The document discusses several angle theorems related to circles: 1) Opposite angles of a cyclic quadrilateral are supplementary. 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. Proofs are provided for each theorem using properties of angles in circles.

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.
A
D

O

B C

4. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A
D

O

B C

5. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D

O

B C

6. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D
 Proof: Join AO and CO
O

B C

7. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D
 Proof: Join AO and CO
 at centre twice at 
O AOC  2α  
 circumferenceon same arc 

B C

8. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D
 Proof: Join AO and CO
 at centre twice at 
O AOC  2α  
 circumferenceon same arc 
  at centre twice at 
B C AOC reflex  2  
 circumferenceon same arc 

9. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D
 Proof: Join AO and CO
 at centre twice at 
O AOC  2α  
 circumferenceon same arc 
  at centre twice at 
B C AOC reflex  2  
 circumferenceon same arc 
 AOC  AOC reflex  2  2  360 revolution 

10. Angle Theorems
(5b) Opposite angles of a cyclic quadrilateral are supplementary.
ABC  ADC  180 opposite' s in cyclic quadrilateral  180 

A Prove :     180
D
 Proof: Join AO and CO
 at centre twice at 
O AOC  2α  
 circumferenceon same arc 
  at centre twice at 
B C AOC reflex  2  
 circumferenceon same arc 
 AOC  AOC reflex  2  2  360 revolution 
    180

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.
A
D
O
B X
C

13. (5c) The exterior angle of a cyclic quadrilateral is equal to the
opposite interior angle.
 exteriorcyclic quadilateral 
DCX  BAD  
  opposite interior 
A
D
O
B X
C

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.
B C
O
A
D

16. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C
O
A
D

17. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
O
A
D

18. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
Proof: Join AO and DO
O
A
D

19. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
Proof: Join AO and DO
O
 at centre twiceat 
AOD  2ABD  
 circumference on same arc 
A
D

20. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
Proof: Join AO and DO
O
 at centre twiceat 
AOD  2ABD  
 circumference on same arc 
A
 at centre twiceat 
D AOD  2ACD  
 circumference on same arc 

21. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
Proof: Join AO and DO
O
 at centre twiceat 
AOD  2ABD  
 circumference on same arc 
A
 at centre twiceat 
D AOD  2ACD  
 circumference on same arc 
 ABD  ACD

22. (6) Angles subtended at the circumference by the same or equal arcs
(or chords) are equal.
ABD  ACD ' s in same segment are 
B C Prove : ABD  ACD
Proof: Join AO and DO
O
 at centre twiceat 
AOD  2ABD  
 circumference on same arc 
A
 at centre twiceat 
D AOD  2ACD  
 circumference on same arc 
 ABD  ACD
Exercise 9C; 1 ace etc, 2 aceg, 3, 4 bd, 6 bdf, 7 bdf, 9, 13, 14