Question

Show that if two diagonals of a parallelogram are equal, then it is a rectangle.

Answer 1

Answer:
The diagonals of a parallelogram are equal.

To show that a given parallelogram is a rectangle, we have to prove that one of its interior angles is 90° and this can be done by the concept of congruent triangles.

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

In ∆ABC and ∆DCB,

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

BC = BC (Common)

AC = DB (Given the diagonals are equal)

∴ ∆ABC ≅ ∆DCB (By SSS Congruence rule)

⇒ ∠ABC = ∠DCB (By CPCT) ------------- (1)

It is known that the sum of the measures of angles on the same side of transversal is 180° (co - interior angles)

∠ABC + ∠DCB = 180° (AB || CD)

⇒∠ABC + ∠ABC = 180° [From equation(1)]

⇒ 2∠ABC = 180°

⇒∠ABC = 90°

Thus, ∠DCB = 90° [From equation (1)]

Hence, ∠B = ∠D = ∠C = ∠A = 90° [Since opposite angles of a parallelogram are equal].

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