Theorem 8.5 Class 9 Maths Explanation with Proof

 Today, we will discuss theorem 8.5 class 9 Maths which is related to Chapter 8 Quadrilaterals Class 9 Mathematics. After understanding theorem 8.5, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

 Theorem 8.5 Class 9

If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

Given 

A quadrilateral ABCD in which ∠A=∠C and ∠B=∠D.

To Prove

ABCD is a parallelogram.

Proof 

⇒  ∠A+∠B=∠C+∠D ……………(i)

We know that the sum of all the angles of a quadrilateral is 360°.

∠A+∠B+ ∠C+∠D=360° 

∠A+∠B+∠A+∠B=360°  from (i)

2(∠A+∠B)=360° 

∠A+∠B=(360°)/2=180° 

⟹ ∠A+∠B=180°= ∠C+∠D from (i)

Now the line segments AD and BC are cut by a transversal AB such that

 ∠A+∠B=180 

So,   AD∥BC  (∵∠A and ∠B are co interior angles)

We can also write ∠A+∠D=∠C+∠B

    ∠A+∠B+ ∠C+∠D=360° (sum of all the angles of a quadrilateral)

∠A+∠A+ ∠D+∠D=360° 

2(∠A+∠D)=360° 

∠A+∠D=180° 

But these are co-interior angels. Hence AB∥DC

As   AB∥DC and AD∥BC 

Hence ABCD is a parallelogram.

Hence proved.

Related Topics

1. Theorem 8.4

2. Theorem 8.6