CONTENT
Table 1. Definitions, characteristics and properties geometric shapes and relationships 4
I. Planimetry
Table 2. Axioms of planimetry 5
Table 3. Angles 6
Table 4. Parallel lines. Perpendicular lines. Perpendicular to line 7
Table 5. Properties of the sides and angles of a triangle 8
Table 6. Equality of triangles 9
Table 7. Median of triangle 10
Table 8. Bisector of triangle 10
Table 9. Height of triangle 11
Table 10. Middle line triangle 11
Table 11. Relationship between the elements of a right triangle 12
Table 12. Relationship between sides and angles in arbitrary triangle 12
Table 13. Transformation of shapes. Movement 13
Table 14. Similarity transformation 14
Table 15. Similarity of triangles 15
Table 16. Parallelogram and its types 16
Table 17. Trapezoid 18
Table 18. Circle, chords and arcs 19
Table 19. Circumference. Tangents and secants. 20
Table 20. Mutual position straight line and circle. The relative position of two circles 21
Table 21, Common tangents of two circles 22
Table 22. Angles in a circle 23
Table 23. Length of the circle and its parts. The area of ​​a circle and its parts is 24
Table 24, Inscribed and circumscribed polygons. Inscribed and circumscribed quadrilaterals. Rectangle. Trapezoid and rhombus. Square 25
Table 25. A circle circumscribed about a triangle and a circle inscribed in a triangle 26
Table 26. Circles described and inscribed in regular polygons 27
Table 27. Areas of triangles 27
Table 28. Areas of quadrilaterals 28
Table 29. Introduction of unknowns when solving calculation problems 29
Table 30. Using the area method when solving problems 30
Table 31. Using an auxiliary circle when solving problems 31
II. Stereometry
Table 32. Problems related to circumscribed or inscribed circles 32
Table 33. Some useful theorems 33
Table 34. Axioms of stereometry 34
Table 35 Parallelism of straight line and plane 34
Table 36. Parallelism of planes 35
Table 37. Image of spatial figures on a plane 36
Table 38. Perpendicularity of a straight line and a plane 37
Table 39. Perpendicular and inclined 38
Table 40. Theorem of three perpendiculars 39
Table 41. Perpendicularity of two planes 39
Table 42. Angles in space 40
Table 43. Distances in space 42
Table 44. Geometric places points (GMT) 43
Table 45. Prism 44
Table 46. Straight prism 45
Table 47. Parallelepiped 46
Table 48. Pyramid 47
Table 49. Correct pyramid - 48
Table 50. Position of height in some types of pyramids 49
Table 51. Truncated pyramid 51
Table 52. Regular polyhedra 52
Table 53. Cylinder 53
Table 54, Sections of a cylinder by planes 54
Table 55. Cone 55
Table 56. Sections of a cone by planes 56
Table 57. Truncated cone 57
Table 58. Sphere and ball 58
Table 59. Section of a ball by plane 58
Table 60. Plane and line tangent to a ball (sphere) 59
Table 61. Sphere circumscribed around prism 60
Table 62. Sphere inscribed in prism 61
Table 63. Ball described near pyramid 62
Table 64. Ball described about rectangular parallelepiped and correct quadrangular pyramid 63
Table 65. Ball inscribed in pyramid 65
Table 66. Solution of stereometric problems on a combination of bodies of revolution 66
Table 67. Finding the distances between crossing lines 67
Table 68. Finding angles between intersecting lines 69
Table 69. Solving stereometric calculation problems 70
Table 70. Solving problems on constructing sections of polyhedra 71
III. Coordinates and vectors
Table 71. Cartesian coordinates 74
Table 72. Vectors 75
Table 73. Operations on vectors 76
Table 74. Vector decomposition 77
Table 75. Translation of geometric facts into vector language and vector relationships into geometric language 78
Table 76. Using coordinates and vectors when solving problems 79

