Theorem 10.9 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 10.9 class 9 Maths which is related to Chapter 10 Circles Class 9 Mathematics. After understanding theorem 10.9, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 

 Theorem 10.9 Class 9

Angles in the same segment of a circle are equal.

Given 

A circle with centre O, in which ∠ACB and∠ADB are two angles subtended by AB in the same segment ACDBA of the circle.

To Prove

∠ACB=∠ADB 

Construction 

Join OA and OB

Proof 

We know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Case – 1 

AB is a minor arc

∠AOB=2∠ACB and ∠AOB=2∠ADB 

⇒2∠ACB=2∠ADB 

⇒∠ACB=∠ADB 

Case- 2

AB is a semicircle

We know that angle in a semicircle is a right angle

∠ACB=90° and ∠ADB=90° 

So,         ∠ACB=∠ADB 

Case-3 

AB is a major arc:

Here reflex   ∠AOB=2∠ACB

And reflex  ∠AOB=2∠ADB

⇒2∠ACB=2∠ADB 

∴∠ACB=∠ADB 

Hence, in all three cases   ∠ACB=∠ADB.

 Related Topics

1. Theorem 10.10

2. Theorem 10.8