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.