Question

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Answer 1

Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that one of its interior angles is 90º.

In ∆ABC and ∆DCB, 

AB = DC (Opposite sides of a parallelogram are equal) 

BC = BC (Common) 

AC = DB (Given) 

∠∆ABC ∠∆DCB (By SSS Congruence rule) 

⇒ ∠ABC = ∠DCB 

It is known that the sum of the measures of angles on the same side of transversal is 180º. 

⇒ ∠ABC + ∠DCB = 180º (AB || CD) 

⇒ ∠ABC + ∠ABC = 180º 

⇒ 2∠ABC = 180º 

⇒ ∠ABC = 90º

Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.