# Interior Angle Bisector Theorem

**Interior Angle Bisector Theorem :**The angle bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

**Given :**A ΔABC in which AD is the internal bisector of ∠A and meets BC in D.

**Prove that :**BD / DC = AB / AC

**Construction :**Draw CE || DA to meet BA produced in E.

Statements |
Reasons |

1) CE || DA | 1) By construction |

2) ∠2 = ∠3 | 2) Alternate interior angles |

3) ∠1 = ∠4 | 3) Corresponding angles |

4) AD is the bisector | 4) Given |

5) ∠1 =∠2 | 5) Definition of angle bisector |

6) ∠3= ∠4 | 6) From (2) and (3) |

7) AE = AC | 7) In ΔACE, side opposite to equal angles are equal |

8) BD / DC = BA / AE | 8) In ΔBCE DA || CE and by BPT theorem |

9) BD / DC = AB / AC | 9) From (7) |

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**Examples**

1) In the given figure, AD is the bisector of ∠A. If BD = 4 cm,

DC = 3 cm and AB = 6 cm, find AC.

**Solution :**

In Δ ABC, AD is the bisector of ∠A.

∴ BD / DC = AB / AC ( Angle bisector theorem)

⇒ 4 / 3 = 6 / AC

⇒ 4 AC = 18

⇒ AC = 18 / 4 = 4.5 cm.

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2) AD is a median of ΔABC. The bisector of ∠ADB and ∠ADC meet AB and AC in E and F respectively.

**Given :**In ΔABC, AD is the median and DE and DF are angle bisectors of ∠ADB and ∠ADC respectively.

**Prove that :**EF || BC

Statements |
Reasons |

1) DE is the angle bisector of ∠ADB | 1) Given |

2) ∴ AD / DB = AE / EB | 2) By interior angle bisector theorem |

3) DF is the angle bisector of ∠ADC | 3) Given |

4) AD / DC = AF / FC | 4) By angle bisector theorem |

5) AD is the median | 5) Given |

6) BD = DC | 6) By definition of median |

7) AD / DB = AF / FC | 7) From (6) |

8) AE / EB = AF / FC | 8) From (2) and (7) |

**Hence EF || BC**

**Similarity in Triangles**

• Similarity in Geometry

• Properties of similar triangles

• Basic Proportionality Theorem(Thales theorem)

• Converse of Basic Proportionality Theorem

• Interior Angle Bisector Theorem

• Exterior Angle Bisector Theorem

• Proofs on Basic Proportionality

• Criteria of Similarity of Triangles

• Geometric Mean of Similar Triangles

• Areas of Two Similar Triangles

• Similarity in Geometry

• Properties of similar triangles

• Basic Proportionality Theorem(Thales theorem)

• Converse of Basic Proportionality Theorem

• Interior Angle Bisector Theorem

• Exterior Angle Bisector Theorem

• Proofs on Basic Proportionality

• Criteria of Similarity of Triangles

• Geometric Mean of Similar Triangles

• Areas of Two Similar Triangles