Theorem 10.11 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 10.11 class 9 Maths which is related to Chapter 10 Circles Class 9 Mathematics. After understanding theorem 10.11, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

 Theorem 10.11 Class 9

The sum of either pair of opposite angles of a cyclic quadrilateral is 180º

Given 

 A cyclic quadrilateral ABCD inscribed in a circle with centre O

To Prove

∠A+∠C=180° 

∠B+∠D=180° 

Construction 

Join OB and OD

Proof 

∠DOB=2∠C …………………..(1)

(Angle subtended by an arc in the centre is double the angle subtended by it at any point on the remaining part of the circle)

Reflex  ∠DOB=2∠A …………………….(2)

From (1) and (2), we get:

2∠C+2∠A=∠DOB+rflex∠DOB 

2(∠C+∠A)=360° (sum of angles at a point)

⇒∠A+ ∠C=180° ……………………(3)

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

⇒180°+∠B+ ∠D=360° 

∠B+ ∠D=180° 

Hence. ∠A+ ∠C=180° and ∠B+∠D=180°  

Related Topics

1. Theorem 10.12

2. Theorem 10.10