## Geometry – Shuvalova

In this post, we will see the book Geometry by E. Z. Shuvalova. The book provides a comprehensive treatment of basic geometrical concepts for high schools. The first chapter deals with properties of triangles. Chapter 2 gives a comprehensive introduction to  structure of solid geometry. The chapter on parallelism in space details the concept in various contexts. Next three chapters dal with space, vectors and planes and their properties including  dihedral and polyhedral angles. The last few chapters treat solids, polyhedrons and their properties such as areas volumes and sections by planes. There are worked problems and exercises for each chapter.

The book was translated from Russian by Leonid Levant and was published in 1980 by Mir.

You can get the book here.

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# Contents

Symbols Used in the Book 9

CHAPTER 1. BASIC RELATIONS BETWEEN THE ELEMENTS OF AN ARBITRARY TRIANGLE. SOLVING OBLIQUE TRIANGLES

Sec. 1. The Law of Sines 11
Sec. 2. The Law of Cosines 13
Sec. 3. Expressing the Tangent of a Half-Angle in Terms of the Sides of a Triangle and Radius of the Inscribed Circle (or the Incircle) 16
Sec. 4. Solving Oblique Triangles 17
Sec. 5. Measuring Distances Between “Inaccessible” 20
Sec. 6. Other Problems on Solving Oblique Triangles 23
Sec. 7. Worked Problems 28
Problems to Chapter 4 32

CHAPTER 2. LOGICAL STRUCTURE OF THE COURSE OF SOLID
GEOMETRY

Sec. 8. Structure of the Course of Solid Geometry. Notation and Terminology 36
Sec. 9. Axioms of Belonging 38
Sec. 10. Axioms of Distance 40
Sec. 11. Axioms of Order 42
Sec. 12. Axiom of Plane Mobility 45
Sec. 13. Axiom of Parallelism 47
Sec. 14. Corollaries Following from the Axioms 48

CHAPTER 3. PARALLELISM IN SPACE

Sec. 15. The Relative Position of Two Straight Lines in
Space. The Relative Position of a Plane and a Straight Line in Space 50
Sec. 16. Parallelism of a Straight Line and a Plane 51
Sec. 17. Parallel Planes 52
Sec. 18. Direction in Space. The Angle Between Two Lines 55
Sec. 19. Parallel Projecting 58
Sec. 20. Representation of Figures in Solid Geometry 61
Sec. 21. Worked Problems 63
Problems to Chapter 3 66

CHAPTER 4. TRANSFORMATION OF SPACE. VECTORS

Sec. 22. Transformation of Space 68
Sec. 23. Translation in Space 69
Sec. 24. The Vector Defined 70
Sec. 25. The Sum of Vectors 72
Sec. 26. Subtraction of Vectors. Multiplication of a Vector by a Number 75
Sec. 27. A Linear Combination of Vectors. The Conditions of Collinearity and Coplanarity 78
Sec. 28. Scalar Product of Vectors 81
Sec. 29. Arithmetic Properties of a Scalar Product 83
Sec. 30. Vector Product of Vectors 85
Problems to Chapter 4

CHAPTER 5. PERPENDICULARITY IN SPACE. DIHEDRAL AND POLYHEDRAL ANGLES

Sec. 31. A Perpendicular to a Plane 91
Sec. 32. An Inclined Line and Its Projection on a Plane. The Distance from a Point toa Plane 92
Sec. 33. The Theorem on Three Perpendiculars 94
Sec. 34. The Angle Between an Inclined Line and a Plane 97
Sec. 35. Dependence Between Parallelism and Perpendicularity of Straight Lines and Planes 98
Sec. 36. The Distance Between Skew Lines 101
Sec. 37. The Triple Product of Three Vectors. The Test for Coplanarity of Three Vectors 102
Sec. 38. Dihedral Angles 104
Sec. 39. The Angle Between Planes. Perpendicular Planes 106
Sec. 40. Orthogonal Projecting 108
Sec. 41. The Length of a Projection of a Line Segment. The Area of a Projection of a Plane Polygon 110
Sec. 42. The Area of a Projection of an Arbitrary Plane 113
Sec. 43. Polyhedral Angles 114
Sec. 44. Worked Problems 118
Problems to Chapter 5 123

CHAPTER 6. THE COORDINATE METHOD

Sec. 45. Rectangular Coordinate System 127
Sec. 46. Expressing a Scalar Product of Vectors in Terms of Their Coordinates. The Equation of a Plane 132
Sec. 47. Expressing the Vector Product of Two Vectors in Terms of Their Coordinates 135
Sec. 48. Expressing the Triple Product of Three Vectors in Terms of Their Coordinates 138
Sec. 49. Worked Problems 138
Problems to Chapter 6. 145

CHAPTER 7. POLYHEDRA, CYLINDERS, CONES

Sec. 50. The Polyhedron Defined 147
Sec. 51. Regular Polyhedra 147
Sec. 52. Eulers Theorem 149
Sec. 53. The Prism. 152
Sec. 54. A Cylindrical Surface. The Cylinder 156
Sec. 55. The Pyramid 159
Sec. 56. A Conical Surface. The Cone 160
Sec. 57. Homothety in бресе 163
Sec. 58. Properties of Parallel Sections of a Cone (Pyramid). Frustums of a Cone (Pyramid) 165
Sec. 59. Sections of Polyhedra 167
Sec. 60. Worked Problems 169

Problems to Chapter 7 174

CHAPTER 8. THE BALL

Sec. 61. A Sphere and a Ball Defined. A Sphere and a Ball Cut by a Plane 181
Sec. 62. A Tangent Plane. 182
Sec. 63. The Concept of a Spherical Triangle 183
Sec. 64. Worked Problems 485
Problems to Chapter 8. 189

CHAPTER 9. MEASURING VOLUMES

Sec. 65. General Properties of Volumes 191
Sec. 66. The Volume of a Rectangular Parallelepiped 192
Sec. 67. The Volume of a Right Cylindrical Solid 194
Sec. 68. The Volume of an Oblique Cylindrical Solid 195
Sec. 69. The General Formula for Computing the Volume of a Figure Using the Areas of Cross-Sections 196
Sec. 70. The Formulas for Computing the Volume of a Cone, a Ball and Its Parts. Simpson’s Formula 199
Sec. 71. Worked Problems 204

Problems to Chapter 9 209

CHAPTER 10. THE AREA OF SURFACE

Sec. 72. The Area of the Surface of a Polyhedron 215
Sec. 73. The Area of an Arbitrary Surface 218
Sec. 74. The Area of the Surface of a Right Circular Cylinder, a Circular Cone, and a Ball 219
Sec. 75. Worked Problems 223

Problems to Chapter 10 226 