1. Different Types of Angle

2. Angles at a Point O A B C are three pairs of adjacent angles with a common vertex O. a, b and c are called angles at a point. a b a and b, c b b and c, a c c and a b a b a b c b c a c a c O

3. A B C a b A B C c [Abbreviation: ∠ s at a pt.] The sum of all angles at a point is 360 O. ∵ round angle = 360 O ∴ a + b + c = 360 O

4. 210 O a Solution a + 300 O = 360 O a = 360 O  300 O Find a in the figure. a + 90 O + 210 O = 360 O ( ∠ s at a pt.) = 60 O Let us try the following question.

5. Follow-up question 2 Find the unknowns in the following figures. (a) y + 10 O y 170 O Solution 2y + 180 O = 360 O 2y = 180 O (a)y + 10 O + y + 170 O = 360 O ( ∠ s at a pt.) y = 90 O

6. Follow-up question 2 (cont’d) Solution (b)5x + 5x + 2x = 360 O ( ∠ s at a pt.) (b) 5x5x 5x5x 2x2x 12x = 360 O x = 30 O

7. Example 3 Solution

8. Example 4 Solution

9. Adjacent Angles on a Straight Line a a b andb are two different angles. A B C O O They have a common arm OB B common arm and a common vertex O. common vertex O They are on the opposite sides of the common arm OB. a and b are called adjacent angles.

10. A a b C O B A C O then a and b are called adjacent angles on a straight line. If OA andOCOC are on the same straight line, a A O b B C

11. ∵ ∠ AOC = 180 O The sum of the adjacent angles on a straight line is 180 O. [Abbreviation: adj. ∠ s on st. line] A C O ∴ a + b = 180 O a b B A C O

12. Let us try the following question. In the figure, XOY is a straight line, find a. X O Y 49 O R a a + 49 O = 180 O (adj. ∠ s on st. line) ∵ XOY is a straight line. ∴ ∠ XOY = 180 O a = 180 O  49 O = 131 O

13. Follow-up question 1 In each of the following figures, XOY is a straight line. Find the unknown. Solution m + 32 O + 90 O = 180 O (adj. ∠ s on st. line) (a) ∵ XOY is a straight line. ∴ ∠ XOY = 180 O m = 180 O  122 O = 68 O m O 32 O (a) X Y

14. 3b = 180 O b = 60 O b + b + b = 180 O (adj. ∠ s on st. line) (b) b b b X O Y R S Follow-up question 1 (cont’d) Solution (b) ∵ XOY is a straight line. ∴ ∠ XOY = 180 O

15. Example 1 Solution

16. Example 2 Solution

17. ID1 (P.157)ID2 (P.158) ID3 (P.159) ID4 (P.160) ID5 (P.160) ID6 (P.162)ID7 (P.163) 170 o 70 o 45 o 22 20 o No. 35 o 30 o

18. b c d a Vertically Opposite Angles and b are opposite to each other without any common arm. Similarly, c and d are another pair of vertically opposite angles. They are called vertically opposite angles. A B C D O AB and CD are two straight lines intersecting at point O. a

19. A B C D O a b c d When two straight lines intersect, the vertically opposite angles formed are equal. i.e. a = b and c = d [Abbreviation: vert. opp. ∠ s]

20. Solution In the figure, AB, CD and EF are straight lines. Find h and k. Let us try the following question. 100 O 44 O h k A B C D E F h = 44 O (vert. opp. ∠ s) k = 100 O (vert. opp. ∠ s)

21. Follow-up question 3 In the following figures, AOB, COD and EOF are straight lines. Find the unknowns. (a) B O A C D X 45 O 120 O z Solution (a) (vert. opp. ∠ s) z + 45 O = 120 O z = 75 O

22. 7d + c + 3d = 180 O (adj. ∠ s on st. line) 10d = 100 O d = 10 O (b) 10d + 80 O = 180 O (b)c = 80 O (vert. opp. ∠ s) Follow-up question 3 (cont’d) Solution 3d3d 80 O 7d7d c A B C D E F

23. Example 5 (vert. opp.  s) Solution

24. Example 6 Solution

25. Example 7 In the figure, AB, CD and EF are straight lines intersecting at O. Find ∠ BOD and ∠ COE.

26. CP (P.164) 70 o 120 o 45 o 60 o No.

27. CP (P.165) 85 o 40 o

28. Angles related to Parallel Lines

29. In the figure, straight line EF intersects 2 straight lines AB and CD. Angles Formed by a Transversal and Two Lines A B C D EF is called the transversal of AB and CD. E F E F

30. They are also on the same side of EF (i.e. on the right of EF). E A B C D F a e a and e are on the same side of AB and CD. a and e are called a pair of corresponding angles. Corresponding Angles Consider the following figure. (i.e. above the lines AB and CD).

31. d h d h c g b f Yes, they are E A B C D F b f b and f, c g c and g, d and h. Can you find any other pairs of corresponding angles in the figure? e a

32. e c E A B C D F c and e They are on the opposite sides of EF. c and e are called a pair of alternate angles. lie between AB and CD. Consider the following figure. Alternate Angles c lies on the left of EF and e lies on the right of EF. c lies below AB and e lies above CD.

