# In ∆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.

Given: In ∆ABC, AB=AC and the bisectors of B and C meet at a point O.

To prove: BO=CO and BAO = CAO

Proof:

In , ∆ABC we have,

OBC = B

OCB = C

But B = C … given

So, OBC = OCB

Since the base angles are equal, sides are equal

OC = OB …(1)

Since OB and OC are bisectors of angles B and C respectively, we have

ABO = B

ACO = C

∴∠ABO = ACO …(2)

Now in ∆ABO and ∆ACO

AB = AC … given

ABO = ACO … from 2

BO = OC … from 1

Thus by SAS property of congruence,

∆ABO ∆ACO

Hence, we know that, corresponding parts of the congruent triangles are equal

BAO = CAO

ie. AO bisects A

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
NCERT | Triangle Inequalities47 mins
Know More About Triangle Inequalities28 mins
Champ Quiz | Area of Triangle53 mins
NCERT | Imp. Qs. on Area of Parallelogram and Triangles43 mins
Quiz | Angle Sum Property of Triangle40 mins
Champ Quiz | Mastering Triangles44 mins
RD Sharma | Important Proofs in Triangles28 mins
Triangles - Important Questions39 mins
Quiz | Area and Parallelogram46 mins
Quiz | Area of Parallelogram and Triangles43 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses