In this section we will discuss theorems on cyclic quadrilateral.

A quadrilateral is called

It has some special properties which other quadrilaterals, in general, need not have. Here we have proved some theorems on cyclic quadrilateral.

1) The opposite angles of a Cyclic - quadrilateral are supplementary.

Statements |
Reasons |

1) ∠ACB = ∠ADB | 1) Angles in the same segment. |

2) ∠BAC = ∠BDC | 2) Angles in the same segment |

3)∠ACB + ∠BAC = ∠ADB + ∠BDC | 3) Addition property |

4) ∠ACB + ∠BAC = ∠ADC | 4) Add ∠ABC on both sides. |

5) ∠ABC + ∠ACB + ∠BAC = ∠ABC + ∠ADC | 5) From Above. |

6) 180^{o} = ∠ABC + ∠ADC |
6) Sum of the angle of a triangle is 180^{o} |

7) ∠B + ∠D = 180^{o} |
7) Opposite angles of cyclic quadrilateral. |

8) ∠A + ∠B + ∠C + ∠D | 8) Measure of a quadrilateral. |

9) ∠A + ∠C = 360^{o} - (∠B + ∠D) |
9) From Above. |

10) ∠A + ∠C = 360^{o} - 180^{o} = 180^{o} |
10) Angle sum property |

11) ∠A + ∠C = 180^{o} and ∠B + ∠D = 180^{o} |
11) From above .So opposite angles are supplementary. |

Statements |
Reasons |

1) ∠ABC + ∠ADC = 180^{o} |
1) Opposite angles of cyclic quadrilateral |

2) ∠ABC + ∠CBE = 180^{o} |
2) Linear Pair angles. |

3) ∠ABC + ∠ADC = ∠ABC + ∠CBE | 3) From above. |

4) ∠ADC = ∠CBE | 4) subtraction property |

• Theorems on Chord

• Theorems on Chord and Subtended Angle

• Theorems on Arc and Angle

• Theorems on Cyclic Quadrilateral

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