Л.С.Атанасян, И.Ф.Бутузов. Геометрия 7-9 сынып, 2002

Introduction

Grade 7. Chapter I. Basic geometric concepts

§1. Line and segment

1. Points, lines, and segments

2. Drawing a straight line on the ground

Practical tasks

§2. Ray and angle

3. Ray

4. Angle

Practical tasks

§3. Comparing segments and angles

5. Congruence of geometric figures

6. Comparing segments and angles

Questions and problems

§4. Measuring segments

7. Length of a segment

8. Units of measurement. Measuring instruments

Practical tasks

Questions and problems

§5. Measuring angles

9. Degree measure of an angle

10. Measuring angles on the ground

Practical tasks

§6. Perpendicular lines

11. Adjacent and vertical angles

12. Perpendicular lines

13. Constructing right angles on the ground

Practical tasks

Questions and problems

Review questions for Chapter I

Additional problems

Chapter II. Triangles

§1. First triangle congruence criterion

14. Triangle

15. First triangle congruence criterion

Practical tasks

§2. Medians, angle bisectors, and altitudes of a triangle

16. Constructing a perpendicular to a line

17. Medians, angle bisectors, and altitudes of a triangle

18. Properties of an isosceles triangle

Practical tasks

§3. Second and third triangle congruence criteria

19. Second triangle congruence criterion

20. Third triangle congruence criterion

Problems

§4. Construction problems

21. Circle

22. Compass-and-straightedge constructions

23. Examples of construction problems

Review questions for Chapter II

Additional problems

Chapter III. Parallel lines

§1. Criteria for two lines to be parallel

24. Definition of parallel lines

25. Criteria for two lines to be parallel

26. Practical methods for constructing parallel lines

§2. Parallel postulate

27. On the axioms of geometry

28. Parallel postulate

29. Angles formed by two parallel lines and a transversal

Questions and problems

Review questions for Chapter III

Additional problems

Chapter IV. Relationships between the sides and angles of a triangle

§1. Sum of the angles of a triangle

30. Theorem on the sum of the angles of a triangle

31. Acute, right, and obtuse triangles

§2. Relationships between the sides and angles of a triangle

32. Theorem on relationships between the sides and angles of a triangle

33. Triangle inequality

§3. Right triangles

34. Some properties of right triangles

35. Congruence criteria for right triangles

36. Corner reflector

Problems

§4. Constructing a triangle from three elements

37. Distance from a point to a line. Distance between parallel lines

38. Constructing a triangle from three elements

Questions and problems

Construction problems

Review questions for Chapter IV

Additional problems

More difficult problems

Problems for Chapter I

Problems for Chapter II

Problems for Chapters III and IV

Construction problems

Grade 8. Chapter V. Quadrilaterals

§1. Polygons

39. Polygon

40. Convex polygon

41. Quadrilateral

§2. Parallelogram and trapezoid

42. Parallelogram

43. Criteria for a parallelogram

44. Trapezoid

Problems

Construction problems

§3. Rectangle, rhombus, square

45. Rectangle

46. Rhombus and square

47. Axial and central symmetry

Questions and problems

Review questions for Chapter V

Additional problems

Chapter VI. Area

§1. Area of a polygon

48. Concept of the area of a polygon

49. Area of a square

50. Area of a rectangle

Questions and problems

§2. Areas of a parallelogram, triangle, and trapezoid

51. Area of a parallelogram

52. Area of a triangle

53. Area of a trapezoid

Problems

§3. Pythagorean theorem

54. Pythagorean theorem

55. Converse of the Pythagorean theorem

Review questions for Chapter VI

Additional problems

Problems solved using a calculator

Chapter VII. Similar triangles

§1. Definition of similar triangles

56. Proportional segments

57. Definition of similar triangles

58. Ratio of areas of similar triangles

§2. Triangle similarity criteria

59. First triangle similarity criterion

60. Second triangle similarity criterion

61. Third triangle similarity criterion

§3. Applying similarity to prove theorems and solve problems

