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Proofs for circle theorems Tuesday, 13 July 2010.

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Presentation on theme: "Proofs for circle theorems Tuesday, 13 July 2010."— Presentation transcript:

1 Proofs for circle theorems Tuesday, 13 July 2010

2 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)  

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

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

5 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 

6 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 22  360 –2  180 – 

7 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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