### 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;

$\angle \mathrm{ADC}=\mathrm{115\xb0}$.

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### Solution

The reflex angle AOC is equal to $115\times 2=230\xb0$, 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 $\mathrm{360\xb0}-\mathrm{130\xb0}-2\times \mathrm{90\xb0}=\mathrm{50\xb0}$

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