11X1 T07 03 angle theorems 2
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Transcript of 11X1 T07 03 angle theorems 2
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
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
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.
C
A
B
O
D
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.
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