Cambridge International Mathematics

3 ABCD is a cyclic quadrilateral. Sides AB and DC areproducedor extended to meet at E. Sides DA

and CB are produced to F.

If angle AFB is 30 oand angle BEC is 20 o, find all angles of quadrilateral ABCD.

4 ABCDE is a pentagon inscribed in a circle with centre O.

BD is a diameter of the circle. AE is parallel to BD, and is

produced to F.

Angle BAC=45o and angle CAE=70o.

a Find the size of angles BCD and BDC.

b Show that BC=CD

c Calculate the size of angle DEF.

5 An alternative method for establishing theopposite angles of

a cyclic quadrilateraltheorem is to use the figure alongside.

Show how this can be done.

6 Answer theOpening Problemon page 547.

TESTS FOR CYCLIC QUADRILATERALS

A quadrilateral is acyclic quadrilateralif one of the following is true:

² one pair of opposite angles

is supplementary

If®+ ̄= 180othen ABCD

is a cyclic quadrilateral.

² one side subtends equal

angles at the other two

vertices

If ®= ̄ then ABCD is a

cyclic quadrilateral.

² an exterior angle is equal to

the opposite interior angle

If ®= ̄ then ABCD is a

cyclic quadrilateral.

D B

b°

d°

a°

c°

C

A

O

A B

D C

a

b

D

A B

C

a b

D

A B

C

b

a

A

B

C

D

E

F

O

45°

70°

Circle geometry (Chapter 27) 559

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Y:\HAESE\IGCSE01\IG01_27\559IGCSE01_27.CDR Thursday, 30 October 2008 2:19:10 PM PETER