11 x1 t13 02 angle theorems 1

Angle Theorems

Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

angle subtended by the arc (or chord) at the circumference.



C A

B

O



arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

C A

B

O




ABCAOC 2 :Prove

C A

B

O




ABCAOC 2 :Prove

Proof: Join BO and produce to X

C A

B

O

X




ABCAOC 2 :Prove


radii , isosceles is OBOAAOB

C A

B

O

X




ABCAOC 2 :Prove



isosceles s'base OABOBA

C A

B

O

X




ABCAOC 2 :Prove




OABOABOBAAOX exterior C

A

B

O

X




ABCAOC 2 :Prove




OABOABOBAAOX exterior

2 OBAAOX C

A

B

O

X




ABCAOC 2 :Prove





2 OBAAOX C

A

B

O

X

methodsimilar by 2 OBCCOX




ABCAOC 2 :Prove





2 OBAAOX

2 ABCAOC

C A

B

O

X

methodsimilar by 2 OBCCOX

(5a) The angle in a semicircle is a right angle.


A

B

O

C


semicircle ain 90 ACB

A

B

O

C



diameter is :Data AOBA

B

O

C



diameter is :Data AOB90 :Prove ACB

A

B

O

C




Proof: straight 180AOB

A

B

O

C





arc sameon ncecircumfere

at twicecentreat 2 ACBAOB

A

B

O

C







90ACB

A

B

O

C







90ACB

A

B

O

C

Exercise 9B; 1 ace etc, 2, 6, 8ac, 9ab, 10ac, 11ac, 12, 13

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