Question

In 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 $A$

Answer 1

(i) In $\Delta ABC$, we have
$AB=AC$
$\Rightarrow \angle C=\angle B$ [ Since angles opposite to equal sides are equal ]
$\Rightarrow \frac{1}{2}\angle B=\frac{1}{2}\angle C$
$\Rightarrow \angle OBC=\angle OCB$
$\Rightarrow \angle ABO=\angle ACO$ …(1)
$\Rightarrow OB=OC$ ∣ Since sides opp. to equal $\angle s$ are equal … (2)
(ii) Now, in $\Delta ABO$ and $\Delta ACO$, we have
$AB=AC$ [Given]
$\angle ABO=\angle ACO$ [From (1)]
$OB=OC$ [From (2)]
$\therefore$ By SAS criterion of congruence, we have
$\Delta ABO\cong \Delta ACO$
$\Rightarrow \angle BAO=\angle CAO$ [Since corresponding parts of congruent triangles are equal]
$\Rightarrow AO$ bisects $\angle A$