Question

In the given figure, if ∠AOC=130°, then the value of ∠ABC

Answer 1

Arc ABC subtends ∠AOC at center and ∠APC on the remaining part of the circle (at the point P).
Since the angle subtended by an arc at the center is double the angle subtended by it on any remaining
point of the circle, we have
$\Rightarrow$ ∠APC = $\frac{1}{2}$∠AOC.
$\Rightarrow$ ∠APC = $\frac{1}{2}$(130°)= 65°
Since APCB is a cyclic quadrilateral, we have its opposite angles to be supplementary.
$\Rightarrow$ ∠ABC + ∠APC = 180°
$\Rightarrow$ ∠ABC + 65° = 180°
$\Rightarrow$ ∠ABC = 180° - 65° =115°