1. 1B_Ch10(1)

2. 10.1 Angles Relating to Intersecting Lines
1B_Ch10(2)
10.1 Angles Relating to Intersecting Lines A
Angles at a Point B
Adjacent Angles on a Straight Line C
Vertically Opposite Angles
Index

3. 10.2 Angles Relating to Parallel Lines
1B_Ch10(3)
10.2 Angles Relating to Parallel Lines A
Angles Formed by Two Lines and a Transversal B
Angles Formed by Parallel Lines and a Transversal
Index

4. In the right figure, x and y are a pair of adjacent angles.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(4)
Example
Angles at a Point A)
In the right figure, x and y are a pair of adjacent angles.
2. In the right figure, angles a, b, c and d are formed with a common vertex O and each are adjacent to the next. These angles are called angles at a point.
Index

5. 3. In the figure, a + b + c + d = 360° 【Reference: ∠s at a pt.】
10.1 Angles Relating to Intersecting Lines
1B_Ch10(5)
Example
Angles at a Point A)
3. In the figure, a + b + c + d = 360°
【Reference: ∠s at a pt.】
Index 10.1
Index

6. In (b) and (c), x and y are a pair of adjacent angles.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(6)
In the following figures, determine x and y are a pair of adjacent angles.
(a) x y
(b) x y
(c) y x
In (b) and (c), x and y are a pair of adjacent angles.
Key Concept
Index

7. Find r in the figure. r + 140° + 70° + 90° = 360° r + 300° = 360°
10.1 Angles Relating to Intersecting Lines
1B_Ch10(7)
Find r in the figure.
r + 140° + 70° + 90° = 360°
∠s at a pt.
r + 300° = 360°
r = 60°
Index

8. Find the three marked angles in the figure.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(8)
Find the three marked angles in the figure.
3p + 4p + 5p = 360°
∠s at a pt.
12p = 360°
p = 30°
Fulfill Exercise Objective
Find unknown angles using ‘angles at a point’.
∴ 3p = 90°, 4p = 120°, 5p = 150°
∴ The three marked angles in the figure are 90°, 120° and 150°.
Key Concept
Index

9. Adjacent Angles on a Straight Line B)
10.1 Angles Relating to Intersecting Lines
1B_Ch10(9)
Example
Adjacent Angles on a Straight Line B)
In the right figure, AOD is a straight line. We call a, b and c adjacent angles on a straight line.
2. In the figure, a + b + c = 180°.
【Reference: adj. ∠s on st. line】
Index

10. Adjacent Angles on a Straight Line B)
10.1 Angles Relating to Intersecting Lines
1B_Ch10(10)
Example
Adjacent Angles on a Straight Line B)
3. If a + b + c = 180°, then the points A, O and B must lie on the same straight line.
Index 10.1
Index

11. Find x in the figure. In the figure, x + 130° = 180° x = 50°
10.1 Angles Relating to Intersecting Lines
1B_Ch10(11)
Find x in the figure.
In the figure,
x + 130° = 180°
adj. ∠s on st. line
x = 50°
Index

12. In the figure, XOY is a straight line. Find .
10.1 Angles Relating to Intersecting Lines
1B_Ch10(12)
In the figure, XOY is a straight line. Find .
32° + 90° +  = 180°
adj. ∠s on st. line
 ° = 180°
 = 58°
Fulfill Exercise Objective
Find unknown angles using ‘adjacent angles on a straight line’.
Index

13. In the figure, AOB is a straight line. Find the largest marked angle.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(13)
In the figure, AOB is a straight line. Find the largest marked angle.
3z + 2z + z = 180°
adj. ∠s on st. line
6z = 180°
z = 30°
Fulfill Exercise Objective
Find unknown angles using ‘adjacent angles on a straight line’.
∠AOC is the largest angle and
∠AOC
= 3z
= 3 × 30°
= 90°
Index

14. (a) Express  in terms of . (b) If  = 32°, find .
10.1 Angles Relating to Intersecting Lines
1B_Ch10(14)
In the figure below, a light ray SP strikes the mirror HK at the point P, then it reflects in the direction PR. Suppose ∠SPH = ∠RPK = , ∠SPR = .
(a) Express  in terms of .
(b) If  = 32°, find .
Index

