The document proves that the alternate segment theorem, which states that any angle in an alternate segment of a circle is equal to an angle formed by the tangent and chord at the point of contact. It does this by joining a radius to the point of contact P to form two right angles, and using angle properties such as angle sums in triangles to show that the two angles in question are equal.

1. Alternate Segment
Theorem

2. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.

3. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
A
B
O
X P

4. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
A
B
O
X P

5. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Prove : APY  ABP
A
B
O
X P Y

6. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O
X P Y

7. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O
Join AZ
X P Y

8. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O
Join AZ
ZPY  90 radius  tangent 
X P Y

9. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O
Join AZ
ZPY  90 radius  tangent 
X P Y APZ  90 - APY

10. Alternate Segment
Theorem
(9) An angle formed by a tangent to a circle with a chord drawn to the
point of contact is equal to any angle in the alternate segment.
BPX  PAB  alternate segment theorem 
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O
Join AZ
ZPY  90 radius  tangent 
X P Y APZ  90 - APY
PAZ  90  in semicircle 

11. Z
A
B
O
X Y
P

12. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 

13. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 
90  AZP  90  APY  180

14. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 
90  AZP  90  APY  180
AZP  APY

15. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 
90  AZP  90  APY  180
AZP  APY
AZP  ABP  's in same segment = 

16. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 
90  AZP  90  APY  180
AZP  APY
AZP  ABP  's in same segment = 
 APY  ABP

17. Z
A
B
O
X Y
P
PAZ  AZP  APZ  180  sum APZ 
90  AZP  90  APY  180
AZP  APY
AZP  ABP  's in same segment = 
 APY  ABP
Exercise 9F; 2ace etc, 3bd, 4bd, 5b, 7b, 9b, 10b, 12, 14, 15b