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The Arc and Angle means the angle is subtended in the given arc.

1) The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.

Statements |
Reasons |

1) ∠POM = ∠OPM + ∠ORP | 1) By exterior angle theorem (∠POM is exterior angle) |

2) OP = OR | 2) Radii of same circle |

3) ∠OPR = ∠ORP | 3) In a Δ two sides are equal then the angle opposite to them are also equal. |

4) ∠POM = ∠ORP + ∠ORP | 4) Substitution property. From (1) |

5) ∠POM = 2∠ORP | 5) Addition property. |

Similarly, when ∠QOM is an exterior angle then ∠QOM = 2∠ORQ

∴ from above, ∠POQ = 2∠PRQ

Statements |
Reasons |

1) ∠POM = ∠OPR + ∠ORP | 1) Exterior angle theorem. |

2) ∠POM = ∠ORP + ∠ORP | 2) As,OP = OR = radius. ∴∠ORQ = ∠ORP |

3) ∠POM = 2∠ORP | 3) Substitution and addition property. |

4) ∠QOM = ∠ORQ + ∠OQR | 4) Exterior angle theorem. |

5) ∠QOM = ∠ORQ + ∠ORQ | 5) As, OQ =OR = radius, ∴ ∠ORQ = ∠OQR |

6) ∠QOM = 2∠ORQ | 6) Substitution and addition property. |

∴ From above, ∠POQ = 2∠PRQ

Statements |
Reasons |

1) ∠POM = ∠OPR + ∠ ORP | 1) By exterior angle theorem. |

2) ∠POM = 2∠ORP | 2) As, OP = OR = radius, ∴∠OPR = ∠ORP and by addition property |

3) ∠ QOM = ∠ORQ + ∠OQR | 3) By exterior angle theorem in ΔQOR |

4) ∠QOM = 2∠ORQ | 4) As, OP = OR = radius, ∴∠ORQ = ∠OQR and by addition property |

5) ∠POM + ∠QOM = 2(∠ORP + ∠ORQ) | 5) From (2) and (4) |

6) Reflex ∠POQ = 2∠PRQ | 6) Reflex of ∠POQ =∠POM + ∠QOM |

Statements |
Reasons |

1) ∠POQ = 2∠PRQ | 1) Angle subtended by an arc of a circle at its center is twice the angle formed by the same arc. |

2) 180 ^{0} = 2∠PRQ |
2) As POQ is a straight line |

3) ∴ ∠PRQ = 90 ^{0}
| 3) Division property |

• Theorems on Chord

• Theorems on Chord and Subtended Angle

• Theorems on Arc and Angle

• Theorems on Cyclic Quadrilateral

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