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_1.jpg" alt=">7th grade. Problem solving."> 7 класс. Решение задач. "Прямоугольный треугольник"!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_2.jpg" alt=">8 9 10 11 13 14 15 16 17 20 21 22 23 24">8 9 10 11 13 14 15 16 17 20 21 22 23 24 26 1 2 3 4 5 6 12 19 25 7 Properties of right triangles. Test tasks theoretical knowledge. ... according to ready-made drawings ... according to ready-made drawings Signs of equality of a rectangular pipeline. 27 28 30 29 31 32 18 33

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_3.jpg" alt=">Right triangle. A B C C a t e t K a i.e."> Прямоугольный треугольник. А В С К а т е т К а т е т Г и п о т е н у з а!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_4.jpg" alt=">Property of a right triangle. A B C In a right triangle, the sum of the acute angles equal to"> Свойство прямоугольного треугольника. А В С В прямоугольном треугольнике сумма острых углов равна 900. 1!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_5.jpg" alt=">Property of a right triangle. A B C In a right triangle, the leg lying against"> Свойство прямоугольного треугольника. А В С В прямоугольном треугольнике катет, лежащий против угла в 300, !} equal to half hypotenuse 300 2

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_6.jpg" alt=">Property of a right triangle. A B C In a right triangle, the leg equal half"> Свойство прямоугольного треугольника. А В С В прямоугольном треугольнике катет, равный половине гипотенузы лежит против угла в 300. 300 3!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_7.jpg" alt=">Signs of equality of right triangles. A B C If the legs of one right triangle respectively equal"> Признаки равенства прямоугольных треугольников. А В С Если катеты одного прямоугольного треугольника соответственно равны катетам другого, то такие треугольники равны. 1 А В С!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_8.jpg" alt=">Signs of equality of right triangles. A B C If a leg and adjacent to it's spicy"> Признаки равенства прямоугольных треугольников. А В С Если катет и прилежащий к нему острый угол одного прямоугольного треугольника соответственно равны катету и прилежащему к нему !} sharp corner other, then such triangles are congruent. 2 A B C

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_9.jpg" alt=">Signs for equality of right triangles. A B C If the hypotenuse and an acute angle one rectangular"> Признаки равенства прямоугольных треугольников. А В С Если гипотенуза и острый угол одного прямоугольного треугольника соответственно равны гипотенузе и острому углу другого, то такие треугольники равны. 3 А В С!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_10.jpg" alt=">Signs of equality of right triangles. A B C If the hypotenuse and leg are the same rectangular"> Признаки равенства прямоугольных треугольников. А В С Если гипотенуза и катет одного прямоугольного треугольника соответственно равны гипотенузе и катету другого, то такие треугольники равны. 4 А В С!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_11.jpg" alt=">A B C A triangle is called isosceles if its two sides are equal. AB = AC"> А В С Треугольник называется равнобедренном если две его стороны равны. АВ = АС !} Isosceles triangle.

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_12.jpg" alt=">A M B K C N Angles at the base. Median, height , bisector. In an isosceles"> А М В К С N Углы при основании. Медиана, высота, биссектриса. В равнобедренном треугольнике углы при основании равны. В равнобедренном тр-ке биссектриса, проведённая к основанию, является медианой и высотой. Свойства равнобедренного треугольника.!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_13.jpg" alt=">A B C External corner triangle equal to the sum two angles of a triangle not adjacent to "> A B C An external angle of a triangle is equal to the sum of two angles of a triangle not adjacent to it. D An external angle of a triangle.

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_14.jpg" alt=">Property of the median drawn from a vertex right angle. A B C In a right triangle "> Property of the median drawn from the vertex of a right angle. A B C In a right triangle, the median drawn from the vertex of a right angle is equal to half the hypotenuse. M

