Presentation of the basic concepts and axioms of stereometry. Presentation of the "axiom of stereometry". first, by strong influence

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Methodological development Savchenko E.M. MOU gymnasium No. 1, Polyarnye Zori, Murmansk region
Subject of stereometry
Axioms of stereometry
Geometry Grade 10

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Planimetry
Stereometry
Explore properties geometric shapes on surface
Explores the properties of figures in space
Translated from Greek, the word "geometry" means "surveying", "geo" - in Greek, the earth, "metreo" - to measure
The word "stereometry" comes from the Greek words "stereos" volumetric, spatial, "metreo" - to measure

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Planimetry
Stereometry
Along with these figures, we will consider geometric bodies and their surfaces. For example, polyhedrons. Cube, parallelepiped, prism, pyramid. bodies of revolution. Ball, sphere, cylinder, cone.
Basic shapes: point, line
Basic shapes: point, line, plane
Other figures: segment, ray, triangle, square, rhombus, parallelogram, trapezoid, rectangle, convex and non-convex n-gons, circle, circle, arc, etc.

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To designate points, we use uppercase Latin letters.
For the designation of straight lines, we use lowercase Latin letters.
Or we denote the line with two capital Latin letters.

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The planes will be denoted by Greek letters.
In the figures, the planes are indicated as parallelograms. The plane as a geometric figure should be imagined as extending indefinitely in all directions.

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When studying spatial figures, in particular geometric bodies, their flat images in the drawing are used. The image of a spatial figure is its projection onto a particular plane. The same figure admits different representations.
Various images of a cone

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Stereometry is widely used in construction, architecture, mechanical engineering, geodesy, and in many other areas of science and technology.
When designing this machine, it was important to obtain such a shape that air resistance was minimal during movement.

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Opera House in Sydney
Danish architect Jorn Utzon was inspired by the sight of sails.

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Eiffel Tower Paris, Champ de Mars
Engineer Gustave Eiffel found an unusual shape for his project. The Eiffel Tower is very stable: a strong wind deflects its top by only 10-12 cm. In the heat, from uneven heating by the sun's rays, it can deviate by 18 cm.

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18,000 iron parts held together with 2,500,000 rivets

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The original idea for the construction of the tower was found by the architects L. Batalov and D. Burdin with the participation of the designer N. Nikitin. Metal cables are stretched inside cylindrical concrete blocks. This design is extremely stable.
The theoretical deviation of the top of the tower at maximum design wind speeds is about 12 meters.

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The main properties of points, lines and planes are expressed in axioms. Of the many axioms, we will formulate only three.
A1. Through any three points that do not lie on the same straight line, there passes a plane, and moreover, only one.
Illustration for axiom A1: a glass plate will lie tightly on three points A, B and C that do not lie on one straight line.
A
B
C

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Illustrations to axiom A1 from life.
The stool with three legs will always fit perfectly on the floor and will not wobble. A four legged stool has stability issues if the legs of the chair are not the same length. The stool swings, that is, it rests on three legs, and the fourth leg (the fourth "point") does not lie in the plane of the floor, but hangs in the air.
A tripod is often used for video cameras, photography and other devices. The tripod's three legs are stable on any indoor floor, on pavement or directly on the lawn on the street, on the sand on the beach or in the grass in the forest. The three tripod legs will always find a plane.

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ABOUT
A
IN
Construction of right angles on the ground using the simplest device called an eker.
Tripod with eker.

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a
A2. If two points of a line lie in a plane, then all points of the line lie in that plane.
A
B

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The property expressed in axiom A2 is used to check the "flatness" of the drawing ruler. The ruler is applied edge to the flat surface of the table. If the edge of the ruler is even, then it is adjacent to the surface of the table with all its points. If the edge is uneven, then in some places a gap is formed between it and the surface of the table.

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It follows from axiom A2 that if a line does not lie in a given plane, then it has at most one common point with it. If a line and a plane have only one common point, then they are said to intersect.

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a
A3. If two planes have a common point, then they have a common line on which all common points of these planes lie.
In this case, the planes are said to intersect in a straight line.

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A clear illustration of Axiom A3 is the intersection of two adjacent walls, the wall and the ceiling of the classroom.

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A1. Through any three points that do not lie on the same straight line, there passes a plane, and moreover, only one.

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Theorem
Through a line and a point not lying on it passes a plane, and moreover, only one.
M
a

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Some consequences of the axioms.
Theorem
A plane passes through two intersecting lines, and moreover, only one
M
a
b
N

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Training exercises
Name the planes in which the straight lines lie PE MK DB AB EC
P
E
A
B
C
D
M
K

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Training exercises
Name the points of intersection of the line DK with the plane ABC, the line CE with the plane ADB.
P
E
A
B
C
D

Lesson 1: What does stereometry study? Stereometry is a branch of geometry that studies the properties of figures in space. The word "stereometry" comes from the Greek words "stereos" - volumetric, spatial and "metreo" - to measure. Many geometric terms are translated from ancient Greek, because geometry originated in Ancient Greece and developed in philosophical schools.

Lesson 2: The main figures of stereometry. There are various ways of depicting a plane: a plane is depicted as a parallelogram; a plane is denoted by a figure bounded by two parallel lines and two arbitrary curves; the plane is transmitted by a figure of arbitrary shape.

Lesson 3: Spatial figures. The lesson is devoted to preparing for the introduction of the axioms of stereometry. Students are offered the following tasks: 1. Draw a line a, a point A lying on it and a point B not lying on it. 2. Draw a plane and two intersecting lines a and b lying on it. 3. Draw a plane, points A and B lying on it, as well as points C and D located on opposite sides of the plane. 4. Draw a plane and a straight line intersecting it. 5. Draw planes that intersect at right angles.

