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This resource explores angle relationships formed by two intersecting circles centered at O and P, specifically at points B and C. It discusses the tangent at B to the circle at P, and the intersections involving line AC and DB. By examining cyclic quadrilaterals and various angle properties, it provides an interactive approach to understanding these relationships. A common chord is suggested to enhance clarity in the diagram. The resource culminates in demonstrating that angle AEB equals angle ABE through geometric proof, inviting further exploration with a GeoGebra file.
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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 The line AC meets the circle centred at P at D.
T E B D P O C A DB meets the circle centred at O again at E.
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.
T E B D P O C A Show that AEB = ABE.
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
T E B D P O C A Now AEBC is a cyclic quadrilateral x
T E B D P O C A BCD = x(exterior angle of cyclic quadrilateral AEBC) x x
T E B D P O C A Now BCD lies in a segment of the circle centre P. x x
T E B D P O C A TBD = x(angle in the alternate segment) x x x
T E B D P O C A EBA = x (vertically opposite) x x x x
T E B D P O C A AEB = ABE. x x
