Question

is an isosceles triangle ABC, with AB = AC the bisectors of angle B and angle C intersect each other at O. join A to O. show that (i) OB = OC (ii) AO bisects angle A

Answer 1

Please check this answer

Answer 2

$\bf{Given}$:
ΔABD is a isosceles triangle
AB = AC
CO bisects $\angle{C}$
BO bisects $\angle{B}$

(i) To be shown,
OB = OC

as, CO is the bisector of $\angle{C}$
$\angle{ACO}$ = $\angle{OCB}$

and as, BO is the bisector of $\angle{B}$
$\angle{ABO}$ = $\angle{OBC}$

$\therefore$ $\angle{OCB}$ = $\angle{OBC}$

$\therefore$ OC = OB
[ if two opposite angles are equal so opposite side are equal ]

(ii) To be shown,
ΔAOC = ΔAOB

AC = AB (given)
AO = AO (common side)
OC = OB { proved in (i) }
$\angle{OCA}$ = $\angle{ABO}$

$\therefore$ by $\bf{SAS}$ congruence condition
ΔAOC $\cong$ ΔAOB