5th lesson: Signs of parallelism of planes. When studying the axioms of stereometry, we recall the first axioms of planimetry and formulate their spatial analogies. As a result, we obtain the following table: Axiom a DrawingFormulation P1P1 Whatever the line in space, there are points in space that belong to this line, and points that do not belong to it. П2П2 Through any two points of space it is possible to draw a straight line, and moreover, only one.

Lesson 6: Parallel design. Consider the consequences of the axioms: DrawingFormulation Sl. 1 Through a line and a point not lying on it, you can draw a plane, and moreover, only one. If two points of a line belong to a plane, then the whole line belongs to that plane. Through three points that do not lie on the same straight line, it is possible to draw a plane, and moreover, only one.

Image of spatial figures on a plane There are seven lessons on the topic: 1. P Parallel design and its main properties; 2. P Parallel design of plane figures; 3. And Image of spatial figures in a parallel projection; 4. C Section of polyhedra; 5. Z golden ratio; 6. Central design and its properties; 7. And Image of spatial figures in the central projection.

Lesson 1: Parallel design and its main properties. The main properties of parallel projection: 1. a parallel projection of a straight line is a straight line or a point; 2. a parallel projection of a segment is a segment or a point; 3. the ratio of the lengths of the segments lying on the same straight line is preserved (in particular, the middle of the segment in parallel projection goes to the middle of the corresponding segment); 4. parallel projection of two parallel lines are parallel lines, or one line, or two points; 5. the ratio of the lengths of segments lying on parallel lines is preserved in parallel design; 6. if the figure lies in a plane parallel to the projection plane, then its parallel projection onto this plane will be a figure equal to the original one.

Lesson 2: Parallel projections of plane figures. The question of the image of flat figures in parallel design is considered. Students must imagine which figures are parallel projections of polygons and a circle. Find out what properties of polygons are preserved during parallel design. Learn how parallel projections of basic plane figures are built.

The golden section in architecture Famous Russian architects M. Kazakov and V. Bazhenov widely used the golden section in their work. For example, the golden ratio can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the First Clinical Hospital was built in Moscow. Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect works of architecture of V. Bazhenov.

Polyhedra. This course includes the following lessons: 1. Regular polyhedra. Regular polyhedra. 2. Semi-regular polyhedra. Semiregular polyhedra. 3. Star polyhedra. Star polyhedrons. 4. Euler's theorem. Euler's theorem.

Lesson 4: Euler's theorem. One of the most interesting properties of convex polyhedra is described by Euler's theorem. Polyhedron name a Number of vertices (B) Number of edges (P) Number of faces (D) Triangular pyramid 464 Quadrangular prism 8126 Pentagonal bipyramid regular dodecahedron n-gonal pyramid n+12n2n n-gonal prism 2n2n3n3nn+2 First, the students examine the polyhedra known to them and fill in the table. Then the theorem itself is deduced: V-P+G=2

Angles between lines and planes in space. When studying this topic, it is desirable to note that the problem of measuring angles dates back to ancient times. The history of the creation of measuring instruments and methods of measurement should be covered as widely as possible. For this, it is proposed to conduct the following classes: 1. The volume of figures in space. Cylinder volume; The volume of figures in space. Cylinder volume; 2. Cavalieri principle; Cavalieri principle; 3. The volume of the cone; The volume of the cone; 4. The volume of the ball. The volume of the ball.

Lesson 1: The volume of figures in space. Cylinder volume. This lesson deals with the problems of measuring the volumes of spatial figures. The main properties of volume are listed: oothe volume of a figure in space is a non-negative number; oothe volume of a cube with edge 1 is 1; oro-equal figures have equal volumes; ooif figure F is composed of figures F 1 and F 2, then the volume of the figure F is equal to the sum of the volumes of the figures F 1 and F 2.

Stereometry

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Stereometry. Pencil. Geometry. Planimetry. Basic concepts of stereometry. Axioms of stereometry. Axioms. Points of a straight line. Planes. Consequences from the axioms. Intersecting lines. Plane. Determination of body volume. Bodies with equal volumes. Volume cuboid. Prism volumes. Two right triangles. The volume of an inclined prism. Perpendicular section. Polyhedron. Rectangles. Image planes. Parallelepiped. Rectangular parallelepiped. Pyramid. Tetrahedron. Figure. Segments. Truncated pyramid. Octahedron. Dodecahedron. Icosahedron. Cylinders. bodies of revolution. Ball sector. - Stereometry.ppt

Fundamentals of stereometry

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On the teaching of stereometry in the humanities classes. What does stereometry study. Angle between lines in space. Parallelepiped. Fourth quarter. Stereometry. Pythagoras. The main figures of stereometry. Spatial figures. Parallelism of lines and planes. Signs of parallel planes. parallel design. The image of spatial figures on the plane. Parallel design and its main properties. Parallel projections of plane figures. Image of spatial figures. Section of polyhedra. Golden section. The Golden Ratio in Sculpture. Golden section in architecture. - Fundamentals of stereometry.ppt

Subject of stereometry

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Axioms of stereometry. Geometry. The concept of the science of stereometry. visual representations. From the history. Stereometry. Egyptian pyramids. Do you remember the Pythagorean theorem. Pythagoras. Pythagorean theorem. Pentagram. Regular polyhedra. Universe. Philosophical school. Euclid. Spatial representations. undefined concepts. Basic concepts of stereometry. Invisible side. Planimetry. Dots. Directions. Today at the lesson. - Subject of stereometry.ppt

Introduction to solid geometry

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School geometry. Arithmetic. geometric knowledge applied. Geometric knowledge helped. Let's translate into the language of squares. Let's take 6 matches. Plane. Planimetry. Crossword. - Stereometry. Polyhedron. Figures. Bodies. Mobile dwellings of the Indians are called Tipi. Magazine "Quantum". Summing up the lesson. - Introduction to stereometry.ppt