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_15.jpg" alt=">Sign of a right triangle. A B C If the median of the triangle is equal to half the side , To"> Признак прямоугольного треугольника. А В С Если медиана треугольника равна половине стороны, к которой она проведена, то этот треугольник прямоугольный. M!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_16.jpg" alt=">1. Answer Hint Property of a right triangle 370 A B C In a right triangle"> 1. Ответ Подсказка Свойство прямоугольного треугольника 370 А В С В прямоугольном треугольнике сумма острых углов равна 900.!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_17.jpg" alt=">2. Answer Hint (3) Property of an isosceles triangle A B C Isosceles triangle"> 2. Ответ Подсказка (3) Свойство равнобедренного треугольника А В С Равнобедренный треугольник Свойство прямоугольного треугольника!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_18.jpg" alt=">3. Answer Hint (2) Property of a right triangle A B C 2x">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_19.jpg" alt=">4. Answer Hint (2) Property of a right triangle A B C 300 4">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_20.jpg" alt=">5. Answer Hint (2) Property of a right triangle A B C 1200 13"> 5. Ответ Подсказка (2) Свойство прямоугольного треугольника А В С 1200 13 Внешний угол треугольника D!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_21.jpg" alt=">6. Answer Hint (2) Property of right triangle A B C 8.4 Property"> 6. Ответ Подсказка (2) Свойство прямоугольного треугольника А В С 8,4 Свойство прямоугольного треугольника 4,2!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_22.jpg" alt=">7. Answer Hint (3) Properties of an isosceles triangle A B C 450 8"> 7. Ответ Подсказка (3) Свойства равнобедренного треугольника А В С 450 8 Свойство прямоугольного треугольника D Свойство медианы…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_23.jpg" alt=">8. Answer Hint (2) Property of a right triangle R E S 1500 9"> 8. Ответ Подсказка (2) Свойство прямоугольного треугольника Р Е С 1500 9 Внешний угол треугольника К!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_24.jpg" alt=">9. Answer Hint (2) Property of a right triangle A D C 250 Sign"> 9. Ответ Подсказка (2) Свойство прямоугольного треугольника A D С 250 Признак прямоугольного треугольника B!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_25.jpg" alt=">It is necessary to write down the condition of the problem using the drawing and answer the question posed. In"> Необходимо по рисунку записать условие задачи и ответить на поставленный вопрос. В задачах подсказки отсутствуют. 11 12 13 10 14 15 Решение задач по готовым чертежам. 16 17!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_26.jpg" alt=">10. Answer A B C Find: 700 ?">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_27.jpg" alt=">11. Answer A B C Find the angles of the triangle. 15.2 cm D 7.6cm">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_28.jpg" alt=">12. Answer A B C Find: AH H 4cm 1200">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_29.jpg" alt=">13. Answer 300 A B C Find: AE 600 7 E">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_30.jpg" alt=">14. Answer A B C 7 D 7 3.5">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_31.jpg" alt=">15. Answer 20 A B C Find: CK 1500 K">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_32.jpg" alt=">16. Answer 700 A B C M">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_33.jpg" alt=">17. Answer 16 A B C K 8 D ?">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_34.jpg" alt=">18. Prove the equality of triangles. A B D Conclusion C Hint Sign of equality rectangular"> 18. Доказать равенство треугольников. А B D Вывод С Подсказка Признак равенства прямоугольных треугольников По гипотенузе и острому углу…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_35.jpg" alt=">19. Prove the equality of triangles. A B D Conclusion C Hint Sign of equality rectangular"> 19. Доказать равенство треугольников. А B D Вывод С Подсказка Признак равенства прямоугольных треугольников По катету и прилежащему к нему острому углу…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_36.jpg" alt=">20. Prove the equality of triangles. A B D Conclusion C Hint Sign of equality rectangular"> 20. Доказать равенство треугольников. А B D Вывод С Подсказка Признак равенства прямоугольных треугольников По катетам…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_37.jpg" alt=">21. Prove the equality of triangles. A B D Conclusion C Hint Sign of equality rectangular"> 21. Доказать равенство треугольников. А B D Вывод С Подсказка Признак равенства прямоугольных треугольников По катету и гипотенузе… О!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_38.jpg" alt=">22. A B Conclusion D Hint (2) Sign of equality of rectangular triangles C Consider"> 22. А B Вывод D Подсказка (2) Признак равенства прямоугольных треугольников C Рассмотреть треугольники BD - биссектриса!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_39.jpg" alt=">23. A K B Conclusion M Hint (4) Sign of equality of rectangular triangles N"> 23. А K B Вывод M Подсказка (4) Признак равенства прямоугольных треугольников N C !} Additional construction Consider triangles Properties of an isosceles triangle MC - median

