Question

In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O.
Prove that : (i) BO = CO
                   (ii) AO bisects angle BAC.

Answer 1

In ΔABC,
AB = AC
⇒ ∠B = ∠C ...( angles opposite to equal sides are equal )

⇒ `1/2 ∠"B" = 1/2∠"C"`

⇒ ∠OBC = ∠OCB       ...[ ∵ OB and OC are bisectors of ∠B and ∠C respectively, ∠OBC = `1/2∠"B" and ∠"OCB" = 1/2∠"C"` ] ...(i)

⇒ OB = OC              ...( Sides opposite to equal angles are equal )  ...(ii)

Now, in ΔABO and ΔACO,
AB = AC                 ...( given )
∠OBC = ∠OCB       ...[ from(i) ]
OB = OC                ...[ from(ii) ] ...( proved )
∴ ΔABO ≅ ΔACO   ...( by SAS congruence criterion )
⇒ ∠BAO = ∠CAO   ...( c.p.c.t. )
⇒  AO bisects ∠BAC  ...(proved)