linking angles
Post on 10-Jan-2016
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Linking AnglesLinking AnglesVisualising Angle Relationships in Circles
Two circles centred at O and P intersect at B and C.
C
B
O P
The tangent at B to the circle centred P meets the circle centred O at A.
A
C
B
O P
T
The line AC meets the circle centred at P at D.
D
A
C
B
O P
T
DB meets the circle centred at O again at E.
E
D
A
C
B
O P
T
DB meets the circle centred at O again at E.
E
D
A
C
B
O P
T
It is often easier to see relationships if the common chord is added to the diagram.
Show that AEB = ABE.
E
D
A
C
B
O P
T
Proof:Proof:Let AEB = x
E
D
A
C
B
O P
T
x
Introducing a variable will make it easier to trace the path of the angle relationships through the diagram.
Now AEBC is a cyclic quadrilateral
E
D
A
C
B
O P
T
x
BCD = x (exterior angle of cyclic quadrilateral AEBC)
E
D
A
C
B
O P
T
x
x
Now BCD lies in a segment of the circle centre P.
E
D
A
C
B
O P
T
x
x
TBD = x(angle in the alternate segment)
E
D
A
C
B
O P
T
x
x
x
EBA = x (vertically opposite)
E
D
A
C
B
O P
T
x
x
xx
AEB = ABE.
E
D
A
C
B
O P
T
xx
Explore this relationship further using this GeoGebra file
Linking Angle Relationships
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