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_40.jpg" alt=">It is necessary to write down the condition of the problem using the drawing and answer the question posed. In"> Необходимо по рисунку записать условие задачи и ответить на поставленный вопрос. В задачах подсказки отсутствуют. 25 26 27 24 Решение задач по готовым чертежам.!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_41.jpg" alt=">24. Conclusion 1 A B C K Prove: ∆ABC = ∆DKP 2 D P"> 24. Вывод 1 А В С K Доказать: ∆ABC = ∆DKP 2 D P По гипотенузе и острому углу…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_42.jpg" alt=">25. Conclusion A B C P Along the sides...">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_43.jpg" alt=">26. A B C P M N">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_44.jpg" alt=">27. A K B M N L C">!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_45.jpg" alt=">Test tasks to test theoretical knowledge. In tasks 28 and 29 you must"> Тестовые задания на проверку теоретических знаний. В заданиях 28 и 29 необходимо выбрать верный ответ. Объяснить. В 30 и 31 заданиях необходимо найти !} degree measures angles 1, 2 and 3. In tasks 32 and 33, find the degree measures of angles 1, 2, 3, 4 and 5. indicate congruent right triangles, explain the answer. 29 30 31 28 32 33

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_46.jpg" alt=">28. 600 A B C Is ∆ABC rectangular? 300 NO YES Think! Why?"> 28. 600 А В С Является ли ∆ABC прямоугольным? 300 НЕТ ДА Подумай! Почему? В прямоугольном треугольнике сумма острых углов равна 900.!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_47.jpg" alt=">29. A B C Is ∆ABC rectangular? 450 NO YES Think about it!"> 29. А В С Является ли ∆ABC прямоугольным? 450 НЕТ ДА Подумай! Почему? По определению, треугольник равнобедренный – углы при основании равны. В прямоугольном треугольнике сумма острых углов равна 900.!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_48.jpg" alt=">30. 500 A B C Find the degree measures of angles 1, 2 and 3 1"> 30. 500 А В С Найдите градусные меры углов 1, 2 и 3 1 500, 400, 500 400, 500, 400 Подумай! Молодец! 2 3 300, 600, 300 450, 450, 450!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_49.jpg" alt=">31. 400 D B C Find the degree measures of angles 1, 2 and 3 1"> 31. 400 D В С Найдите градусные меры углов 1, 2 и 3 1 400, 500, 400 500, 400, 500 Подумай! Молодец! 2 3 300, 600, 300 450, 450, 450!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_50.jpg" alt=">32. 400 A B C Find the degree measures of angles 1.2 ,3,4,5. 1 500, 650, 650,"> 32. 400 А В С Найдите градусные меры углов 1,2,3,4,5. 1 500, 650, 650, 250, 250 250, 250, 650, 650, 500 Подумай! Молодец! 2 3 300, 300, 600, 600, 300 450, 450, 450, 450, 550 D F 4 5!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_51.jpg" alt=">32. 400 A B C Specify equal rectangular pipes. 1 ∆FDB = ∆ADB Think about it!"> 32. 400 А В С Укажите равные прямоугольные тр-ки. 1 ∆FDB = ∆ADB Подумай! Молодец! 2 3 ∆DAB = ∆CAB ∆FDB = ∆ABC D F 4 5 Почему? По гипотенузе и острому углу…!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_52.jpg" alt=">33. 500 A B C Find the degree measures of angles 1.2 ,3,4,5. 1 250, 250, 650,"> 33. 500 А В С Найдите градусные меры углов 1,2,3,4,5. 1 250, 250, 650, 700, 400 200, 200, 700, 700, 400 Подумай! Молодец! 2 3 300, 300, 600, 600, 300 450, 450, 450, 450, 550 D F 4 5!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_53.jpg" alt=">33. Specify equal rectangular tr-ki. ∆FDB = ∆ ADB Think about it! ∆DAB = ∆CAB ∆FDB"> 33. Укажите равные прямоугольные тр-ки. ∆FDB = ∆ADB Подумай! Молодец! ∆DAB = ∆CAB ∆FDB = ∆ABC Почему? По гипотенузе и острому углу… 500 А В С 1 2 3 D F 4 5!}

