Theorems on Circles at both the Intermediate and Higher tier are difficult areas for students, which teachers often find equally difficult to deliver.Here, I have explained some Circle theorems based on circle so that student can understand them easily.

1) If two arcs of a circle are congruent, then corresponding chords are equal.

Statements |
Reasons |

1) BC = PQ | 1) Given |

2) OP = AB | 2) Radius of a circle |

3) OQ = AC | 3) Radius of a circle |

4) ΔPOQ = ΔBAC | 4) SSS Postulate |

5) ∠POQ = ∠BAC | 5) CPCTC |

Statements |
Reasons |

1) OP = OQ | 1) Radii of the same circle. |

2) OM = OM | 2) Reflexive (common) |

3) ∠OMP = ∠OMQ | 3) Each 90^{0} |

4) ΔPMO = ΔQMO | 4) HL postulate(RHS) |

5) PM = MQ | 5) CPCTC |

Statements |
Reasons |

1) AB = AC | 1) Given |

2) ∠BAM = ∠CAM | 2) Given |

3) AM = AM | 3) Reflexive (Common) |

4) ΔBAM = ΔCAM | 4) SAS Postulate |

5) BM = CM and ∠BMA = ∠CMA | 5) CPCTC |

6) ∠BMA + ∠CMA = 90^{0} |
6) Linear pair angles |

7) AM = BM and ∠BMA = ∠CMA = 90^{0} |
7) From (6) |

8) AM is the perpendicular bisector of BC | 8) Definition of perpendicular bisector. |

9) AM passes through the center O. | 9) Perpendicular bisector of a chord always passes through the center. |

• Theorems on Chord

• Theorems on Chord and Subtended Angle

• Theorems on Arc and Angle

• Theorems on Cyclic Quadrilateral

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