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

9 October 2015 Ronald HUI

9 October 2015 Ronald HUI

Angles in the Alternate Segment

∠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.

Angles in the Alternate Segment B 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

Angles in the Alternate Segment B 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.

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.

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

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?

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

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

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

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

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

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 =

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

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

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

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

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

9 October 2015 Ronald HUI