Angle Theorems
Jul 13, 2015
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.
( )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