33. Yes, d and f. Can you find another pair of alternate angles in the figure? e c a b h g E A B C D F f d f d

34. c f A B C D F They are on the same side of EF. lie between AB and CD. E c and f c and f are called a pair of interior angles on the same side. Consider the following figure. Interior Angles on the Same Side c lies below AB and f lies above CD. Both c and f lie on the left of EF.

35. Yes, d and e. Can you find another pair of interior angles on the same side? a b h g A B C D F e d c f E e d e d

36. If AB and CD are parallel, can you see any relationships among the angles as shown? e c a b h g E A B C D F f d Angles Formed by a Transversal and Parallel Lines e h g E C D F f a d c A B b

37. Corresponding Angles e E C D F a A B If AB // CD, then a = e. [Abbreviation: corr. ∠ s, AB // CD]

38. b 40 O 140 O a A B C D Consider the following example. In the figure, AB // CD. 140 O Then, a = 40 O (corr. ∠ s, CD // AB) and b =b =

39. Alternate Angles If AB // CD, then c = e. e E C D F c A B [Abbreviation: alt. ∠ s, AB // CD]

40. b 65 O 115 O a A B C D Consider the following example. In the figure, AB // CD. 115 O Then, a  65 O (alt. ∠ s, CD // AB) and b b 

41. Interior Angles on the Same Side e E C D F d A B If AB // CD, then d + e = 180 O. [Abbreviation: int. ∠ s, AB // CD]

42. 105 O a A B C D Then, a + 105 O = 180 O a = 75 O Consider the following example. In the figure, AB // CD. (int. ∠ s, CD // AB)

43. Follow-up question 4 Find the unknown in the following figure. (a) 84 O x A B C D E G F H Q R P S Solution (a) ARG = AQE = 84 O (corr. ∠ s, GH // CD) AQE = CPE = 84 O (corr. ∠ s, AB // CD) x = ARG = 84 O (vert. opp. ∠ s)

44. Find the unknown in the following figure. (b) 7y7y 5y5y A B C D Solution Follow-up question 4 (cont’d) (b) 5y + 7y = 180 O (int. ∠ s, CD // AB) 12y = 180 O y = 15 O

45. Example 8 Find x and y in the figure. Solution

46. Example 9 Find x and y in the figure.

47. Example 10 Find t in the figure. Solution

48. Example 11 Find a and b in the figure.

49. Solution Example 12 In the figure, AB // DC // FE and ED // CBG. Find ∠ ABG.

50. Solution Example 13 In the figure, AB // CD and EF // GH. Find x and y.

51. Solution Example 14 In the figure, AB // DE. Find x.

52. Identifying Parallel Lines

53. Two lines are parallel if any one of the following conditions is satisfied. AB C D E F [Abbreviation: corr. ∠ s equal] Condition I: If a a = b b,b, thenAB //CD.

54. Condition II: AB C D E F [Abbreviation: alt. ∠ s equal] c If c = b b,b, Condition III: AB C D E F [Abbreviation: int. ∠ s supp.] d Ifd + b b = 180 O, thenAB //CD. thenAB //CD.

55. In the figure, is BC parallel to DE? Give the reason. i.e. The alternate angles ∠ BCD and ∠ CDE are not equal. ∵ ∠ BCD ≠ ∠ CDE ∴ BC is not parallel to DE. 130 O 120 O 60 O A B C D E F 50 O

56. ∵ ∠ ABC + ∠ BCD = 120 O + 60 O = 180 O In the figure, is AB parallel to EF? ∴ AB // DC (int. ∠ s supp.) ∴ DC // EF (int. ∠ s supp.) ∵ AB // DC and DC // EF ∴ AB // EF ∵ ∠ CDE + ∠ DEF = 50 O + 130 O = 180 O 130 O 120 O 60 O A B C D E F 50 O

57. 136 O Follow-up question 5 In the following figure, is AB parallel to CD? Give the reason. (a) Solution 44 O A B C D E F G H (a) ∠ HFB = ∠ AFE(vert. opp. ∠ s) = 44 O ∵ ∠ HFB + ∠ EGD = 44 O + 136 O = 180 O ∴ AB // CD(int. ∠ s supp.)

58. Follow-up question 5 (cont’d) In the following figure, is AB parallel to CD? Give the reason. (b) Solution A B C D E F G H 154 O 26 O (b) ∠ CFH + ∠ CFE = 180 O (adj. ∠ s on st. line) ∵ ∠ AGH = ∠ CFH = 154 O ∴ AB // CD(corr. ∠ s equal) ∠ CFH = 180 O – 26 O = 154 O

59. Example 15 Solution

60. Example 16 Solution

61. Example 17

62. Solution Example 17

63. Example 7 (Extra) Solution

64. Example 14 (Extra) In the figure, BA // EF. Find x.

65. Solution

67. Example 17 (Extra)

68. Solution