Axioms of geometry

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Axioms of stereometry. To get acquainted with the axioms of stereometry. Planimetry. Dots. You can draw a straight line and only one. Of the three points, only one lies between the other two. Each segment has a certain length. The straight line divides the plane into two half-planes. Each angle has a certain degree measure. You can postpone a segment of a given length and only one. An angle can be plotted on any half-line from the starting point. Triangle. You can draw at most one straight line in a plane. Stereometry. Axioms. Points in space. Different planes have a common point. You can draw a plane, and moreover, only one. - Axioms of Geometry.pptx

Axioms of stereometry

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Axioms of stereometry. 1. Concepts of stereometry 2. Image of a plane 3. Axioms of stereometry 4. Consequences from the axioms of stereometry. The system of axioms of stereometry consists of the axioms of planimetry and three axioms of stereometry. Stereometry is a branch of geometry that studies the properties of figures in space. The picture shows two generally accepted images of the plane. Planes are denoted by small Greek letters: a, b, g, ... There is at least one straight line and at least one plane. The distance from point A to point B is equal to the distance from point B to point A: AB=BA. Consequences from the axioms of stereometry. - Axioms of stereometry.ppt

Axioms of stereometry Grade 10

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Axioms of stereometry. A, B, C? one straight line A, B, C? ? ? is the only plane. In any plane of space, all the axioms and theorems of planimetry are valid. Consequences from the axioms of stereometry. A plane passes through two intersecting straight lines, and moreover, only one. 1. Do they lie on a plane? points B and C? 2. Does the point D lie on the plane (MOB)? 3. Name the line of intersection of the planes (MOB) and (ADO). Name the different ways to calculate the area of ​​a rhombus. The problem is the intersection of two planes ABCDA1B1C1D1 - a cube, K belongs to DD1, DK=KD1. Provide answers to the questions below with the necessary justifications. - Axioms of stereometry Grade 10.ppt

Basic axioms of stereometry

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Consequences from the axioms of stereometry

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Geometry slides. Axioms of stereometry and some consequences from them. Stereometry. Planimetry. Geometry section. Axioms of stereometry. Various planes. Various lines. Axioms of planimetry. Draw a cube image. Explain the answer. The existence of a plane. Explanation of new material. oral work. Find the line of intersection of the planes. What planes does the point belong to. Plane. Proof. Cube elements. Intersection of a line with a plane. Plane and line. How many edges pass through one, two, three, four points. Lines that intersect at a point. - Consequences from the axioms of stereometry.ppt

Spatial figures on the plane

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The image of spatial figures on the plane. The purpose of the lesson. True False. One of two parallel lines intersects the plane. By the plane intersection lemma. Is it true that two non-intersecting lines in space are parallel. Parallel and skew lines do not have common points. If two lines are parallel to some plane, then they are parallel to each other. Lines can be not only parallel, but also intersect. Two planes are crossed by two parallel lines. There are no conditions for fulfilling the sign of parallelism of planes. Gerard Desargues. - Spatial figures on the plane.ppt

Mutual arrangement of lines in space

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Mutual arrangement of lines in space. Crossing straight lines. Introduce the definition of skew lines. Introduce formulations and prove the sign and property of skew lines. Location of lines in space: They lie in the same plane! Given a cube ABCDA1B1C1D1. Are lines AA1 and DD1 parallel? AA1 and CC1? 2. Are AA1 and DC in parallel? Sign of intersecting lines. Given: AB?, CD? ? = C, C AB. Consolidation of the studied theorem: Determine the relative position of the lines AB1 and DC. 2. Indicate the relative position of the DC line and the AA1B1B plane. - Mutual arrangement of lines in space.ppt

Problems in stereometry

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Tasks. Find the volume of the pyramid. Find the volume V of the cylinder. Find the surface area of ​​the polyhedron. Circumference. Find the area of ​​the trapezoid. Find the ordinate of point A. Find the angle of the polyhedron. Find the square of the distance between the vertices. The volume of the sphere and its parts. circular sector. Lead ball diameter. - Tasks on stereometry.pptx

"Tasks in geometry" Grade 11

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Use of ICT. Problem. Project technology. The relevance of the project. Application of presentations. Content. Preface. Polyhedra inscribed in a sphere. Prism. We will answer verbally. A sphere is described near a triangular prism, the center of which lies outside the prism. Combination of sphere and prism. Measurements of a cuboid. A sphere with a radius of 5 cm is described near a regular hexagonal prism. Pyramid. Near any triangular pyramid, a sphere can be described. A combination of a sphere and a pyramid. The base of the triangular pyramid is right triangle. Let's build an axial section. Polyhedra circumscribed about a ball. - "Tasks in geometry" Grade 11.ppt

Plane equation

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Linear Algebra and Analytic Geometry. Subject: Plane. Plane. CONCLUSIONS: 1) The plane is a surface of the first order. Investigation of the general equation of the plane. Equation (3) is called the equation of a plane in segments. ?1: by+cz = 0 (intersection with the oyz plane) ?2: ax+by = 0 (intersection with the oxy plane). A) the plane cuts off segments a and b on the axes ox and oy, respectively, and is parallel to the axis oz; A) the plane cuts off the segment a on the ox axis and is parallel to the axes oy and oz (i.e., parallel to the oyz plane); Comment. Let the plane? does not go through O(0;0;0). 2. Other forms of writing the equation of the plane. - Plane Equation.pps

Planes in space

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Analytic geometry. Part 2 Geometry in space. Analytical geometry in space. Plane equations. 1. Equation of a plane with respect to a point and a normal vector. Given: point and normal vector Equation of the plane: Let point Then. 2. General equation of the plane. The equation of the form is called the general equation of the plane. Coefficients A,B,C in the equation determine the coordinates of the normal vector: Theorem. 5. Coefficients A=B=0 (Fig. 5) 6. Coefficients A=C=0 (Fig. 6) 7. Coefficients B=C=0 (Fig. 7). 8. Coefficients A=B=D=0 9. Coefficients A=C=D=0 10. Coefficients B=C=D=0. -

The cycle of lessons on the topic: "Axioms of stereometry" consists of the following lessons:

1. The subject of stereometry. Axioms of stereometry"

2. Some consequences from the axioms.

3;4. Solving problems on the application of axioms and their consequences.

