Question

in ∆ABC,ab=ac and the bisector of <b and<c meet at at a point O . prove that bo =co and the ray ao is the bisector of <A​

Answer 1

Answer:

It is given that AB=AC and the bisectors ∠B and ∠C meet at a point O

Consider △BOC

So we get

∠BOC=

2

1

∠B and ∠OCB=

2

1

∠C

It is given that AB=AC so we get ∠B=∠C

So we get

∠OBC=∠OCB

We know that if the base angles are equal even the sides are equal

So we get OB=OC.(1)

∠B and ∠C has the bisectors OB and OC so we get

∠ABO=

2

1

∠B and ∠ACO=

2

1

∠C

So we get

∠ABO=∠ACO..(2)

Considering △ABO and △ACO and equation (1) and (2)

It is given that AB=AC

By SAS congruence criterion

△ABO≅△ACO

∠BAO=∠CAO(c.p.c.t)

Therefore, it is proved that BO=CO and the ray AO is the bisector of ∠A.