Cambridge International Mathematics

Proof: Opposite angles of a cyclic quadrilateral

Join OA and OC.

Let ADCb =®o and AbBC= ̄o

) AOCb =2®o fangle at the centreg

and reflex AOCb =2 ̄o fangle at the centreg

But 2 ®+2 ̄= 360 fangles at a pointg

) ®+ ̄= 180

) ABCb +ADCb = 180o

and since the angles of any quadrilateral add to 360 o,

BADb +BbCD= 180o

Proof: Exterior angle of a cyclic quadrilateral

This theorem is an immediate consequence of the opposite angles

of a cyclic quadrilateral being supplementary.

Let AbBC=μo

) ADCb = (180¡μ)o fopp. angles of cyclic quad.g

) CDEb =μo fangles on a lineg

) AbBC=CDEb

Example 3 Self Tutor

Solve forx:

The angles given are opposite angles of a cyclic quadrilateral.

) (x+ 15) + (x¡21) = 180

) 2 x¡6 = 180

) 2 x= 186

) x=93

O

b°

a°

2°a

2°b

A

B C

D

O

q°

A

B C

D

E

()180¡-¡q°

q°

()x-21°

()x¡+¡15°

Circle geometry (Chapter 27) 557

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Y:\HAESE\IGCSE01\IG01_27\557IGCSE01_27.CDR Monday, 27 October 2008 2:40:46 PM PETER