Question

In the given figure, chord $AB$ and chord $BC$ are equal. The measure of $\angle ABO$ is:

• A. $\angle ABO={45}^o$
• B. $\angle ABO={30}^o$
• C. $\angle ABO={55}^o$
• D. $\angle ABO={40}^o$

Answer 1

Answer: (A)

$\angle ABO={45}^o$

$O$ is the centre of the circle and
$OA=OC=OB$ radius of the circle.
$\angle ABC$ is the angle at the circumference subtented by daimeter AC.
$\therefore \angle ABC={90}^o$
Now,
In $△ABO$ and $△CBO$
$AB=BC$(Given)
$AO=CO$(radius)
$BO=BO$(common side)
$\therefore △ABO\cong △CBO$(SSS congruence)
$\therefore \angle ABO=\angle CBO$
$\Rightarrow \angle ABO+\angle CBO={90}^o$
$\Rightarrow 2\angle ABO={90}^o$
$\Rightarrow \angle ABO=\frac{90}{2}$
$\Rightarrow \angle ABO={45}^o$