5. Solving problems on the application of the axioms of stereometry and their consequences. Independent work.

A presentation has been prepared for each lesson.

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A cycle of lessons on the topic: "Axioms of stereometry and their consequences."

Lesson 1. The subject of stereometry. Axioms of stereometry.

Lesson Objectives:

1. to acquaint students with the content of the course of stereometry;
2. study the axioms relative position points, lines and planes in space;
3. learn to apply the axioms of stereometry in solving problems.

During the classes:

slide 1.

1. Organizational moment.

2. Learning new material.

Teacher: For three years, starting from the 7th grade, we have been studying school course geometry.

Slide 2. Questions for students:

What is geometry? (Geometry is the science of the properties of geometric shapes)

What is planimetry? (Planimetry is a section of geometry that studies the properties of figures on a plane)

What are the basic concepts of planimetry? (point, line)

Teacher: Today we are starting to study a new section of geometry - stereometry.

Slide 3. Stereometry is a branch of geometry that studies the properties of figures in space. (Students write in notebook)

slide 4. Basic concepts of space: point, line, plane.

The concept of a plane is given by the smooth surface of a table, wall, floor, ceiling, etc. The plane, as a geometric figure, must be represented as extending in all directions, infinite. Planes are denoted by Greek letters α, β, γ, etc.

1. Name the points lying in the plane β; not lying in the plane β.

2. Name the lines: lying in the plane β; not lying in the plane β.

Slide 5. About the basic concepts (point, line, plane) we have a visual representation and definitions are not given to them. Their properties are expressed in axioms.

Along with a point, a straight line, a plane, geometric bodies (a cube, a parallelepiped, a cylinder, a tetrahedron, a cone, etc.) are considered in stereometry, their properties are studied, and their areas and volumes are calculated. The idea of ​​geometric bodies is given by the objects around us.

slide 6. Questions for students:

What geometric bodies do the objects depicted in these drawings remind you of.

Name objects from your environment (our classroom) that remind you of geometric bodies.

Slide 7. Practical work (in notebooks)

1. Draw a cube in a notebook (visible lines - a solid line, invisible - a dotted line).

2. Label the vertices of the cube capital letters ABCDA 1 V 1 S 1 D 1

3. Highlight with a colored pencil:

• vertices A, C, B 1, D 1 ; segments AB, CD, B 1 S, D 1 WITH; diagonals of the square AA 1 V 1 V.

Draw students' attention to the visible and invisible lines in the figure; image of a square AA 1 in 1 In in space.

slide 8. Questions for students:

What is an axiom? What axioms of planimetry do you know?

In space, the basic properties of points, lines and planes, concerning their mutual arrangement, are expressed in axioms.

slide 9. Students write and draw in notebooks.

Axiom 1. (A1) Through any 3 points that do not lie on one straight line, there passes a plane and, moreover, only one.

slide 10. Note that if we take not 3, but 4 arbitrary points, then not a single plane may pass through them, that is, 4 points may not lie in the same plane.

Slide 11. Axiom 2. (A2) If 2 points of a line lie in a plane, then all points of the line lie in this plane. In this case, the line is said to lie in a plane or the plane passes through the line.

slide 12. Question for students:

How many points do a line and a plane have in common? (Fig.1 - infinitely many; Fig.2 - one)

slide 13. Axiom 3. (A3) If two planes have a common point, then they have a common straight line on which all common points of these planes lie.

In this case, the planes are said to intersect in a straight line.

3. Consolidation of the studied material.

slide 14. Solving problems from textbook No. 1 (a, b), 2 (a).

Students read the condition of the tasks and, according to the picture on the slide, give an answer with an explanation.

Task 1.

a) P, E (ADV) RE (ADV) according to A 2

Similar to MK (VDS)

V, D (ADV) and (VDS) VD (ADV) and (ICE)

Similarly AB (ADV) and (ABC)

C, E (ABC) and (DEC) CE (ABC) and (DEC)

b) C (DK) and (ABC) DK ∩ (ABC) = S. Since. there is at most one point of intersection of a line and a plane (the line does not lie in the plane), then this is the only point.

Similarly, SE ∩ (ADV) = E.

Task 2(a)

In the DSS plane 1: D, C, C 1, D 1 , K, M, R. In the BQC plane: B 1, B, P, Q, C 1, M, C.

slide 15. 4. Summing up the lesson.Questions for students:

1. What is the name of the section of geometry that we will study in grades 10-11?
2. What is stereometry?
3. Formulate with the help of the picture the axioms of stereometry that you studied today in the lesson.

slide 16. 5. Homework.

Lesson 2. Some consequences from the axioms.

Lesson Objectives:

Repeat the axioms of stereometry and their application in solving problems homework;

To acquaint students with the consequences of the axioms;

To teach how to apply the consequences of the axioms in solving problems, as well as to consolidate the ability to apply the axioms of stereometry in solving problems;

Review formulas for calculating the area of ​​a rhombus.

During the classes.

slide 1. 1. Organizational moment.Presentation of the topic and objectives of the lesson.

Slide 2.

1) Formulate the axioms of stereometry and arrange the drawings on the board.

2) No. 1 (c, d); 2(b, e).

Students orally from the place on the picture on the slide answer the questions of homework.

Slide 3. 3. Learning new material.Consider and prove the consequences of the axioms.

Theorem 1. A plane passes through a line and a point not lying on it, and moreover, only one.

Students write the wording in a notebook and, answering the teacher's questions, make appropriate notes and drawings in a notebook.

