Theorem 8.8 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 8.8 class 9 Maths which is related to Chapter 8 Quadrilaterals Class 9 Mathematics. After understanding theorem 8.8, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

 Theorem 8.8 Class 9

A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Given 

 A quadrilateral ABCD in which AB=CD and AB∥CD

To Prove

ABCD is a parallelogram.

Construction

Join AC

Proof  

In ∆s ABC and CDA, we have

AB=DC                   (Given)

AC=AC                  (common side)

 And ∠BAC=∠DCA (∵AB∥CD alternate interior angles are equal)

So, by SAS criterion of congruence, we have 

∆ABC≅∆CDA 

⟹ ∠BCA= ∠DAC        (CPCT) 

This line AC intersects AB and DC at A and C respectively such that

∠DAC= ∠BCA 

Alternate interior angles are equal.

AD∥BC 

Thus AB∥CD and   AD∥BC 

Hence , quadrilateral ABCD is a parallelogram.

Hence proved