Src="https://present5.com/presentacii-2/20171208%5C14538-geo7_prjam_tr.ppt%5C14538-geo7_prjam_tr_54.jpg" alt=">Resources used: N.F. Gavrilova " Lesson-based developments in geometry, grade 7."> Resources used: N.F. Gavrilova "Lesson developments in geometry, grade 7. Universal edition. Moscow "Waco" 2006. 2. Picture: http://fotki.yandex.ru/users/val- pryanikova/view/543773/?page=3

Geometry. Tasks and exercises on ready-made drawings. 7-9 grades. Rabinovich E.M.

M.: 2016. - 60 p.

The proposed manual is an additional collection of geometry problems for students in grades 7-9 and is focused on the textbook by A.V. Pogorelov "Geometry 7-11". The manual is compiled in the form of tables and contains more than 400 problems and exercises in geometry for students in grades 7-9. The tasks of each table correspond to a specific topic school course planimetry.

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Content
Preface 3
7th grade
7.1. Measuring segments 5
7.2. Angle measurement 6
7.3. Adjacent angles 7
7.4. Related and vertical angles 8
7.5. Tests for equivalence of triangles 9
7.6. Isosceles triangle 10
7.7. Signs of parallel lines 11
7.8. Signs of parallel lines 12
7.9. Sum of triangle angles 13
7.10. Sum of triangle angles 14
7.11. Right Triangle 15
7.12. Circle 16
7.13. Circle and tangent 17
8th grade
8.1. Definition and characteristics of a parallelogram 18
8.2. Definition and characteristics of a parallelogram 19
8.3. Properties of a parallelogram 20
8.4. Properties of a parallelogram 21
8.5. Properties of a parallelogram 22
8.6. Trapezoid 23
8.7. Thales' Theorem 24
8.8. Midline of triangle and trapezoid 25
8.9. Triangle inequality 26
8.10. Solving Right Triangles 27
8.11. Pythagorean theorem. Proportional segments in a right triangle 28
8.12. Cartesian coordinates on a plane 29
8.13. Cartesian coordinates on plane 30
8.14. Symmetry about point 31
8.15. Symmetry about a straight line 32
8.16. Vectors on a plane 33
8.15. Vectors on a plane 34
9th grade
9.1. Similar triangles 35
9.2. The first sign of similarity of triangles 36
9.3. The second and third signs of similarity of triangles 37
9.4. Inscribed angles 38
9.5. Inscribed angles. Angle between tangent and chord 39
9.6. Solving triangles 40
9.7. Solving triangles 41
9.8. Regular polygons 42
9.9. Area of ​​triangle 43
9.10. Area of ​​quadrilateral 44
9.11. Area of ​​quadrilateral 45
9.12. Areas of figures 46
9.13. Areas of figures 47
9.14. Area of ​​a circle and its parts 48
Answers. Directions. Solutions 49
List of used literature 58

IN textbook logically ordered and systematized basic and additional information from the school geometry course (planimetry and stereometry), which allow you to solve the most complex geometric problems offered at graduation and entrance exams(during the state final certification or in Unified State Exam assignments in mathematics).

Method of traces for constructing sections.
Contents of the method. First, they construct the line of intersection of the cutting plane with the plane of some face (the trace of the cutting plane on this face), and then find the points of intersection of the cutting plane with the corresponding edges of the polyhedron (or with their extensions). Sometimes for this it is necessary to consider certain auxiliary planes, for which a trace of the cutting plane is also constructed (or a trace of this auxiliary plane on the plane of some face). To obtain the trace (that is, line b) of plane β on plane a (see figure), it is enough to find the points of intersection of two straight lines of plane β with plane a (since two points, for example A and C, uniquely define line b).

