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Proofs for circle theorems Tuesday, 13 July 2010

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1. Angle subtended at the centre A B C The angle subtended at the centre from an arc is double the angle at the circumference. x x y y 180 – 2x 180 – 2y 2x+2y = 2(x+y)

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2. Angles subtended from the same arc. A B C Angles subtended from the same arc are equal.

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3. Angles in a semi-circle. A B C 90 The largest angle in a semi-circle will always be 90

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4. The Angle between a Tangent and its radius. A Tangent C 90 Definition: A tangent is a line that will touch the circle at one point only. (i.e. it does not cut the circle) Definition: A tangent is a line that will touch the circle at one point only. (i.e. it does not cut the circle) The angle between a tangent an its radius will always be 90

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5. Angles in a cyclic quadrilateral. Definition: A cyclic quadrilateral is any four-sided polygon whose four corners touch the circumference of the circle Definition: A cyclic quadrilateral is any four-sided polygon whose four corners touch the circumference of the circle Opposite angles in a cyclic quadrilateral add up to 180 a b c d 22 360 –2 180 –

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4. The Angle between a Tangent and a chord. Tangent Definition: A chord is any straight line which touches the circumference at two points. The largest chord possible is called the diameter. The angle between a tangent a chord is equal to the angle in the alternate segment. Chord 90 90 – =180 – 90 – (90 – ) =

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