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
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.
∠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 )
2) ABCD is a cyclic-quadrilateral. AB and DC are produced to meet in E. Prove that ΔEBC ~ ΔEDA.
ABCD is a cyclic-quadrilateral.
Prove that :
ΔEBC ~ ΔEDA.
| 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
• Parts of Circle
• Arc and Chords
• Equal Chords of a Circle
• Arc and Angles
• Cyclic Quadrilaterals
• Tangent to Circle