Cambridge International Mathematics

A

B

D

C

O

Part 2: Angles subtended by the same arc theorem

What to do:

1 Use a compass to draw a circle with centre O. Mark on it points A, B, C and D.

2 Draw angles ACB and ADB with a ruler.

3 Measure angles ACB and ADB. What do you notice?

4 Repeat the above steps with another circle.

5 Copy and complete: “Angles subtended by an arc on the circle are ...... in size.”

You should have discovered the following theorems:

Name of theorem Statement Diagram

Angle at the centre The angle at the centre of a circle is twice the angle on the circle subtended by the same arc.

AbOB=2£AbCB

Angles subtended by the same arc

Angles subtended by an arc on the circle are equal in size.

AbDB=AbCB

Note: ² The following diagrams show other cases ofthe angle at the centre theorem. These cases can be easily shown using thegeometry package.

² Theangle in a semi-circle theoremis a special case of the angle at the centre theorem.

GEOMETRY PACKAGE

A

B

C

O

A

B

D C

2a a

a

2a

a

O O

O

180°

90° O

GEOMETRY PACKAGE

Circle geometry (Chapter 27) 553

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Y:\HAESE\IGCSE01\IG01_27\553IGCSE01_27.CDR Monday, 27 October 2008 2:40:35 PM PETER