Theorem 10.10 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 10.10 class 9 Maths which is related to Chapter 10 Circles Class 9 Mathematics. After understanding theorem 10.10, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 Theorem 10.10 Class 9

 

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).

Given 

A, B, C, and D are four points.

AB is a line segment that subtends equal angles at two points C and D. That is 

∠ACB=∠ADB 

To Prove 

A, B, C, and D are concyclic

Proof 

To show that points A, B, C, and D lie on a circle, let us draw a circle through points A, C, and B.

Suppose it does not pass through point D.

Then it will intersect AD at a point, say E. Join EB.

If points A, C, E, and B lie in a circle,

∠ACB=∠AEB (Angles in the same segment)

But is given that  ∠ACB=∠ADB 

Therefore ∠AEB=∠ADB 

This is not possible unless E coincides with D.

Similarly, E^' should also coincide with D.

Thus, our assumption that point D does not lie on the circle was wrong.

Hence A, B, C, and D are concyclic points.

Related Topics

1. Theorem 10.11

2. Theorem 10.9