1. RONALD HUI TAK SUN SECONDARY SCHOOL
HKDSE Mathematics
RONALD HUI
TAK SUN SECONDARY SCHOOL

2. 9 October 2015
Ronald HUI

3. 9 October 2015
Ronald HUI

4. Angles in the Alternate Segment

5. ∠ATP and ∠ATQ are called tangent-chord angles.
Angles in the Alternate Segment A
chord
tangent-chord angle
tangent-chord angle T P Q
tangent at T
∠ATP and ∠ATQ are called tangent-chord angles.

6. Angles in the Alternate SegmentB A
alternate
segment C T P Q
Segment ABT lies on the opposite side of ∠ATQ.
It is called the alternate segment corresponding
to ∠ATQ.
is an angle in the alternate segment
corresponding to ∠ATQ.
∠ABT

7. Angles in the Alternate SegmentB A
alternate
segment C T P Q
Segment ACT lies on the opposite side of ∠ATP.
It is called the alternate segment corresponding
to ∠ATP.
∠ACT
is an angle in the alternate segment
corresponding to ∠ATP.

8. In fact, any angle at the circumference in segment ABT is called an P Q
In fact, any angle at the circumference
in segment ABT is called an
angle in the alternate segment
corresponding to ∠ATQ.

9. E B A D C T P Q
∠ADT and ∠AET are two other examples of angles in the alternate segment corresponding to ∠ATQ .

10. In each of the following, PQ is the tangent to the circle at T.b B a T P Q A B b a
Yes No
Is a an angle in the alternate
segment corresponding to b?

11. In the figure, PQ is the tangent to the circle at T.
NT is a diameter of the circle.
in the alternate segment corresponding to ∠ATQ.
∠ANT is an angle
tangent  radius P T Q N A O
70 m n
∵ ∠OTQ = 90
∴ m + 70 = 90
m = 20
∠ sum of △
In △ANT,
n + m + 90 = 180
n = 70
∠ANT = ∠ATQ

12. Consider two other angles in the alternate segment corresponding to ∠ATQ,
e.g. ∠ABT
and ∠ADT.
∠s in the same
segment N
∵ ∠ABT = ∠ANT D A
∴ b = 70 n d
∠s in the same
segment
∵ ∠ADT = ∠ANT B b O
∴ d = 70 m
70 P Q T

13. Do you think these results
Consider two other angles in the alternate segment corresponding to ∠ATQ,
∠ABT
and ∠ADT. N
∠ABT = ∠ADT = ∠ATQ D A n
Do you think these results
still hold when ∠ATQ
is not 70? d B b O m
70 P Q T

14. ∠ANT = ∠ATQ ∠ABT = ∠ATQ ∠ADT = ∠ATQO
∠ADT = ∠ATQ P Q T
In fact, angles in the alternate segment corresponding to ∠ATQ are always equal to ∠ATQ.

15. Theorem 2.5 A tangent-chord angle of a circle is equal to
an angle in the alternate segment.
Abbreviation:
 in alt. segment A B T P Q A C T P Q
∠ATQ = ∠ABT
∠ATP = ∠ACT

16. Example:
In the figure, PQ is the tangent to the circle at T. If AB // PQ, find x.
alt. s, AB // PQ x A B Q P T
35
BAT = 35
x = ∠BAT
 in alt. segment
35 =

17. Yes, the converse of Theorem 2.5
In the figure, A, B and T are points on the circle. Q is a point outside the circle such that ATQ = ABT.
Is TQ the tangent to
the circle at T? A B
Yes, the converse of Theorem 2.5
is also true. Q T

18. Theorem 2.6 (Converse of Theorem 2.5)
If TP is a straight line such that ∠ATP = ∠ABT,
then TP is the tangent to the circle at T. A
Abbreviation:
converse of
 in alt. segment B P T

19. Refer to the figure. Is TQ the tangent to the circle at T?
Example:
Refer to the figure. Is TQ the tangent to the circle at T? T
∵ TB = TA Q
∴ TBA
= TAB
base s, isos. △
4 cm
4 cm
BTQ
= TBA
alt. s, TQ // AB
∴ BTQ = TAB A B
∴ TQ is the tangent to the circle at T.
converse of  in
alt. segment

20. With the notation in the figure below, we have:
Theorem 2.5
PQ is the tangent to the circle at T.
x = y
Theorem 2.6
(converse of Theorem 2.5) A B y x P Q T

21. converse of  in alt. segment
Follow-up question
In the figure, TP is the tangent to the circle at P.
Prove that TQ is the tangent to the circle at Q.
In △PQT, Q
PQT = 180  70  40
70
= 70
 sum of △ S
70
PSQ = QPT
70
 in alt. segment
40
= 70 T P
∵ PQT = PSQ
∴ TQ is the tangent to the circle at Q.
converse of  in alt. segment

22. 9 October 2015
Ronald HUI