15. (a) Since HPK is a straight line,
10.1 Angles Relating to Intersecting Lines
1B_Ch10(15)
Back to Question
(a) Since HPK is a straight line,
 +  +  = 180°
adj. ∠s on st. line
∴  = 180° – 2
(b) Since  = 32°, 
= 180° – 2 × 32°
= 116°
Fulfill Exercise Objective
Find unknown angles using ‘adjacent angles on a straight line’.
Key Concept
Index

16. ∴ Yes, ACE is a straight line.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(16)
The figure shows two set squares ABC and CDE, in which ∠ACB = ∠ECD = 90°. Is ACE a straight line? ∴
∠ACB + ∠ECD
= 90° + 90°
= 180°
∴ Yes, ACE is a straight line.
Index

17. It is known that ∠ACB = 60°, ∠GCF = 90° and ∠ECD = 30°.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(17)
It is known that ∠ACB = 60°, ∠GCF = 90° and ∠ECD = 30°.
Is BCD a straight line? A B C D E G F
60o
30o ∴
∠ACB + ∠GCF + ∠ECD
= 60° + 90° + 30°
= 180°
∴ Yes, BCD is a straight line.
Key Concept
Index

18. Vertically Opposite Angles C)
10.1 Angles Relating to Intersecting Lines
1B_Ch10(18)
Example
Vertically Opposite Angles C)
In the right figure, the lines AB and CD interest at point O. We call a and b a pair of vertically opposite angles.
2. In the figure, a = b.
【Reference: vert. opp. ∠s】
Index 10.1
Index

19. Find x in the figure. In the figure, x = 45°
10.1 Angles Relating to Intersecting Lines
1B_Ch10(19)
Find x in the figure.
In the figure, x = 45°
vert. opp. ∠s
Index

20. Consider the straight line CG.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(20)
In the figure, the straight lines AE, BF and CG intersect at point O, AE  CG. Find p.
In the figure, ∠BOA = 4p.
vert. opp. ∠s
Consider the straight line CG.
∴∠COB + ∠BOA + 90° = 180°
adj. ∠s on st. line
p + 4p + 90° = 180°
5p = 90°
p = 18°
Fulfill Exercise Objective
Find unknown angles using ‘vertically opposite angles’.
Index

21. In the figure, CAD, CBE and GABF are straight lines. Find x and y.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(21)
In the figure, CAD, CBE and GABF are straight lines. Find x and y. ∴
GF is a straight line.
∴ (40° – x) + 142° = 180°
adj. ∠s on st. line
x = 40° + 142° – 180°
= 2°
∴∠ABC
= 40° – x
= 38°
Index

22. GF and CD are intersecting straight lines.
10.1 Angles Relating to Intersecting Lines
1B_Ch10(22)
Back to Question ∴
GF and CD are intersecting straight lines.
∴ ∠BAC = ∠DAG = 76°
vert. opp. ∠s ∴
∠BAC + ∠ABC + ∠BCA = 180°
∠sum of 
∴ 76° + 38° + 3y = 180°
114° + 3y = 180°
3y = 66°
y = 22°
Fulfill Exercise Objective
Miscellaneous questions.
Key Concept
Index

23. Angles Formed by Two Lines and a Transversal A)
10.2 Angles Relating to Parallel Lines
1B_Ch10(23)
Angles Formed by Two Lines and a Transversal A)
When two lines AB and CD are cut by another line TS, the line TS is called a transversal of AB and CD.
Index

24. Angles Formed by Two Lines and a Transversal A)
10.2 Angles Relating to Parallel Lines
1B_Ch10(24)
Example
Angles Formed by Two Lines and a Transversal A)
Each of a and e ; b and f ; c and g ; d and h is a pair of corresponding angles.
Each of c and e ; d and f is a pair of alternate angles.
Each of c and f ; d and e is a pair of interior angles on the same side of the transversal.
Index 10.2
Index

25. L is a transversal of lines m1 and m2. Find all
10.2 Angles Relating to Parallel Lines
1B_Ch10(25)
L is a transversal of lines m1 and m2. Find all a b c d e f g h m1 m2 L
(a) corresponding angles,
(b) alternate angles and
(c) interior angles on the same side of the transversal
in the figure.
(a) a and e, b and f, c and g, d and h
(b) d and f, c and e
Key Concept
(c) d and e, c and f
Index

26. Angles Formed by Parallel Lines and a Transversal B)
10.2 Angles Relating to Parallel Lines
1B_Ch10(26)
Example
Angles Formed by Parallel Lines and a Transversal B)
Corresponding Angles of Parallel Lines
In the figure, if AB // CD, then a = b.
【Reference: corr. ∠s, AB // CD】
Index

