Question

In the given figure, $E$ is any point in the interior of the circle with centre $O$. Chord $AB=AC$. If $\angle OBE=20°$, then find the value of $x$.

Answer 1

Given that, the chords $AB$ and $AC$ are equal.
And we know that the equal chords subtend equal angles at the centre.
i.e., $\angle AOB=\angle AOC$
Also, $BOC$ is a straight line,
$\therefore \angle AOB+\angle AOC=180°$
$\Rightarrow \angle AOC=\frac{180°}{2}$
$\Rightarrow \angle AOC=90°$
$\angle BOE$ is vertically opposite to $\angle AOC$,
So, $\angle BOE=90°$
In $\Delta BOE$, using the angle sum property of all the internal angles of a triangle, we get:
$\angle OBE+\angle BEO+\angle BOE=180°$
$\Rightarrow 20°+x+90°=180°$
$\Rightarrow x=180°-20°-90°$
$\Rightarrow x=70°$