data) 

∠AOB = ∠COD (Vertically opposite angles) 

∴ ∆AOB ≅ ∆COD (SAS Postulate) 

AB = CD …………. (i) 

∠ABO = ∠CDO 

∴ AB || CD ………… (ii) 

From (i) and (ii) ABCD is a parallelogram. 

Now, in ∆AOD and ∆COD, 

AO = OC (Data) 

∠AOD = ∠COD = 90° (Data) 

OD is common 

∴ ∆AOD ≅ ∆COD (SAS Postulate) 

AD = CD …………. (iii) 

AD = BC ………….. (iv) 

From (ii), (iii) and (iv) 

AB = BC = CD = AD 

Four sides of a quadrilateral is they are equal each other and bisect each other at right angles, then it is a square. 

∴ ABCD is a square.

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