27. Angles Formed by Parallel Lines and a Transversal B)
10.2 Angles Relating to Parallel Lines
1B_Ch10(27)
Example
Angles Formed by Parallel Lines and a Transversal B)
2. Alternate Angles of Parallel Lines
In the figure, if AB // CD, then a = b.
【Reference: alt. ∠s, AB // CD】
Index

28. Angles Formed by Parallel Lines and a Transversal B)
10.2 Angles Relating to Parallel Lines
1B_Ch10(28)
Example
Angles Formed by Parallel Lines and a Transversal B)
3. Interior Angles on the Same Side of the Transversal
In the figure, if AB // CD, then a + b = 180°.
【Reference: int. ∠s, AB // CD】
Index 10.2
Index

29. Find a in the figure. In the figure, a = 54°
10.2 Angles Relating to Parallel Lines
1B_Ch10(29)
Find a in the figure.
In the figure, a = 54°
corr. ∠s, AB // CD
Index

30. Find x in the figure. ∠AEF + 60° = 180° ∠AEF = 120° ∴ x = ∠AEF = 120°
10.2 Angles Relating to Parallel Lines
1B_Ch10(30)
Find x in the figure. x
60° A B C D E F
∠AEF + 60° = 180°
adj. ∠s on st. line
∠AEF = 120°
∴ x
= ∠AEF
= 120°
corr. ∠s, AB // CD
Key Concept
Index

31. Find x in the figure. In the figure, x = 135°
10.2 Angles Relating to Parallel Lines
1B_Ch10(31)
Find x in the figure.
In the figure, x = 135°
alt. ∠s, AB // CD
Index

32. Find y in the figure. ∠BGD + 95° = 180° ∠BGD = 85° ∴ y = ∠BGD = 85° C
10.2 Angles Relating to Parallel Lines
1B_Ch10(32)
Find y in the figure.
95° E A C D G B F y
∠BGD + 95° = 180°
adj. ∠s on st. line
∠BGD = 85°
∴ y
= ∠BGD
= 85°
alt. ∠s, AB // CD
Key Concept
Index

33. Find x in the figure. In the figure, x + 50° = 180° ∴ x = 130°
10.2 Angles Relating to Parallel Lines
1B_Ch10(33)
Find x in the figure.
In the figure,
x + 50° = 180°
int. ∠s, PQ // RS
∴ x = 130°
Index

34. Find the unknown angles a and b in the figure.
10.2 Angles Relating to Parallel Lines
1B_Ch10(34)
Find the unknown angles a and b in the figure.
a + 110° = 180°
int. ∠s, AD // BE
Fulfill Exercise Objective
Find unknown angles associated with parallel lines.
a = 70°
∠EBC = ∠DAC
corr. ∠s, AD // BE
b + 60° = 110°
b = 50°
Index

35. Find the unknown angles x and y in the figure.
10.2 Angles Relating to Parallel Lines
1B_Ch10(35)
Find the unknown angles x and y in the figure.
With the notation in the figure,
x = 54°
alt. ∠s, AB // CD
Index

36. m + 120° = 180° m = 60° In CQR, m + x + y = 180°
10.2 Angles Relating to Parallel Lines
1B_Ch10(36)
Back to Question
m + 120° = 180°
int. ∠s, AB // CD
m = 60°
In CQR,
m + x + y = 180°
∠sum of 
∴ 60° + 54° + y = 180°
y = 66°
Fulfill Exercise Objective
Find unknown angles associated with parallel lines.
Index

37. 10.2 Angles Relating to Parallel Lines
1B_Ch10(37)
In the figure, AB, QR and CD are parallel while PQ and RS are also parallel. If ∠RSA = 57°, find ∠QPD.
Index

38. With the notation in the figure,
10.2 Angles Relating to Parallel Lines
1B_Ch10(38)
Back to Question
With the notation in the figure,
m + 57° = 180°
int. ∠s, AB // QR
m = 123° ∴
n = m
alt. ∠s, PQ // RS ∴
n = 123° ∴
 + n = 180°
int. ∠s, QR // CD
 + 123° = 180°
Fulfill Exercise Objective
Find unknown angles associated with parallel lines.
 = 57° ∴
∠QPD = 57°
Index

39. Find reflex ∠TRS in the figure.
10.2 Angles Relating to Parallel Lines
1B_Ch10(39)
Find reflex ∠TRS in the figure.
Draw from T the line AT such that AT // PQ.
Index

