Data : ABCD is a parallelogram and diagonals AC and BD bisects at right angles at O’.
To Prove: ABCD is a rhombus.
Proof: Here, AC and BD bisect each other at right angles.
∴ AO = OC
BO = OD
and ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°
If sides are euqal to each other, then ABCD is said to be a rhombus.
Now, ∆AOD and ∆COD, AO = OC (Data)
∠AOD = ∠COD = 90° (Data) OD is common.
∴ ∆AOD ≅ ∆COD (SAS Postulate)
∴ AD = CD …………… (i)
Similarly, ∆AOD = ∆AOB
AD = AB ………… (ii)
∆AOB ≅ ∆COB
∴ AB = BC ……….. (iii)
∆COB ≅ ∆COD
∴ BC = CD ……………. (iv)
From (i), (ii), (iii) and (iv),
AB = BC = CD = AD
All 4 sides of parallelogram ABCD are equal, then it is rhombus.
