# 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 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.

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

• Cyclic Quadrilaterals

• Tangent to Circle

• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle