Cyclic Quadrilateral


 
 
Concept Explanation
 

Cyclic Quadrilateral

CYCLIC QUADRILATERAL

DEFINITION  A quadrilateral is called a cyclic quadrilateral if its all vertices lie on a circle.

A cyclic quadrilateral has some special properties which other quadrilaterals, in general, need not have . We shall state and prove these properties as theorems.

Theorem 1 The sum of either pair of opposite angles of a cyclic quadrilaterals is large 180^{circ}

                                 OR

The opposite angles of a cyclic quadrilateral are supplementary.

Given   A cyclic quadrilateral ABCD.

TO PROVE  large angle A + angle C = 180^{circ} and angle B + angle D = 180^{circ}

CONSTRUCTION   Join AC and BD.

PROOF  Consider side AB of quadrilateral ABCD as the chord of the circle. Clearly, large angle ACB ;; and;; angle ADB are angles in the same segment determined by chord AB of the circle.

large therefore angle ACB = angle ADB         [ since angles in the same segment are equal] ....(i)

Now, consider the side BC of quadrilateral ABCD as the chord of the circle. We find that large angle BAC ;; and ;;angle BDC    are angles in the same segment

large therefore angle BAC = angle BDC      [large because Angles in the same segment are equal ] ....(ii)

Adding equations (i) and (ii) , we get

large Rightarrow ;;;angle ACB + angle BAC = angle ADB + angle BDC

large Rightarrow ;;; angle ACB + angle BAC = angle ADC

large Rightarrow ;;; angle ABC + angle ACB + angle BAC = angle ABC + angle ADC

large Rightarrow ;;; 180^{circ}= angle ABC + angle ADC      [large because The sum of the angle of a triangle is large 180^{circ}]

large Rightarrow ;;; angle ABC + angle ADC = 180^{circ}

large Rightarrow ;;; angle B + angle D = 180^{circ}

large But, ;;; angle A + angle B + angle C + angle D = 360^{circ}

large therefore ;;;angle A + angle C = 360^{circ}-(angle B +angle D)

large Rightarrow ;;;angle A + angle C = 360^{circ} -180^{circ}=180^{circ}

Hence,   large angle A + angle C = 180^{circ};;and;;angle B + angle D = 180^{circ}

The converse of this theorem is also true as given below.

Converse Theorem: If the sum of any pair of opposite angles of a quadriletral is 180^0, then the quadrilateral is cyclic.

Given: A quadrilateral ABCD in which angle B + angle D = 180^0

Tp Prove: ABCD is a cyclic quadrilateral

Construction: Draw a circle passing through A, B and C.

Proof: Suppose the circle meets CD or CD produced at E. Join AE

Now ABCE is a cyclic Quadrilateral.

As Opposite angles of a cyclic quadrilateral are supplementary

Rightarrow angle B + angle E = 180^0     .........(1)    

But, angle B + angle D = 180^0    ........(2)           [Given]

From Equation (1) and (2) we get

angle D = angle E

But this is not possible angle E is an exterior angle to the Delta ADE in acse (1) or angle D is an exterior angle to the Delta ADE in case (2)

And we know that the exterior angle of a triangle is equal to sum of interior opposite angles

That is exterior angle is always greater than interior opposite angle

Thus our assumption is wrong

Hence ABCD is a cyclic quadrilateral.

 

Case (1)

Case (2)

Illustration: In the figure ACDF is a cyclic quadrilateral. A circle passing through A and F meets AC and DF in the points B and E respectively. Prove that BE || CD.

Solution: ACDF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary

angle ACD + angle AFD = 180^0       .....................(1)

Also ABEF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary

 angle ABE + angle AFE = 180^0     ..................(2)

angle AFE ;is; same; as ; angle AFD

Rightarrow;; angle ABE + angle AFD = 180^0   ..............(3)

From Equation 1 and 3 we get

Rightarrow;; angle ABE = angle ACD

But they are corresponding angles when BE and CD are two straight lines and BC is the transversal

As they are equal, So BE || CD

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

From the figure below, identify the correct relation

Right Option : B
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Explanation
Question : 2

Identify the correct relationship from the figure below

Right Option : D
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Explanation
 
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