Question

If diagonals of paralleloram are equal and intersect at right angle then parallelogram is a square prove

Answer 1

Consider a parallelogram ABCD
In ABCD the diagonals are AC and BD both intersect at right angle at O. So In triangle AOB and BOC
AO=OC ( Diagonals are equal so half are also equal)
Angle AOB=Angle BOC (90°)
OB=OB(Common)
So by SAS Triangle AOB=BOC so AB=BC
Similarly DOC and AOD Are also congruent so AD=DC
SO IF, AD=DC and AB=BC
SO We can conclude AD=AB also
So now in triangle ACB and ADC
AD=BC
Angle ADC=ABC(cpct)
AB=DC
SO IF ABC AND ADC ARE CONGRUENT so We can conclude that Angle A and B are equal and D and C are equal
So If A=B
So A+B=180
A+A=180 (Because both are equal)
2A=180
A=180÷2
A=90
SO, B is also 90
So If C=D
So C+D=180
C+C=180 (Because both are equal)
2C=180
C=180÷2
C=90
SO D is also 90
So all angle of square are 90° hence proved it is a square

Answer 2

I have proved this theorem.


Just have look here.