Prove that a diagonal of a parallelogram divides it into two congruent triangles.

Solution:

Consider a parallelogram ABCD

Join the diagonal AC of the parallelogram

We have to prove that a diagonal of a parallelogram divides it into two congruent triangles.

The diagonal AC divides the parallelogram ABCD into two triangles ABC and ADC.

Considering triangles ABC and ADC,

We know that the opposite sides of a parallelogram are parallel and congruent.

So, AD = BC

AD||BC

We know that the alternate interior angles are equal.

∠BAC = ∠DCA

∠BCA = ∠DAC

Common side = AC

We observe that one side and two angles made on this side are equal.

By ASA criteria, the triangles ABC and ADC are similar.

Therefore, △ABC ≅ △ADC.

✦ Try This: In ΔABC, AB=BC, AD⊥BC, CE⊥AB. Prove that AD=CE

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8

NCERT Exemplar Class 9 Maths Exercise 8.4 Sample Problem 2

Prove that a diagonal of a parallelogram divides it into two congruent triangles.

Summary:

It is proven that a diagonal of a parallelogram divides it into two congruent triangles by ASA criterion which states that two triangles are congruent, if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle

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