The document discusses several angle theorems related to cyclic quadrilaterals: 1) Opposite angles of a cyclic quadrilateral are supplementary. This is proven using properties of angles on a circumference. 2) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. 3) Angles subtended at the circumference by the same or equal arcs are equal. This is proven using properties of angles on a circumference.

1. Angle Theorems

2. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.

3. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
C
A
B
O
D



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
2reflexAOC

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
2reflexAOC

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
2reflexAOC

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

13. 








interioropposite
alquadilatercyclicexterior
BADDCX
(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