62. Midline of a triangle

63. Proportional segments in a right triangle

64. Practical applications of triangle similarity

65. On similarity of arbitrary figures

Questions and problems

Construction problems

§4. Relationships between the sides and angles of a right triangle

66. Sine, cosine, and tangent of an acute angle in a right triangle

67. Values of sine, cosine, and tangent for 30°, 45°, and 60°

Problems

Review questions for Chapter VII

Additional problems

Problems solved using a calculator

Chapter VIII. Circle

§1. Tangent to a circle

68. Relative positions of a line and a circle

69. Tangent to a circle

Problems

§2. Central and inscribed angles

70. Degree measure of a circular arc

71. Inscribed angle theorem

Problems

§3. Four notable points of a triangle

72. Properties of an angle bisector and the perpendicular bisector of a segment

73. Theorem on the concurrency of the altitudes of a triangle

Problems

§4. Incircle and circumcircle

74. Incircle

75. Circumcircle

Problems

Review questions for Chapter VIII

Additional problems

Chapter IX. Vectors

§1. Concept of a vector

76. Concept of a vector

77. Equality of vectors

78. Constructing a vector from a given point

Practical tasks

Questions and problems

§2. Addition and subtraction of vectors

79. Sum of two vectors

80. Laws of vector addition. Parallelogram rule

81. Sum of several vectors

82. Subtraction of vectors

Practical tasks

Questions and problems

§3. Multiplying a vector by a scalar. Applying vectors to solve problems

83. Multiplication of a vector by a scalar

84. Applying vectors to solve problems

85. Midline of a trapezoid

Practical tasks

Problems

Review questions for Chapter IX

Additional problems

More difficult problems

Grade 9. Chapter X. Coordinate method

§1. Coordinates of a vector

86. Decomposing a vector into two non-collinear vectors

87. Coordinates of a vector

Problems

§2. Basic coordinate problems

88. Relationship between vector coordinates and the coordinates of its endpoints

89. Basic coordinate problems

Problems

§3. Equations of a circle and a line

90. Equation of a curve in the plane

91. Equation of a circle

92. Equation of a line

Problems

Review questions for Chapter X

Additional problems

Chapter XI. Relationships between the sides and angles of a triangle. Dot product of vectors

§1. Sine, cosine, and tangent of an angle

93. Sine, cosine, and tangent

94. Fundamental trigonometric identity. Reduction formulas

95. Formulas for calculating point coordinates

Problems

§2. Relationships between the sides and angles of a triangle

96. Theorem on the area of a triangle

97. Law of sines

98. Law of cosines

99. Solving triangles

100. Measurement work

Problems

§3. Dot product of vectors

101. Angle between vectors

102. Dot product of vectors

103. Dot product in coordinates

104. Properties of the dot product

Problems

Review questions for Chapter XI

Additional problems

Solving problems using MK-54–MK-57 programmable calculators

Chapter XII. Circumference and area of a circle

§1. Regular polygons

105. Regular polygon

106. Circumcircle of a regular polygon

107. Incircle of a regular polygon

108. Formulas for the area, side length, and inradius of a regular polygon

109. Constructing regular polygons

Questions and problems

§2. Circumference and area of a circle

110. Circumference

111. Area of a circle

112. Area of a circular sector

Questions and problems

Review questions for Chapter XII

Additional problems

Solving problems using MK-54–MK-57 programmable calculators

Chapter XIII. Transformations

§1. Concept of a transformation

113. Mapping the plane onto itself

114. Concept of a transformation

115. Superpositions and transformations

Problems

§2. Translation and rotation

116. Translation

117. Rotation

Problems

Review questions for Chapter XIII

Additional problems

More difficult problems

Appendix 1. Axioms of plane geometry

Appendix 2. Examples of using trigonometric-function tables

Appendix 3. Some information on the development of geometry

Appendix 4. Some notable theorems of plane geometry

Answers and hints