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