Question

ABCD is a cyclic quadrilateral in a circle with centre O. Prove that angle A = angle C=180°

Answer 1

 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 180

 0 

OR 
The opposite angles of cyclic quadrilateral are supplementary.
 
∠A + ∠C = 180 0 and ∠B + ∠D = 180 0 

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 = 160 0 , find the measure of ∠BPD and ∠BCD.

Solution : 
 
∠BOD = 160 0 ( given and it’s a central angle )

∴ ∠BAD = ½ ∠BOD = 80 0 

As ABCD is a cyclic-quadrilateral, 

⇒ ∠BAD + ∠BPD = 180 0 

80 + ∠BPD = 180

⇒ ∠BPD = 100 0 

⇒ ∠BCD = 100 0 ( 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.

 
StatementsReasons

1) ABCD is cyclic-quadrilateral.1) Given2) ∠EBC = ∠EDA2) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.3) ∠ECB = ∠EAD3) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.4) ∠E = ∠E4) Reflexive (common )5) ΔEBC ~ ΔEDA5) AAA Postulate