Theorem 10.12 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 10.12 class 9 Maths which is related to Chapter 10 Circles Class 9 Mathematics. After understanding theorem 10.12, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

 Theorem 10.12 Class 9

If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

Given 

 A quadrilateral ABCD and ∠A+∠C=180°

To Prove 

ABCD is a cyclic quadrilateral.

Construction 

If ABCD is not a cyclic quadrilateral. Let a circle passing through the point D, A, and B meet BC at E 

or BC produced at E. so that DABE is a cyclic quadrilateral.

Proof 

As ABED is a cyclic quadrilateral, the sum of opposite angles is 180°.

∴∠A+∠E=180° but ∠A+∠C=180° 

⟹∠A+∠E=∠A+∠C⇒∠E=∠C 

But ∠E≠∠C because the exterior angle of a triangle cannot be equal to the interior opposite angle.

∴our supposition is wrong. 

Hence, ABCD is a cyclic quadrilateral.

Related Topics

1. Theorem 10.11

2. Theorem 10.1