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Linking Angles

Linking Angles. Visualising Angle Relationships in Circles. B. P. O. C. Two circles centred at O and P intersect at B and C. T. B. P. O. C. A. The tangent at B to the circle centred P meets the circle centred O at A. T. B. D. P. O. C. A.

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Linking Angles

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  1. Linking Angles Visualising Angle Relationships in Circles

  2. B P O C Two circles centred at O and P intersect at B and C.

  3. T B P O C A The tangent at B to the circle centred P meets the circle centred O at A.

  4. T B D P O C A The line AC meets the circle centred at P at D.

  5. T E B D P O C A DB meets the circle centred at O again at E.

  6. T E B D P O C A DB meets the circle centred at O again at E. It is often easier to see relationships if the common chord is added to the diagram. In this case, joining AE is also helpful.

  7. T E B D P O C A Show that AEB = ABE.

  8. T E B D P O C A Proof:Let AEB = x Introducing a variable will make it easier to trace the path of the angle relationships through the diagram. x

  9. T E B D P O C A Now AEBC is a cyclic quadrilateral x

  10. T E B D P O C A BCD = x(exterior angle of cyclic quadrilateral AEBC) x x

  11. T E B D P O C A Now BCD lies in a segment of the circle centre P. x x

  12. T E B D P O C A TBD = x(angle in the alternate segment) x x x

  13. T E B D P O C A EBA = x (vertically opposite) x x x x

  14. T E B D P O C A AEB = ABE. x x

  15. Explore this relationship further using this GeoGebra file

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