Question

prove that the diagonals of a parallelogram divides the parallelogram into two congruent triangles.

Answer 1

Let there be parallelogram ABCD and diagonal AC bisecting it.

Now according to the properties of a parallelogram,  AB ll CD, AC ll BD & AB=CD, AD=BC. As the diagonal is a common side for both triangles. Hence using the SSS property, we can  say that both the triangles are congruent.

Answer 2

Statement : A diagonal of a parallelogram divides it into two congruent triangles.

Given : A parallelogram ABCD.

To prove : ΔBAC ≅ ΔDCA

Construction : Draw a diagonal AC.

Proof :

In ΔBAC and ΔDCA,

∠1 = ∠2 [alternate interior angles]
∠3 = ∠4 [alternate interior angles]
AC = AC [common]

ΔBAC ≅ ΔDCA [ASA]

Hence, it is proved.