It must be remembered that the point of intersection of any line a of the plane β with the plane a always lies on the trace of the plane β on the plane a (that is, on the line b).
If we consider parallel (or central) projection, then in order to find the point of intersection of a line with the projection plane, it is enough to find the point of intersection of the line with its projection onto this plane.

CONTENT
Table 1. Definitions, characteristics and properties of geometric figures and relationships 4
I. Planimetry
Table 2. Axioms of planimetry 5
Table 3. Angles 6
Table 4. Parallel lines. Perpendicular lines. Perpendicular to line 7
Table 5. Properties of the sides and angles of a triangle 8
Table 6. Equality of triangles 9
Table 7. Median of triangle 10
Table 8. Bisector of triangle 10
Table 9. Height of triangle 11
Table 10. Middle line of triangle 11
Table 11. Relationship between the elements of a right triangle 12
Table 12. The relationship between sides and angles in an arbitrary triangle 12
Table 13. Transformation of shapes. Movement 13
Table 14. Similarity transformation 14
Table 15. Similarity of triangles 15
Table 16. Parallelogram and its types 16
Table 17. Trapezoid 18
Table 18. Circle, chords and arcs 19
Table 19. Circumference. Tangents and secants 20
Table 20. The relative position of the straight line and the circle. The relative position of two circles 21
Table 21, Common tangents of two circles 22
Table 22. Angles in a circle 23
Table 23. Length of the circle and its parts. The area of ​​a circle and its parts is 24
Table 24, Inscribed and circumscribed polygons. Inscribed and circumscribed quadrilaterals. Rectangle. Trapezoid and rhombus. Square 25
Table 25. A circle circumscribed about a triangle and a circle inscribed in a triangle 26
Table 26. Circles circumscribed and inscribed in regular polygons 27
Table 27. Areas of triangles 27
Table 28. Areas of quadrilaterals 28
Table 29. Introduction of unknowns when solving calculation problems 29
Table 30. Using the area method when solving problems 30
Table 31. Using an auxiliary circle when solving problems 31
II. Stereometry
Table 32. Problems related to circumscribed or inscribed circles 32
Table 33. Some useful theorems 33
Table 34. Axioms of stereometry 34
Table 35 Parallelism of straight line and plane 34
Table 36. Parallelism of planes 35
Table 37. Image of spatial figures on a plane 36
Table 38. Perpendicularity of a straight line and a plane 37
Table 39. Perpendicular and inclined 38
Table 40. Theorem of three perpendiculars 39
Table 41. Perpendicularity of two planes 39
Table 42. Angles in space 40
Table 43. Distances in space 42
Table 44. Geometric locations of points (GMT) 43
Table 45. Prism 44
Table 46. Straight prism 45
Table 47. Parallelepiped 46
Table 48. Pyramid 47
Table 49. Regular pyramid - 48
Table 50. Position of height in some types of pyramids 49
Table 51. Truncated pyramid 51
Table 52. Regular polyhedra 52
Table 53. Cylinder 53
Table 54, Sections of a cylinder by planes 54
Table 55. Cone 55
Table 56. Sections of a cone by planes 56
Table 57. Truncated cone 57
Table 58. Sphere and ball 58
Table 59. Section of a ball by plane 58
Table 60. Plane and line tangent to a ball (sphere) 59
Table 61. Sphere circumscribed around prism 60
Table 62. Sphere inscribed in prism 61
Table 63. Ball described near pyramid 62
Table 64. A ball circumscribed about a rectangular parallelepiped and a regular quadrangular pyramid 63
Table 65. Ball inscribed in pyramid 65
Table 66. Solution of stereometric problems on a combination of bodies of revolution 66
Table 67. Finding the distances between crossing lines 67
Table 68. Finding angles between intersecting lines 69
Table 69. Solving stereometric calculation problems 70
Table 70. Solving problems on constructing sections of polyhedra 71
III. Coordinates and vectors
Table 71. Cartesian coordinates 74
Table 72. Vectors 75
Table 73. Operations on vectors 76
Table 74. Vector decomposition 77
Table 75. Translation of geometric facts into vector language and vector relationships into geometric language 78
Table 76. Using coordinates and vectors when solving problems 79.

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