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. 3) If two non-parallel sides of trapezoid ( trapezium ) are equal, it is cyclic. 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

⇒ ∠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

Circles

Circles
Parts of Circle
Arc and Chords
Equal Chords of a Circle
Arc and Angles