40. With the notation in the figure,
10.2 Angles Relating to Parallel Lines
1B_Ch10(40)
Back to Question
With the notation in the figure,
y + 136° = 180°
int. ∠s, PQ // AT
y = 44°
x + y = 2x – 36° ∴
alt. ∠s, RS // AT ∴
x + 44° = 2x – 36°
44° + 36° = 2x – x
x = 80°
Fulfill Exercise Objective
Add auxiliary lines to find unknown angles.
Reflex ∠TRS
= 360° – (2x – 36°)
= 360° – 2 × 80° + 36°
= 236°
Key Concept
Index

41. Conditions for Two Lines to be Parallel
10.3 Conditions for Parallel Lines
1B_Ch10(41)
Example
Conditions for Two Lines to be Parallel
1. In the figure, if a = b, then AB // CD.
【Reference: corr. ∠s equal】
Index

42. Conditions for Two Lines to be Parallel
10.3 Conditions for Parallel Lines
1B_Ch10(42)
Example
Conditions for Two Lines to be Parallel
2. In the figure, if a = b, then AB // CD.
【Reference: alt. ∠s equal】
Index

43. Conditions for Two Lines to be Parallel
10.3 Conditions for Parallel Lines
1B_Ch10(43)
Example
Conditions for Two Lines to be Parallel
3. In the figure, if a + b = 180°, then AB // CD.
【Reference: int. ∠s supp.】
Index

44. In the figure, is PQ // RS? ∠QBC ∴ = ∠SCD = 120° ∴ Yes, PQ // RS.
10.3 Conditions for Parallel Lines
1B_Ch10(44)
In the figure, is PQ // RS?
∠QBC ∴
= ∠SCD
= 120° ∴
Yes, PQ // RS.
corr. ∠s equal
Index

45. In the figure, is RS // TU? ∠MNR + 80° = 180° ∠MNR = 100° ∠MNR ∴
10.3 Conditions for Parallel Lines
1B_Ch10(45)
100° M S N R T U
80°
In the figure, is RS // TU?
∠MNR + 80° = 180°
adj. ∠s on st. line
∠MNR = 100°
∠MNR ∴
= ∠NUT
= 100° ∴
Yes, RS // TU.
corr. ∠s equal
Key Concept
Index

46. In the figure, is AB // CD? ∠AEF ∴ = ∠DFE = 60° ∴ Yes, AB // CD.
10.3 Conditions for Parallel Lines
1B_Ch10(46)
In the figure, is AB // CD?
∠AEF ∴
= ∠DFE
= 60° ∴
Yes, AB // CD.
alt. ∠s equal
Index

47. In the figure, is AB // CD? ∠ABC + 273° = 360° ∠ABC = 87° ∠ABC ∴
10.3 Conditions for Parallel Lines
1B_Ch10(47)
In the figure, is AB // CD?
273° A B C D
87°
∠ABC + 273° = 360°
∠s at a pt.
∠ABC = 87°
∠ABC ∴
= ∠BCD
= 87° ∴
Yes, AB // CD.
alt. ∠s equal
Key Concept
Index

48. In the figure, is AB // DC? = 140° + 40° = 180° ∠ABC + ∠DCB ∴
10.3 Conditions for Parallel Lines
1B_Ch10(48)
In the figure, is AB // DC?
= 140° + 40°
= 180°
∠ABC + ∠DCB ∴
Yes, AB // DC.
int. ∠s supp.
Index

49. 10.3 Conditions for Parallel Lines
1B_Ch10(49)
In the figure, PQR and ABC are both straight lines, AP // BQ, ∠AQP = 2x + 18°, ∠PAQ = 30°, ∠QAB = 78° and ∠AQB = x. Is PR parallel to AC?
Index

50. x = 30° = 2x + 18° = 2 × 30° + 18° = 78° ∠PQA ∠PQA ∴ = ∠QAB = 78° ∴
10.3 Conditions for Parallel Lines
1B_Ch10(50)
Back to Question
x = 30°
alt. ∠s, AP // BQ
= 2x + 18°
= 2 × 30° + 18°
= 78°
∠PQA
∠PQA ∴
= ∠QAB
= 78° ∴
Yes, PR // AC.
alt. ∠s, equal
Fulfill Exercise Objective
Determine if two given lines are parallel.
Key Concept
Index