# arc theroem .Please explain it

**Given :**Arc PQ of a circle C(O, r) and point r on the remaining part of the circle.

**Prove that:**∠ POQ = 2∠PRQ

**Construction:**join RO and produce it to point M outside the circle.

**Case 1:**When arc PQ is a minor arc (figure (i))

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

**Case 2:**When arc PQ is a semicircle.[figure(ii)]

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. |

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