Question

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

Answer 1

R.E.F image

In $△ABC$ if $AB=AC$

then $\angle ACB=\angle ABC$ ___(1)

In $△OBC,\frac{\angle ACB}{2}=\angle OCB,\frac{\angle ABC}{2}=\angle OBC$

From (1). $\angle OCB=\angle OBC$

$\to OB=OC$

Now In $△AOB$ & $△AOC$

$AO=AO$ (common)

$\angle OBA=\angle OCA(\because \angle OBA=\frac{1}{2}\angle ABC,\angle OCA=\frac{1}{2}\angle ACB)$

$AB=AC$ (given)

$△AOB\cong △AOC$ (SAS criterion)

then $\angle OAB=\angle OAC$

$\therefore$ OA bisects $\angle A$