Home
Class 9
MATHS
In A B C ,\ A B=A C , and the bisectors...

In ` A B C ,\ A B=A C ,` and the bisectors of angles `B\ a n d\ C` intersect at point `Odot` Prove that `B O=C O\ ` and the ray `A O` is the bisector of angle `B A C`
Since the angles opposite to equal sides are equal,
`AB=AC`
`=>/_C=/_B`
`=>2/_B=2/_C`
Since BO and CO are bisectors of `/_B and /_C`, we also have
`/_ABO=2/_B`
`/_ACO= 2/_C`
`/_ABO=2/_B=2/_C ​=/_ACO.`
Consider `/_\ BCO:/_OBC=/_OCB`
`BO=CO`(Sides opposite to equal angles are equal)
Finally, consider triangles ABO and ACO.
`BA=CA`(given)
`BO=CO`(proved)
`/_ABO=/_ACO` (proved)
Hence, by S.A.S postulate`/_\ABO~=/_\ACO`
`/_BAO=/_CAO`
`AO bisects /_A.`

Text Solution

Verified by Experts


Since the angles opposite to equal sides are equal,
`AB=AC`
`=>/_C=/_B`
`=>2/_B=2/_C`
Since BO and CO are bisectors of `/_B and /_C`, we also have
`/_ABO=2/_B`
`/_ACO= 2/_C`
`/_ABO=2/_B=2/_C ​=/_ACO.`
Consider `/_\ BCO:/_OBC=/_OCB`
`BO=CO`(Sides opposite to equal angles are equal)
Finally, consider triangles ABO and ACO.
`BA=CA`(given)
`BO=CO`(proved)
`/_ABO=/_ACO` (proved)
Hence, by S.A.S postulate`/_\ABO~=/_\ACO`
`/_BAO=/_CAO`
`AO bisects /_A.`
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    RD SHARMA|Exercise All Questions|306 Videos

  • CONSTRUCTION

    RD SHARMA|Exercise All Questions|29 Videos

Similar Questions

Explore conceptually related problems

In o+ABC,AB=AC, and the bisectors of angles B and C intersect at point O. Prove that BO=CO and the ray AO is the bisector of angles BAC.

In Delta ABC,AB=AC and the bisectors of /_B and /_C meet at a point O. Prove that BO=CO and the ray AO is the bisector of /_A.

In A B C ,/_B=2/_C and the bisector of /_B intersects A C at Ddot Prove that (B D)/(D A)=(B C)/(B A)

In a ABC, it is given that AB=AC and the bisectors of /_B and C intersect at O. If M is a point on BO produced,prove that /_MOC=/_ABC

In a quadrilateral A B C D , bisectors of angles A\ a n d\ B intersect at O such that /_A O B=75^0, then write the value of /_C+\ /_D

In a triangle ABC, the bisector of angle A meets the opposite side at D.Using vectors prove that BD:DC=c:b.

In a hat harr ABC, it is given that AB=AC and the bisectors of /_B and /_C intersect at O, If M is a point on BO produced,prove that /_MOC=/_ABC.

If the bisectors of two adjacent angles A\ a n d\ B of a quadrilateral A B C D intersect at a point O such that /_C+/_D=k/_A O B , then find the value of k

RD SHARMA-CONGRUENT TRIANGLE -All Questions
  1. In Figure, it is given that /A=/C\ a n d\ A B=B Cdot Prove that A B...

    Text Solution

    |

  2. A D\ a n d\ B C are equal perpendiculars to a line segment A B . Sho...

    Text Solution

    |

  3. In A B C ,\ A B=A C , and the bisectors of angles B\ a n d\ C inter...

    Text Solution

    |

  4. In Figure, it is given that A B=E F ,\ B C=D E ,\ A B\ |B D\ a n d\ ...

    Text Solution

    |

  5. In Figure, it is given that A B=B C\ a n d\ A D=E Cdot Prove that a...

    Text Solution

    |

  6. In Figure, l||m and M is the mid-point of the line segment A B . Pro...

    Text Solution

    |

  7. In Figure, line l is the bisector of angle A\ a n d\ B is any point ...

    Text Solution

    |

  8. In Figure, A D is a median and B L ,\ C M are perpendiculars drawn f...

    Text Solution

    |

  9. In Figure, B M\ a n d\ D N are both perpendiculars to the segments A...

    Text Solution

    |

  10. In a right angled triangle, one acute angle is double the other. Pro...

    Text Solution

    |

  11. A triangle A B C is an isosceles triangle if any one of the followin...

    Text Solution

    |

  12. A triangle A B C is an isosceles triangle if any one of the followi...

    Text Solution

    |

  13. In Figure, A P\ a n d\ B Q are perpendiculars to the line segment A ...

    Text Solution

    |

  14. In Figure, A B C is an isosceles triangle with A B=A C ,\ B D\ a n d...

    Text Solution

    |

  15. In Figure, A D=A E\ a n d\ D\ a n d\ E are points on B C such that B D...

    Text Solution

    |

  16. In Figure, if A B=A C\ a n d\ B E=C D , prove that A D=A Edot

    Text Solution

    |

  17. In Figure, P S=P R ,\ \ /T P S=\ /Q P Rdot Prove that P T=P Q

    Text Solution

    |

  18. In Figure, If P Q=P T\ a n d\ /T P S=/Q P R , prove that \ P R S is is...

    Text Solution

    |

  19. In Figure, A B C\ a n d\ D B C are two isosceles triangles on the sa...

    Text Solution

    |

  20. In Figure, A B C\ a n d\ D B C are two triangles on the same base ...

    Text Solution

    |