4. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
C
A
B
O
D
5. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
C
A
B
O
D
6. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
Proof: Join AO and CO
C
A
B
O
D
7. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
Proof: Join AO and CO
arcsamenceoncircumfere
attwicecentreat
2αAOC
C
A
B
O
D
8. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
Proof: Join AO and CO
arcsamenceoncircumfere
attwicecentreat
2αAOC
C
A
B
O
D
arcsamenceoncircumfere
attwicecentreat
2reflexAOC
9. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
Proof: Join AO and CO
arcsamenceoncircumfere
attwicecentreat
2αAOC
revolution36022
reflexAOCAOC
C
A
B
O
D
arcsamenceoncircumfere
attwicecentreat
2reflexAOC
10. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
180ralquadrilatecyclicins'opposite180 ADCABC
180:Prove
Proof: Join AO and CO
arcsamenceoncircumfere
attwicecentreat
2αAOC
revolution36022
reflexAOCAOC
180
C
A
B
O
D
arcsamenceoncircumfere
attwicecentreat
2reflexAOC
11. (5c) The exterior angle of a cyclic quadrilateral is equal to the
opposite interior angle.
12. (5c) The exterior angle of a cyclic quadrilateral is equal to the
opposite interior angle.
C
A
B
O
D
X
14. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
15. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
A
B
O
C
D
16. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
A
B
O
C
D
17. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
A
B
O
C
D
18. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
Proof: Join AO and DO
A
B
O
C
D
19. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
Proof: Join AO and DO
arcsameonncecircumfere
attwicecentreat
2 ABDAOD
A
B
O
C
D
20. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
Proof: Join AO and DO
arcsameonncecircumfere
attwicecentreat
2 ABDAOD
A
B
O
C
D
arcsameonncecircumfere
attwicecentreat
2 ACDAOD
21. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
Proof: Join AO and DO
arcsameonncecircumfere
attwicecentreat
2 ABDAOD
ACDABD
A
B
O
C
D
arcsameonncecircumfere
attwicecentreat
2 ACDAOD
22. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
aresegmentsameins'ACDABD
ACDABD :Prove
Proof: Join AO and DO
arcsameonncecircumfere
attwicecentreat
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
arcsameonncecircumfere
attwicecentreat
2 ACDAOD