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.

C A

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

C A

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

C A

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

Proof: Join BO and produce to X

C A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

C A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

isosceles s'base OABOBA

C A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

isosceles s'base OABOBA

OABOABOBAAOX exterior C

A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

isosceles s'base OABOBA

OABOABOBAAOX exterior

2 OBAAOX C

A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

isosceles s'base OABOBA

OABOABOBAAOX exterior

2 OBAAOX C

A

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 :Prove

Proof: Join BO and produce to X

radii , isosceles is OBOAAOB

isosceles s'base OABOBA

OABOABOBAAOX exterior

2 OBAAOX

2 ABCAOC

C A

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 ACB

A

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

90ACB

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

90ACB

A

B

O

C

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