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.

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.

CA

B

O

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

CA

B

O

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :Prove

CA

B

O

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

CA

B

O

X

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB

CA

B

O

X

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB( )∆∠∠=∠∴ isosceles s'base OABOBA

CA

B

O

X

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB( )∆∠∠=∠∴ isosceles s'base OABOBA

( )OABOABOBAAOX ∆∠∠+∠=∠ exterior CA

B

O

X

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB( )∆∠∠=∠∴ isosceles s'base OABOBA

( )OABOABOBAAOX ∆∠∠+∠=∠ exterior

2 OBAAOX ∠=∠∴CA

B

O

X

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB( )∆∠∠=∠∴ isosceles s'base OABOBA

( )OABOABOBAAOX ∆∠∠+∠=∠ exterior

2 OBAAOX ∠=∠∴CA

B

O

X

( )methodsimilar by 2 OBCCOX ∠=∠∴

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.( )arc sameon ncecircumfereat twicecentre,at 2 ∠∠∠=∠ ABCAOC

ABCAOC ∠=∠ 2 :ProveProof: Join BO and produce to X

( )radii , isosceles is ==∆ OBOAAOB( )∆∠∠=∠∴ isosceles s'base OABOBA

( )OABOABOBAAOX ∆∠∠+∠=∠ exterior

2 OBAAOX ∠=∠∴

2 ABCAOC ∠=∠∴

CA

B

O

X

( )methodsimilar by 2 OBCCOX ∠=∠∴

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

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

A

B

O

C

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

( )semicircle ain 90 ∠=∠ ACBA

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOBA

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOB90 :Prove =∠ACB

A

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOB90 :Prove =∠ACB

Proof: ( )∠=∠ straight 180AOB

A

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOB90 :Prove =∠ACB

Proof: ( )∠=∠ straight 180AOB

∠∠

∠=∠arc sameon ncecircumfere

at twicecentreat 2 ACBAOB

A

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOB90 :Prove =∠ACB

Proof: ( )∠=∠ straight 180AOB

∠∠

∠=∠arc sameon ncecircumfere

at twicecentreat 2 ACBAOB

90=∠ACB

A

B

O

C

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

( )semicircle ain 90 ∠=∠ ACB

diameter is :Data AOB90 :Prove =∠ACB

Proof: ( )∠=∠ straight 180AOB

∠∠

∠=∠arc sameon ncecircumfere

at twicecentreat 2 ACBAOB

90=∠ACB

A

B

O

C

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