А.Д.Александров, А.Л.Вернер, Ж.Нұрпейіс. Геометрия 7-9 сынып, 2003

Dear Grade 7 students!

Chapter I. Foundations of geometry

§ 1. What geometry studies

1.1. Geometric figures

1.2. First problems of geometry. Constructions

1.3. On the origin of geometry

§ 2. Segments

2.1. Concept of an axiom

2.2. Axiom of drawing a segment. Interior points of a segment

2.3. Extending a segment

2.4. Ray

2.5. Line

§ 3. Operations with segments

3.1. Equality of segments. Constructing segments

3.2. Comparing segments

3.3. Adding segments

3.4. Subtracting segments

3.5. Dividing a segment into equal parts

§ 4. Length of a segment. Distance

4.1. Concept of segment length. Distance

4.2. Numerical value of segment length

4.3. Measuring the length of a segment

4.4. Relation between operations on segments and their lengths

§ 5. Circumference and circle

5.1. Definitions of circumference and circle

5.2. Parts of a circle

5.3. Constructions with compass and ruler

§ 6. Angles

6.1. Definition of an angle. Adjacent angles

6.2. Equality of angles. Constructing an angle. Property of equal angles

6.3. Criterion for equality of angles. Constructing an angle equal to a given one

6.4. Comparing angles. Right angle. Angle bisector

6.5. Constructing an angle bisector. Constructing the midpoint of a segment

6.6. Vertical angles. Mutually perpendicular lines

6.7. Operations with angles

6.8. Measuring angles

§ 7. Parallelism of lines

7.1. Parallel lines and segments

7.2. Axiom of parallelism

7.3. Axioms and theorems

7.4. Proof by contradiction method

7.5. Equality of angles formed by parallel lines and a transversal

Chapter II. Triangles

§ 8. First theorems about triangles

8.1. Vertices, sides and angles of a triangle

8.2. Sum of the angles of a triangle

8.3. Classification of triangles

8.4. Medians, angle bisectors and altitudes of a triangle

8.5. Concept of congruent figures. Congruence of triangles

8.6. First criterion for congruence of triangles

8.7. Second criterion for congruence of triangles

§ 9. Isosceles triangle

9.1. Definition and properties of an isosceles triangle

9.2. Criterion for an isosceles triangle

9.3. Converse statements

9.4. Perpendicular bisector of a segment

9.5. Criteria for congruence of right triangles

§ 10. Relations between sides and angles of a triangle

10.1. Comparing sides and angles of a triangle

10.2. Distance from a point to a figure

10.3. Distance between figures

Chapter III. Circumference

§ 11. Chords and tangents

11.1. Properties of chords. Symmetry of a circle

11.2. Classification of relative positions of a line and a circle

11.3. Tangency of a line and a circle

§ 12. Triangles and circles

12.1. Circumcircle of a triangle

12.2. Incircle of a triangle

Chapter IV. Quadrilaterals with parallel sides

§ 13. Quadrilaterals

13.1. Quadrilateral and its elements

13.2. Convex and non-convex quadrilaterals

13.3. Convex figures

13.4. Trapezoid

§ 14. Parallelogram

14.1. Definition and properties of a parallelogram

14.2. Criteria for a parallelogram

14.3. Rectangle

14.4. Rhombus. Square

§ 15. Thales' theorem

15.1. Midline of a triangle

15.2. Midline of a trapezoid

15.3. Thales' theorem

15.4. Thales

15.5. Intersection points of altitudes and medians of triangles

15.6. Dividing segments into equal parts

Chapter V. Area of polygonal figures

§ 16. Area of a rectangle

16.1. Polygonal figures

16.2. Concept of area. Calculating area

16.3. Area of a rectangle

§ 17. Pythagorean theorem

17.1. An important theorem of geometry

17.2. Pythagoras

17.3. Algebraic applications of the Pythagorean theorem

17.4. Calculations in triangles. Square root

17.5. Oblique lines and projections

17.6. Triangle inequality

§ 18. Calculating the area of a polygonal figure

18.1. Area of a triangle

18.2. Area of a trapezoid and parallelogram

Chapter VI. Foundations of trigonometry

§ 19. Sine and its application

19.1. Ratio of perpendicular and oblique line

19.2. Definition of sine

19.3. Properties of sine and its graph

19.4. Solving right triangles

19.5. Calculating the area of a triangle

19.6. Law of sines

§ 20. Cosine and its application

20.1. Definition of cosine

20.2. Fundamental trigonometric identity

20.3. Cosine of an acute angle of a right triangle

20.4. Properties of cosine

20.5. Law of cosines

§ 21. Tangent and its application

21.1. Tangent

21.2. Overview of the history of trigonometry

Chapter VII. Vectors

§ 22. Concept of a vector

22.1. Scalar and vector quantities

22.2. Directed segments

22.3. Codirectional rays and vectors

22.4. Equality of vectors

22.5. Angle between vectors

§ 23. Addition of vectors

23.1. Triangle rule. Definition of vector addition

23.2. Parallelogram rule

23.3. Properties of vector addition

23.4. Subtraction of vectors. Opposite vector

23.5. Decomposition of a vector into components

§ 24. Multiplication of a vector by a number

24.1. Definition of multiplying a vector by a number

24.2. Criterion for collinearity of vectors

24.3. Distributive laws of multiplying a vector by a number

24.4. Vector algebra and vector field

§ 25. Coordinates of a vector

25.1. Vectors on a coordinate axis

25.2. Vectors on the coordinate plane

25.3. Calculating vector coordinates

25.4. Operations on vectors given by coordinates

§ 26. Dot product of vectors

26.1. Dot product of vectors

26.2. Expressing the dot product of vectors through coordinates

26.3. Properties of the dot product

26.4. Application of the dot product

§ 27. Vectors and coordinates

27.1. Concept of the equation of a figure

27.2. Equation of a circle

27.3. Defining figures by inequalities

27.4. Coordinate method

27.5. Apollonius circle

27.6. Equation of a line

Chapter VIII. Transformations

§ 28. Motion and congruence of figures

28.1. Transformations of figures

28.2. Motions of figures

28.3. Properties of motion

28.4. Congruence of figures

§ 29. Types of motions

29.1. Translation

29.2. Method of translation

29.3. Reflection in a line (axial symmetry)

29.4. Method of symmetry

29.5. Rotation

29.6. Method of rotation

29.7. On congruence of triangles

§ 30. Symmetry of figures

30.1. Types of symmetry

30.2. Symmetry of unbounded figures

30.3. On symmetry

§ 31. Similarity transformation

31.1. Definition of a similarity transformation

31.2. Homothety

31.3. Properties of homothety

31.4. Properties of similarity

31.5. Criteria for similarity of triangles

31.6. Method of similarity

31.7. Similarities in constructing images of plane figures

Chapter IX. Polygons and circles

§ 32. Polygons

32.1. Broken line

32.2. Polygons

32.3. Circumcircle and incircle of a polygon

32.4. Convex polygons

§ 33. Regular polygons

33.1. Definition of a regular polygon

33.2. Center of a regular polygon

33.3. Symmetry of a regular polygon

33.4. Constructing regular polygons

§ 34. Measurements in a circle

34.1. Degree measure of an arc

34.2. Inscribed angle

34.3. Length of a curve

34.4. Circumference

34.5. Length of an arc

34.6. About the number π

34.7. Area of a figure

34.8. Area of a circle and sector

Summary. Foundations of plane geometry

Answers

Table of trigonometric functions

Name index