What is given in the theorem? (a line and a point not lying on it)

What needs to be proven? (passes the plane; one)

What can be used for proof? (axioms of stereometry)

Which of the axioms allows you to construct a plane? (A1, a plane passes through three points, and moreover, only one)

What is in this theorem and what is missing for using A1 (we have - a point; two more points are needed)

Where will we build two more points? (on this line)

What conclusion can we draw? (Through three points we build a plane)

Does a line belong to this plane? (Yes)

On what basis can such a conclusion be drawn? (based on A2: if two points of a line belong to the plane, then the whole line belongs to the plane)

How many planes can be drawn through a given line and a given point? (one)

Why? (since a plane passing through a line and a plane passes through a given point and two points on a line, then this plane is the only one by A1)

slide 4. Theorem 2. A plane passes through two intersecting lines, and moreover, only one.

Students prove the theorem on their own, then listen to several proofs and make additions and clarifications (if necessary)

Pay attention to the fact that the proof is not based on axioms, but on Corollary 1.

Slide 5. 4. Consolidation of the studied material.

Task 6 (from study guide)

Students work in notebooks, offer their solutions, then compare their solution with the solution on the screen. Two cases are considered: 1) the points do not lie on one straight line; 2) the points lie on the same straight line.

Slide 6.7. task on the slide. Students read the condition, make a drawing and the necessary notes in notebooks. The teacher conducts frontal work with the class on the issues of the problem. In the course of solving the problem, we repeat the formulas for calculating the area of ​​a rhombus.

Given: ABSD is a rhombus, AC∩VD=O, M, (A, D, O) ; AB \u003d 4cm, A \u003d 60º.

Find: (B, C) ; D (MOV); (MOV)∩(ADO); S ABSD.

Solution:

Pay attention to the fact that if two planes have common points, then they intersect along a straight line passing through these points.

5. Summing up:

Formulate the axioms of stereometry.

Formulate consequences from the axioms.

The goal of the lesson has been achieved. The axioms of stereometry were repeated, they got acquainted with the consequences of the axioms and applied them in solving problems.

Marking (with comments)

slide 8. 6. Setting homework:

Lesson 3

Lesson Objectives:

Repeat the axioms of stereometry and their consequences;

To form the skill of applying the axioms of stereometry and their consequences in solving problems;

Students know the axioms of stereometry and their consequences and are able to apply them in solving problems.

During the classes.

slide 1. 1. Organizational moment.Presentation of the topic and objectives of the lesson.

2. Actualization of students' knowledge.

1) Checking homework on student questions.

Before the lesson, have several students take notebooks with homework to check.

2) Two students prepare a proof of the consequences of the axioms at the blackboard.

3) Two students (level 1) and two students (level 2) work on individual survey cards. Slide.

4) Frontal work with students.

Slide 2. Given: cube ABCDA1V1S1D1

Find:

1. Several points that lie in the α plane; (A, B, C, D)
2. Several points that do not lie in the α plane; (A 1 , V 1 , S 1 , D 1 )
3. Several lines that lie in the plane α; (AB, BC, SD, AD, AC, VD)
4. Several lines that do not lie in the plane α; (A 1 B 1, B 1 C 1, C 1 D 1, A 1 D 1, A 1 C 1, B 1 D 1, AA 1, BB 1, SS 1, DD 1)
5. Several lines that intersect the line BC; (BB 1 , SS 1 )
6. Several lines that do not intersect line BC. (AD, AA 1 …)

Slide 3. Fill in the gaps to get the correct statement:

slide 4. Do straight lines AA lie 1 , AB, AD in the same plane? (Direct AA 1 , AB, AD pass through point A, but do not lie in the same plane)

3. Problem solving.

Slide 5. Students solve problems No. 7, 10, 14 from the textbook, making appropriate drawings and notes on the board and in notebooks.

Task number 7.

2) Do all the lines passing through the point M lie in the same plane?

Solution: By Corollary 2:

2) All lines passing through the point M do not necessarily lie in the same plane. (see example from slide 4)

Problem 10. Students solve the problem on their own (similar to problem number 7). The teacher selectively takes notebooks for checking and provides individual assistance in solving the problem to students who have not completed the task.

Problem number 14. Solution: All lines a, b, c lie in the same plane. In this case, by Corollary 2, one can draw a plane, and one plane passes through three lines.

One of the three lines, for example c, does not lie in the plane α defined by the lines a and b. In this case, three different planes pass through the given three lines, defined by the pairs of lines a and b, a and c, b and c.

slide 6. Students make a drawing and the necessary constructions and notes in notebooks. When building, students pronounce the axioms, the result of the construction is recorded using symbols.

Task. Given: ABCDA cube 1 B 1 C 1 D 1

t.M lies on the edge of explosives 1 , point N lies on the edge CC 1 and the point K lies on the edge DD 1

a) Name the planes in which the points M lie; N.

b) find t.F-point of intersection of lines MN and BC. What property does point F have?

c) find the point of intersection of the line KN and the plane ABC.

d) find the line of intersection of the MNK and ABC planes.

Solution:

Slide 7. To solve the following problem, we repeat the formula for calculating the area of ​​a quadrilateral. The derivation of the formula is disassembled by slide.

Students write the formula in their notebook.

Slide 8. Prove that all vertices of a quadrilateral ABCD lie in the same plane if its diagonals AC and BD intersect.

Calculate area of ​​a quadrilateral if AC┴VD, AC = 10cm, VD = 12cm.

Answer: 60 cm 2

4. Summing up the lesson.

What caused the difficulty? The teacher announces the marks for the lesson with a commentary.

slide 9.

Lesson 4

Lesson Objectives:

Carry out a control of knowledge of the axioms of stereometry and their consequences;

To consolidate the formed skill of applying the axioms of stereometry and their consequences in solving problems;

Review: the Pythagorean theorem and its application; formulas for calculating the area of ​​an equilateral triangle, rectangle.

