# Angle Side Angle Postulate

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**Angle side angle postulate (ASA)**- > If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent by angle-side-angle postulate.

∠B ≅ ∠E, BC ≅ EF and ∠C ≅ ∠F

∴ ΔABC ≅ Δ DEF by ASA

**Examples :**

1)

**Given :**∠BAC = ∠DAC and ∠BCA = ∠DCA

**Prove that :**AB = AD and CB = CD.

Statements |
Reasons |

1) ∠BAC = ∠DAC | 1) Given |

2) AC = AC | 2) Reflexive |

3) ∠BCA = ∠DCA | 3) Given |

4) ΔBAC ≅ ΔDAC | 4) By ASA (angle side angle postulate) |

5) AB = AD | 5) CPCTC |

6) CB = CD | 6) CPCTC |

2)

**Given :**∠BCD = ∠ADC and ∠ACB = ∠BDA

**Prove that :**AD = BC and ∠A = ∠B

Statements |
Reasons |

1) ∠BCD = ∠ADC | 1) Given |

2) ∠ACB = ∠BDA | 2) Given |

3) ∠BCD + ∠ACB = ∠ADC + ∠BDA | 3) Adding (1) and (2) |

4) ∠ACD = ∠BDC | 4) Addition property |

5) CD = CD | 5) Reflexive |

6) ΔACD ≅ ΔBDC | 6) By ASA postulate |

7) AD = BC | 7) CPCTC |

8) ∠A = ∠B | 8) CPCTC |

3)

**Given :**AC = BC , ∠DCA = ∠ECB and ∠DBC = ∠EAC

**Prove that :**i) ΔDBC ≅ ΔEAC

(ii) DC = EC and (iii) BD = AE

Statements |
Reasons |

1) ∠DCA = ∠ECB | 1) Given |

2) ∠DCA + ∠ECD = ∠ECB + ∠ECD | 2) Adding angle ∠ECD both sides in (1) |

3) ∠ECA = ∠DCB | 3) Addition property |

4) BC = AC | 4) Given |

5) ∠DBC = ∠EAC | 5) Given |

6) ΔDBC ≅ ΔEAC | 6) By ASA postulate |

7) DC = EC | 7) CPCTC |

8) BD = AE | 8) CPCTC |

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**Side Angle Side Postulate**

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**Side Side Side Postulate**

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**Angle Angle Side Postulate**

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**Angle Side Angle Postulate**

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**HL postulate(Hypotenuse – Leg OR RHS)**

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