Download - 11X1 T07 03 angle theorems 2
Angle Theorems
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
C
A
B
O
D
α
β
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
C
A
B
O
D
α
β
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βα
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A
B
O
D
α
β
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βαProof: Join AO and CO
C
A
B
O
D
α
β
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βαProof: Join AO and CO
∠∠
=∠arc samenceon circumfere
at twicecentreat 2αAOC
C
A
B
O
D
α
β
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βαProof: Join AO and CO
∠∠
=∠arc samenceon circumfere
at twicecentreat 2αAOC
C
A
B
O
D
α
β
∠∠
=∠arc samenceon circumfere
at twicecentreat 2βreflexAOC
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βαProof: Join AO and CO
∠∠
=∠arc samenceon circumfere
at twicecentreat 2αAOC
( )revolution 36022 =+=∠+∠∴ βαreflexAOCAOC
C
A
B
O
D
α
β
∠∠
=∠arc samenceon circumfere
at twicecentreat 2βreflexAOC
Angle Theorems(5b) Opposite angles of a cyclic quadrilateral are supplementary.
( ) 180 ralquadrilate cyclicin s'opposite 180 =∠=∠+∠ ADCABC
180 :Prove =+ βαProof: Join AO and CO
∠∠
=∠arc samenceon circumfere
at twicecentreat 2αAOC
( )revolution 36022 =+=∠+∠∴ βαreflexAOCAOC
180=+∴ βα
C
A
B
O
D
α
β
∠∠
=∠arc samenceon circumfere
at twicecentreat 2βreflexAOC
(5c) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
(5c) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
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A
B
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X
∠=∠
∠=∠interior opposite
alquadilater cyclicexterior BADDCX
(5c) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
C
A
B
O
D
X
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
A
B
O
C
D
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
A
B
O
C
D
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :Prove
A
B
O
C
D
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :ProveProof: Join AO and DO
A
B
O
C
D
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :ProveProof: Join AO and DO
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ABDAOD
A
B
O
C
D
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :ProveProof: Join AO and DO
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ABDAOD
A
B
O
C
D
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ACDAOD
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :ProveProof: Join AO and DO
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ABDAOD
ACDABD ∠=∠∴
A
B
O
C
D
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ACDAOD
(6) Angles subtended at the circumference by the same or equal arcs (or chords) are equal.
( )=∠∠=∠ aresegment samein s' ACDABD
ACDABD ∠=∠ :ProveProof: Join AO and DO
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ABDAOD
ACDABD ∠=∠∴
Exercise 9C; 1 ace etc, 2 aceg, 3, 4 bd, 6 bdf, 7 bdf, 9, 13, 14
A
B
O
C
D
∠∠
∠=∠arc sameon ncecircumfere
at twicecentreat 2 ACDAOD