During the classes.

slide 1. 1. Organizational moment.Presentation of the topic and objectives of the lesson.

Slide 2. 2. Checking homework.

Before the lesson, have several students take notebooks with homework to check.

Two students are preparing at the blackboard for solving problems from homework - No. 9, 15.

The rest of the students answer the questions of the mathematical dictation on the slide.

Slide 3. 3. Problem solving (frontal work with the class)

Task number 1.

Given a tetrahedron MABC, each edge of which is 6 cm.

1. Name the line along which the planes intersect: a) MAB and MFC; b) MSF and ABC.
2. Find the length of CF and SABC
3. How to construct the point of intersection of the line DE with the plane ABC?

Questions for students (if necessary):

Which points belong to both planes at the same time. Based on what axiom can you draw a conclusion?

Formulate the property of the median of an isosceles triangle.

Formulate the Pythagorean theorem.

Why can the Pythagorean theorem be applied in this case?

How can you calculate the area of ​​an equilateral triangle?

Is it always possible to construct the point of intersection of the line DE with the plane ABC?

Slide 4. Task number 2.

1. How to construct the point of intersection of the plane ABC with the line D 1 R?
2. How to draw a line of intersection of the AD plane 1 P and ABB 1 ?
3. Calculate the length of the segments AP and AD 1 if AB = a

Solution:

Slide 5. Task number 3.

Given : Points A, B, C do not lie on the same line.

Prove that the point P lies in the plane ABC.

With the help of animation on the slide, students draw the appropriate constructions and the necessary conclusions. They make notes in notebooks using mathematical symbols, pronouncing the corresponding axioms and consequences from the axioms.

Questions for students (as needed):

Knowing that points A, B, C do not lie on one straight line, what conclusion can be drawn?

If points A and B lie in a plane, what conclusion can be drawn about the line AB?

What conclusion can be drawn about point M?

If points A and C lie in a plane, what conclusion can be drawn about the line AC?

What conclusion can be drawn about point K?

Knowing that the points M and K lie in a plane, what conclusion can be drawn about the straight line MK?

What conclusion can be drawn about point P?

Solution (another way of proof):

AB∩AC=A. By the second corollary, the lines AB and AC define the plane α. The point M belongs to AB, and hence belongs to the plane α, and the point K belongs to AC, and hence to the plane α. By axiom A2: MK lies in the plane α. The point P belongs to the MC, and hence to the plane α.

Slide 6. Task number 4.

The planes α and β intersect along the straight line c. The line a lies in the plane α and intersects the plane β. Do lines a and c intersect? Why?

Questions for students (if necessary):

Knowing that the line a intersects the plane β, what conclusion can be drawn? (The line and the plane have a common point, for example, point B)

What property does point B have? (Point B belongs to the line a, the plane α, and the plane β)

If a point belongs to two planes at the same time, then what can we say about the relative position of the planes? (planes intersect in a straight line, for example c)

What is the relative position of point B and line c? (point B belongs to line c)

Knowing that point B belongs to both line a and line c, what conclusion can be drawn about these lines? (lines intersect at point B)

Slide 7. Task number 5.

Given a rectangle ABCD, O is the point of intersection of its diagonals. It is known that points A, B, O lie in the plane α. Prove that points C and D also lie in the plane α. Calculate the area of ​​the rectangle if AC = 8 cm, AOB = 60º.

The task is for independent solution with a discussion of the solution and the provision of individual assistance to students. It is useful to discuss the various ways of finding the area of ​​a rectangle:

Ask students to solve the problem in different ways. Answer: 16 cm 2.

4. Summing up the lesson:

What axioms and theorems did we use in the lesson when solving problems? Formulate.

What tasks were the most interesting, the most difficult?

What was useful for you personally in the lesson?

What caused the difficulty?

Marking for the lesson (with commenting on each mark)

slide 8. 5. Statement of homework:

Lesson 5 Independent work (20 min.)

Lesson Objectives:

To consolidate the assimilation of theoretical issues in the process of solving problems;

To check the level of preparedness of students by conducting independent work of a controlling nature.

During the classes.

slide 1. 1. Organizational moment.

Presentation of the topic and objectives of the lesson.

Slide 2. 2. Checking homework.

Before the lesson, have several students take notebooks with homework to check.

Task 1.

Lines a and b intersect at point O, A a, B b, P AB. Prove that the lines a and b and the point P lie in the same plane.

Solution:

Slide 3. Task 2.

In this figure, the plane α contains points A, B, C, D, but does not contain point M. Construct point K - the intersection point of the line AB and the MSD plane. Does the point K lie in the plane α.

Solution:

Slides 4, 5, 6 3. Oral problem solving for repeating theory (according to slides)

Slides 7.8 4. Independent work(multi-level, controlling nature) Students choose their level of difficulty.

5. Summing up.

1) Collect notebooks with independent work.

2) Announcement of marks with commenting.

slide 9. 6. Homework.

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Slides captions:

Lesson 1 Topic: "Subject of stereometry. Axioms of stereometry."

What is geometry? Geometry - the science of the properties of geometric shapes "Geometry" - (Greek) - "land surveying" - What is planimetry? Planimetry is a branch of geometry that studies the properties of figures on a plane. A a Basic concepts of planimetry: point straight line - Basic concepts of planimetry?

Stereometry - a branch of geometry that studies the properties of figures in space

Basic figures in space: point straight plane α β Designation: A; IN; WITH; …; M; ... a A B M N R Designation: a, b, c, d ..., m, n, ... (or two capital Latin) Designation: α, β, γ ... Answer the questions on the picture: 1. Name the points, lying in the plane β ; not lying in the plane β . 2. Name the lines lying in the plane β; not lying in the plane β

Some geometric bodies. A B C D D 1 C 1 B 1 A 1 cube A B C D A 1 B 1 C 1 D 1 parallelepiped A B C D tetrahedron cylinder cone

Name which geometric bodies the objects depicted in these pictures remind you of: Name objects from the environment around you (our classroom) that remind you of geometric bodies.

