11 x1 t13 06 tangent theorems 2 (2012)

Alternate Segment
Theorem

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.

Alternate Segment
Theorem

A
B
O

X P

Alternate Segment
Theorem
BPX  PAB  alternate segment theorem 

A
B
O

X P

Alternate Segment
Theorem
Prove : APY  ABP
A
B
O

X P Y

Alternate Segment
Theorem
Z Prove : APY  ABP
A Proof: Join PO and produce to meet the
B circumference at Z
O

X P Y

Alternate Segment
Theorem
O
Join AZ

X P Y

Alternate Segment
Theorem
O
Join AZ
ZPY  90 radius  tangent 
X P Y

Alternate Segment
Theorem
O
Join AZ
X P Y APZ  90 - APY

Alternate Segment
Theorem
O
Join AZ
X P Y APZ  90 - APY
PAZ  90  in semicircle 

Z
A
B
O

X Y
P
PAZ  AZP  APZ  180  sum APZ 

Z
A
B
O

X Y
P
90  AZP  90  APY  180

Z
A
B
O

X Y
P
90  AZP  90  APY  180
AZP  APY

Z
A
B
O

X Y
P
90  AZP  90  APY  180
AZP  APY
AZP  ABP  's in same segment = 

Z
A
B
O

X Y
P
90  AZP  90  APY  180
AZP  APY
 APY  ABP

Z
A
B
O

X Y
P
90  AZP  90  APY  180
AZP  APY
 APY  ABP

Exercise 9F; 2ace etc, 3bd, 4bd, 5b, 7b, 9b, 10b, 12, 14, 15b

11 x1 t13 06 tangent theorems 2 (2012)

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