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Angles in the Same Segment are Equal

Circles III - Concepts

Angle at the Centre is Double The Angle at Circumference

Angles in the Same Segment are Equal

Angle in a Semicircle

Cyclic Quadrilateral

Class - 9th IMO Subjects

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

â€‹â€‹

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
A. $120^o;60^o$		B. $60^o,;60^o$
C. $60^o,;120^o$		D. $120^o,;120^o$
Right Option : C
View Explanation

Question : 2

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

$120^o$

$240^o$

$60^o$

None of these

Right Option : A

View Explanation

Question : 3

If AB is a chord of a circle, P and Q are the points on the circle different from A and B, then
A. $angle APB=angle AQB$		B. $angle APB+angle AQB=180^o ; or ; angle APB=angle AQB$
C. $angle APB+angle AQB=90^o$		D. $angle APB+angle AQB=180^o$
Right Option : B
View Explanation

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Language - English

Chapters

Polynomials II

Direct and Inverse Variation

Linear Equations in One Variable

Introduction to Trigonometry

Introduction to Euclids Geometry

Herons Formula Complete

Polynomial I

Surface Area and Volume II

Surface Area and Volume I

Coordinate Geometry

Constructions

Areas Of Parallelograms And Triangles II

Areas Of Parallelograms And Triangles I

Linear Equations in Two Variables

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Angles in the Same Segment are Equal

Angles in the Same Segment are Equal

Students / Parents Reviews [10]

Cheshta

Barkha Arora

Ayan Ghosh

Shreya Shrivastava

Hiya Gupta

Aman Kumar Shrivastava

Shivam Rana

Manish Kumar

Yatharthi Sharma

Archit Segal