Angles in the Same Segment are Equal


 
 
Concept Explanation
 

Angles in the Same Segment are Equal

Theorem 1 Angles in the same segment of a circle are equal.

Given    A circle C(O,r), an arc PQ and two angles large angle PRQ;;and;;angle PSQ    in the same segment of the circle.

To prove large angle PRQ = angle PSQ.

Construction Join OP and OQ.

PROOF We know that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point in the remaining part  of the circle . So, in figure, we have

         large angle POQ = 2angle PRQ ;;and;;angle POQ = 2angle PSQ

large Rightarrow               large 2angle PRQ = 2angle PSQ

large Rightarrow                 large angle PRQ = angle PSQ

Hence proved.

Converse Theorem: If a line segmentjoining 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 the circle.

Given: A line segment AB and two points C and D such that angle ACB = angle ADB

To Prove: The points A, B, C and D lie on a circle or the points are concyclic

Construction: Draw a circle Which passes through A, B C

Proof  As the circle passes through A, B and C, let us assume that it does not pass through D.

Then either the circle will intersect AD at F or will cut the line segment AD extended to E

Case I: THe circle intersects AD at F . Then A,C,F,B lie on the circle

Rightarrow angle ACB = angle AFB               ....... 1  [ Angle in the same segment are equal]

but Rightarrow angle ACB = angle ADB         ........2  [ Given]

From Equation 1 and 2 we get

angle AFB = angle ADB

This is not possible because angle AFB is an exterior angle to Delta DBF and we know that exterior angle of a triangle can not be equal to interior opposite angle. Hence our Assumption is wrong. 

Therfore D and F should coincide

Case 2: The circle cuts the line segment AD extended to E Then A,C,E,B lie on the circle

Rightarrow angle ACB = angle AEB               ....... 3  [ Angle in the same segment are equal]

but Rightarrow angle ACB = angle ADB         ........4  [ Given]

From Equation 3 and 4 we get

angle AEB = angle ADB

This is not possible because angle ADB is an exterior angle to Delta DBE and we know that exterior angle of a triangle can not be equal to interior opposite angle. Hence our Assumption is wrong. 

Therfore D and E should coincide

Hence the circle has to pass through A, B, C and C or These points are concyclic

  Illustration: If in the figure  angle ABC=65^o,;then ;find ; angle CDA

  Solution: In the figure we have a circle with center O and AC is the arc

  The arc AC subtends angle at point B and D on the circle

   Now large angle ABC ;and ; angle CDA are in the same segment because they are on the same side of the arc

  Therefore  large angle ABC ;= ; angle CDA                                     [Angles in the same segment are equal]          

   As large angle ABC=65^o,;then ; angle CDA= 65^0        

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Sample Questions
(More Questions for each concept available in Login)
Question : 1

 

AB is chord of a circle with centre O and C and D lie on the same side of AB.If  angle ACB=60^o, ; then ; angle ADB;and; angle AOB are respectively equal to 

Right Option : C
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Explanation
Question : 2

In the figure, if  angle AOB=120^o, ; then ; angle ADB+angle ACB is equal to

 

Right Option : A
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Explanation
Question : 3

Find  angle AOB, if in the figure given above, angle AEB+angle ACB+angle ADB=3x.

 

Right Option : D
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Explanation
 
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