Circle Theorems

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Find ∠ABC

A B C D O 115°

A, C and D are point on a circle centred O;
AB and CB are tangents to the circle;
AD and CD are chords in the circle;



A B C D O 115°

The reflex angle AOC is equal to 115×2=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, ∠ABC is 360°-130°-2×90°=50°


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