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 and extending it to meet the circle at Z. It then uses properties of angles on tangents, in semicircles, and angle sums in triangles to show that ∠APY = ∠ABP.

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