Loading....
Coupon Accepted Successfully!

 

Theorem 6 - Angles in the Same Segment of a circle are Equal

Theorem 6 : Angles in the same segment of a circle are equal.
Given : Two angles ACB and ADB are in the same segment of a circle C(O, r).
To Prove :
ACB = ADB
Construction : Join OA and OB.

 


Proof : We know that, angle subtended by an arc of a circle at the centre is double the angle subtended by the arc in the alternate segment.
Hence,
AOB = 2 ACB
          
AOB = 2 ADB
So,      
ACB = ADB
In Fig. (ii), we have,
Reflex
AOB = 2ACB
and Reflex
AOB = 2ADB
2ACB = 2 ADB
⇒ ∠ACB = ADB. 
 

Example

In the Fig. P is a point on the chord BC such that AB = AP. Prove that CP = CQ.

Solution :

AB = AP [given]
ABP =APB [Angles opposite to equal sides of a triangle are equal]
In
ΔABP and ΔAQC, ABP = AQC [Angle in the same segment of a circle]
In
ΔABP and ΔCPQ, APB = CPQ [Vertically opposite angles are equal]
...CPQ = AQC
... CQ = PC [Sides opposite equal angles are equal]
or CP = CQ


 





Test Your Skills Now!
Take a Quiz now
Reviewer Name