Practical work. 1. Draw a cube in a notebook (visible lines - a solid line, invisible - a dotted line). 2. Mark the vertices of the cube in capital letters ABCDA 1 B 1 C 1 D 1 A B C D D 1 C 1 B 1 A 1 3. Highlight with a colored pencil: vertices A, C, B 1, D 1 segments AB, CD, B 1 C, D 1 C diagonals of the square AA 1 B 1 B

What is an axiom? An axiom is a statement about the properties of geometric figures, it is taken as the starting points, on the basis of which theorems are further proved and, in general, the whole geometry is built. Axioms of planimetry: - through any two points you can draw a straight line and, moreover, only one. of the three points of a straight line, one, and only one, lies between the other two. there are at least three points that do not lie on the same line...

Axioms of stereometry. A B C A1 . Through any three points that do not lie on the same straight line, there passes a plane and, moreover, only one. α

If the legs of the table are not the same in length, then the table stands on three legs, i.e. rests on three "points", and the end of the fourth leg (fourth point) does not lie in the plane of the floor, but hangs in the air.

Axioms of stereometry. A B α A2. If two points of a line lie in a plane, then all points of this line lie in that plane. They say: a line lies in a plane or a plane passes through a line.

a M A line lies in a plane A line intersects a plane How many common points do a line and a plane have?

Axioms of stereometry. α β A3. If two planes have a common point, then they have a common line on which all common points of these planes lie. They say that planes intersect in a straight line. A a

Solve problems: No. 1 (a, b); 2(a) A B C D R E K M A B C D A 1 B 1 C 1 D 1 Q P R K M Name from the figure: a) the planes in which the lines DW, AB, MK, PE, EC lie; b) points of intersection of the line DK with the plane ABC, the line CE with the plane ADV. a) points lying in the planes DSS 1 and B Q C No. 1 (a, b) No. 2 (a)

Let's summarize the lesson: 1) What is the name of the section of geometry that we will study in grades 10-11? 2) What is stereometry? 3) Formulate with the help of the picture the axioms of stereometry that you have studied today in the lesson. A A B B α α A α β

Theorem 1. A plane passes through a line and a point not lying on it, and moreover, only one. Given: a, M ¢ a Prove: (a, M) c α α is unique a M α Proof: 1 . P, O with a; ( Р,О,М ) ¢а Р О By axiom A1: a plane passes through the points Р, О, М. According to axiom A2: because two points of a line belong to the plane, then the whole line belongs to this plane, i.e. (a, M) with α 2 . Any plane passing through the line a and the point M passes through the points P, O, and M, which means that, according to the axiom A1, it is unique. Ch.t.d. Some consequences of the axioms:

Theorem 2. A plane passes through two intersecting lines, and moreover, only one. Given: a ∩ b Prove: 1. (a ∩ b) c α 2. α is the only one a b M H α Proof: 1. The plane α passes through a and H a, H b. (M, H) α , (M, H) b , so by A2 all points b belong to the plane. 2. The plane passes through a and b and it is unique, because any plane passing through the lines a and b also passes through H, which means that α is the only one.

Solve problem No. 6 A B C α Three given points are connected in pairs by segments. Prove that all segments lie in the same plane. Proof: 1. (A, B, C) α , so along A1 through A, B, C there is only one plane. 2. Two points of each segment lie in a plane, which means that according to A2, all points of each of the segments lie in the plane α. 3. Conclusion: AB, BC, AC lie in the plane α 1 case. A B C α 2 case. Proof: Since 3 points belong to one line, then by A2 all points of this line lie in the plane.

Task. A B C D M O ABCD is a rhombus, O is the point of intersection of its diagonals, M is a point in space that does not lie in the plane of the rhombus. Points A, D, O lie in the plane α. Determine and justify: Do points B and C lie in the plane α? Does the point D lie in the MOB plane? Name the line of intersection of the MOV and ADO planes. Calculate the area of ​​a rhombus if its side is 4 cm and the angle is 60 º. Suggest different ways to calculate the area of ​​a rhombus.

oral work. A B C D A 1 B 1 C 1 D 1 α Given: cube ABCDA 1 B 1 C 1 D 1 Find: Several points that lie in the plane α ; Several points that do not lie in the α plane; Several lines that lie in the plane α; Several lines that do not lie in the plane α; Several lines that intersect the line BC; Several lines that do not intersect line BC. Task 1.

oral work. Problem 2. α A M B a b c Fill in the gaps to get the correct statement:

oral work. A B C D A 1 B 1 C 1 D 1 α Straight lines AA 1 , AB, AD pass through point A, but do not lie in the same plane Do the lines AA 1 , AB, AD lie in the same plane?

Solve the problems from the textbook: p. 8 No. 7, 10, 14. Students work on the board and in notebooks:

Problem 1 A B C D A 1 B 1 C 1 D 1 M N F K edge DD 1 a) name the planes in which the points M lie; N. b) find point F - the point of intersection of lines M N and BC. What property does point F have? c) find the point of intersection of the straight line K N and the plane ABC O d) find the line of intersection of the planes M N K and ABC

Task (oral) A B C D M O ABCD is a rhombus, O is the point of intersection of its diagonals, M is a point in space that does not lie in the plane of the rhombus. Points A, D, O lie in the plane α. Determine and justify: 1. What other points lie in the plane α ? Do the points B and M lie in the plane α? Does point B lie in the MOD plane? Name the line of intersection of the MOS and ADO planes. Point O is a common point of the MOV and MOS planes. Is it true that these planes intersect along the line MO? Name three lines that lie in the same plane; not lying in the same plane.

