Cyclic Quadrilaterals

A quadrilateral is called Cyclic Quadrilaterals if its all vertices lie on a circle.

It has some special properties which other quadrilaterals, in general, need not have. We shall state and prove these properties as theorems. They are as follows :

1) The sum of either pair of opposite angles of a cyclic- quadrilateral is 180⁰

OR
The opposite angles of cyclic quadrilateral are supplementary.

∠A + ∠C = 180⁰ and ∠B + ∠D = 180⁰

Converse of the above theorem is also true.

If the opposite angles are supplementary then the quadrilateral is a cyclic-quadrilateral.
2) If one side of a cyclic-quadrilateral is produced, then the exterior angle is equal to the interior opposite angle.

ABCD is a cyclic-quadrilateral then ∠CBE = ∠ADC
3) If two non-parallel sides of trapezoid ( trapezium ) are equal, it is cyclic.
If AD = CB then the trapezoid ABCD is a cyclic-quadrilateral.
Some solved examples on the above results

1) ABCD is a cyclic-quadrilateral; O is the center of the circle. If ∠BOD = 160⁰, find the measure of ∠BPD and ∠BCD.

Solution :

∠BOD = 160⁰ ( given and it’s a central angle )

∴ ∠BAD = ½ ∠BOD = 80⁰

As ABCD is a cyclic-quadrilateral,

⇒ ∠BAD + ∠BPD = 180⁰

80 + ∠BPD = 180

⇒ ∠BPD = 100⁰

⇒ ∠BCD = 100⁰ ( Angles in the same segment )
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2) ABCD is a cyclic-quadrilateral. AB and DC are produced to meet in E. Prove that ΔEBC ~ ΔEDA.
Solution :
Given : ABCD is a cyclic-quadrilateral.

Prove that : ΔEBC ~ ΔEDA.

Statements	Reasons
1) ABCD is cyclic-quadrilateral.	1) Given
2) ∠EBC = ∠EDA	2) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.
3) ∠ECB = ∠EAD	3) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.
4) ∠E = ∠E	4) Reflexive (common )
5) ΔEBC ~ ΔEDA	5) AAA Postulate

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Circles

• Circles
• Parts of Circle
• Arc and Chords
• Equal Chords of a Circle
• Arc and Angles
• Cyclic Quadrilaterals
• Tangent to Circle

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