Circle Theorems

Find $\mathrm{\angle ABC}$

$A$, $C$ and $D$ are point on a circle centred $O$;
$\mathrm{AB}$ and $\mathrm{CB}$ are tangents to the circle;
$\mathrm{AD}$ and $\mathrm{CD}$ are chords in the circle;
$\angle \mathrm{ADC}=\mathrm{115°}$.

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Solution

The reflex angle AOC is equal to $115\times 2=230°$, because the angle subtended by an arc at the centre is twice the angle subtended at the circumference.

So the obtuse angle AOC is then 130° as angles around a point sum to 360°.

The quadrilateral ABCO has internal angles at A and C of 90°, because tangents and radii meet at 90°.

So, $\mathrm{\angle ABC}$ is $\mathrm{360°}-\mathrm{130°}-2\times \mathrm{90°}=\mathrm{50°}$

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