Task (oral) A B C M Sides AB and AC triangle ABC lie in a plane. Prove that the median also lies in the plane.

C D W E F O M Task (oral) What is the error in the drawing, where O E F . Give an explanation. What should the correct drawing look like?

Level 1 A B C S K M N 1. Using this figure, name: a) four points lying in the plane S AB; b) the plane in which the straight line M N lies; c) the straight line along which the planes S AC and S BC intersect. 2. Point C is a common point of the plane u. The line c passes through the point C. Is it true that the planes and intersect along the line c. Explain the answer. 3. Through the line a and the point A, two different planes can be drawn. What is the relative position of the line a and point A. Explain the answer. Level 2 S A B C D E F 1. Using this figure, name: a) two planes containing the straight line DE; b) a straight line along which the planes AE F and S BC intersect; c) the planes intersected by the line S B. 2. The lines a, b and c have a common point. Is it true that these lines lie in the same plane? Justify the answer. 3. The planes and intersect along the straight line c. The line a lies in the plane and intersects the plane. What is the relative position of lines a and c?

A B C D A 1 B 1 C 1 D 1 Level 3 (on cards) 1. Using this figure, name: a) two planes containing a line B 1 C; b) a straight line along which the planes B 1 SD and AA 1 D 1 intersect; c) a plane that does not intersect with the straight line SD 1. 2. Four lines intersect in pairs. Is it true that if any three of them lie in the same plane, then all four lines lie in the same plane? Explain the answer. 3. The vertex C of the plane quadrilateral ABCD lies in the plane, while the points A, B, D do not lie in this plane. Straight lines AB and AD intersect the plane at points B 1 and D 1, respectively. What is the relative position of points C, B 1 and D 1? Explain the answer.

Homework: repeat the material from planimetry and make notes in notebooks on the following issues: Definition of parallel lines Mutual position of two lines on a plane Construction of a line parallel to a given Axiom of parallel lines

• What is stereometry?
• The emergence and development of stereometry
• Basic figures in space
• Designation of points and examples of their models
• Line designation
• Examples of models of straight lines
• Designation of planes and examples of their models
• What else does stereometry study?
• Objects around us and geometric bodies
• The image of geometric bodies in the drawings
• Practical (applied) value of stereometry
• Axioms of stereometry
• Consequences from the axioms of stereometry
• Anchoring
• Used Books

What is stereometry?

Stereometry is a branch of geometry that studies the properties of figures in space.

The emergence and development of stereometry.

• The development of stereometry began much later than planimetry.
• Stereometry developed from observations and solutions to issues that arose in the process of human practical activity.

• Already primitive man, having taken up agriculture, made attempts to estimate, at least in rough terms, the size of the harvest he had gathered by the masses of grain piled up in heaps, stacks or stacks.
• The builder of even the most ancient primitive buildings had to somehow take into account the material that he had at his disposal, and be able to calculate how much material would be needed to build a particular building.

• Stonecutting among the ancient Egyptians and Chaldeans required familiarity with the metric properties of at least the simplest geometric bodies.
• The need for agriculture, navigation, orientation in time pushed people to astronomical observations, and the latter - to the study of the properties of the sphere and its parts, and consequently the laws of the relative position of planes and lines in space.

Basic figures in space.

A plane is a geometric figure that extends indefinitely in all directions.

Designation of points and examples of their models.

Points are indicated by capital Latin letters A, B, C, ...

Examples of point models are:

atoms and molecules

planets across the universe

Designation of lines.

• The lines are marked:
• lowercase Latin letters a, b, c, d, e, k, ...
• two capital Latin letters AB, CD ...

Examples of straight lines.

Examples of straight lines are:

aircraft contrails

Designation of planes and examples of their models.

Planes are denoted by Greek letters α, β, γ,…

Examples of plane models are:

water surface

table surface

What else does stereometry study?

Along with a point, a straight line and a plane, stereometry studies geometric bodies and their surfaces.

Objects around us and geometric bodies.

The objects around us give ideas about geometric bodies.

And by studying the properties of geometric shapes - imaginary objects, we get information about geometric properties real objects and we can use these properties in practice.

crystals - polyhedrons

tin can - cylinder

candy packaging - cone

Images of geometric bodies in the drawings.

• The image of a spatial figure is its projection onto a particular plane.
• Invisible parts of the figure are shown with dashed lines.

Practical (applied) value of stereometry.

• Geometric bodies are fictitious objects
• By studying the properties of geometric shapes, we get ideas about the geometric properties of real objects (their shape, relative position, etc.)
• Stereometry is widely used in construction, architecture, mechanical engineering and other fields of science and technology.

Axioms of stereometry.

• Axiom- this is a statement about the properties of geometric shapes, is taken as the starting points, on the basis of which theorems are further proved and, in general, the whole geometry is built.

Axioms of stereometry.

A1 . Through any three points that do not lie on the same straight line, there passes a plane and, moreover, only one.

Axioms of stereometry.

A2 . If two points of a line lie in a plane, then all points of this line lie in that plane.

In this case, the line is said to lie in a plane or the plane passes through the line.

Axioms of stereometry.

A3. If two planes have a common point, then they have a common line on which all common points of these planes lie.

Planes are said to intersect in a straight line.

Consequences from the axioms.

Theorem 1: A plane passes through a line and a point not lying on it, and moreover, only one.

Theorem 2: A plane passes through two intersecting lines, and only one.

Consolidation.

1. Name the planes in which the lines lie:

Consolidation.

2. Name the point of intersection of the line CE with the plane ADB.

3. Name the lines along which the planes intersect:

Used Books

• Geometry. Grades 10-11: textbook. For general education institutions: basic and profile. levels/hp Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - 21st ed. - M .: Education, 2012. - 255 p.: ill.
• Geometry: Toolkit for higher pedagogical institutions and teachers high school: part 2 Stereometry / ed. Prof. I.K. Andronov.