The geometry course is broad, voluminous and multifaceted: it includes many different topics, rules, theorems and useful knowledge. One can imagine that everything in our world consists of simple things, even the most complex ones. Points, straight lines, planes - all this is in your life. And they are amenable to the existing laws in the world regarding the relationship of objects in space. To prove this, you can try to prove the parallelism of lines and planes.
A straight line is a line that connects two points along the shortest path, without ending and extending on both sides to infinity. A plane is a surface formed by the kinematic movement of a generatrix of a straight line along a guide. In other words, if any two lines have an intersection point in space, they can lie in the same plane. However, how can we express direct ones if this data is not enough for such a statement?
The main condition for the parallelism of a straight line and a plane is that they do not have common points. Unlike lines, which, in the absence of common points, may not be parallel, but divergent, the plane is two-dimensional, which excludes the concept of divergent lines. If this parallelism condition is not met, it means that the straight line intersects the given plane at one point or lies entirely within it.
What does the condition of parallelism between a straight line and a plane show us most clearly? The fact that at any point in space the distance between a parallel line and a plane will be constant. If there is even the slightest slope, billionths of a degree, the straight line will sooner or later intersect the plane due to mutual infinity. That is why parallelism between a straight line and a plane is possible only if this rule is observed, otherwise its main condition - the absence of common points - will not be met.
What can you add when talking about the parallelism of lines and planes? The fact is that if one of the parallel lines belongs to the plane, then the second is either parallel to the plane or also belongs to it. How to prove this? The parallelism of a line and a plane containing a line parallel to the given one is proven very simply. have no common points - therefore, they do not intersect. And if a straight line does not intersect the plane at one point, it means that it is either parallel or lies on the plane. This once again proves the parallelism of a straight line and a plane that do not have intersection points.
There is also a theorem in geometry that states that if there are two planes and a straight line is perpendicular to both of them, then the planes are parallel. A similar theorem states that if two lines are perpendicular to any one plane, they will necessarily be parallel to each other. Is the parallelism of lines and planes expressed by these theorems true and provable?
It turns out that this is true. A straight line perpendicular to a plane will always be strictly perpendicular to any straight line that lies in a given plane and also has an intersection point with another straight line. If a straight line has similar intersections with several planes and is perpendicular to them in all cases, it means that all these planes are parallel to each other. A clear example a children's pyramid can serve: its axis will be the desired perpendicular straight line, and the rings of the pyramid will be planes.
Therefore, it is quite easy to prove the parallelism of a line and a plane. This knowledge is acquired by schoolchildren when studying the basics of geometry and largely determines the further learning of the material. If you know how to competently use the knowledge acquired at the beginning of training, you will be able to operate with a much larger number of formulas and skip unnecessary logical connections between them. The main thing is to understand the basics. If it is not there, then studying geometry can be compared to building without a foundation. That is why this topic requires close attention and thorough research.
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Theorem
If a line that does not belong to a plane is parallel to some line in this plane, then it is parallel to the plane itself.
Proof
Let α be a plane, a a line not lying in it, and a1 a line in the α plane parallel to line a. Let us draw the plane α1 through the lines a and a1. Planes α and α1 intersect along straight line a1. If line a intersected plane α, then the intersection point would belong to line a1. But this is impossible, since the lines a and a1 are parallel. Consequently, line a does not intersect the plane α, and therefore is parallel to the plane α. The theorem has been proven.
18. PLANES
If two parallel planes intersect with a third, then the straight lines of intersection are parallel(Fig. 333).
Indeed, according to the definition Parallel lines are lines that lie in the same plane and do not intersect. Our straight lines lie in the same plane - the cutting plane. They do not intersect, since the parallel planes containing them do not intersect.
This means that the lines are parallel, which is what we needed to prove.
Properties
§ If the plane α is parallel to each of two intersecting lines lying in another plane β, then these planes are parallel
§ If two parallel planes are intersected by a third, then their lines of intersection are parallel
§ Through a point outside a given plane it is possible to draw a plane parallel to the given one, and, moreover, only one
§ Segments of parallel lines limited by two parallel planes, are equal
§ Two angles with respectively parallel and identically directed sides are equal and lie in parallel planes
19.
If two straight lines lie in the same plane, the angle between them is easy to measure - for example, using a protractor. How to measure angle between straight line and plane?
Let the straight line intersect the plane, not at a right angle, but at some other angle. This line is called inclined.
Let's drop a perpendicular from some inclined point onto our plane. Let's connect the base of the perpendicular to the intersection point of the inclined and plane. We got inclined projection onto a plane.
The angle between a line and a plane is the angle between a line and its projection onto a given plane.
Please note that we choose an acute angle as the angle between the straight line and the plane.
If a line is parallel to a plane, then the angle between the line and the plane is zero.
If a line is perpendicular to a plane, its projection onto the plane will be a point. Obviously, in this case the angle between the straight line and the plane is 90°.
A line is perpendicular to a plane if it is perpendicular to any line lying in this plane.
This is the definition. But how to work with it? How to check that a given line is perpendicular to all lines lying in the plane? After all, there are infinitely many of them.
In practice it is used sign of perpendicularity of a line and a plane:
A line is perpendicular to a plane if it is perpendicular to two intersecting lines lying in this plane.
21.Dihedral angle- spatial geometric figure, formed by two half-planes emanating from one straight line, as well as a part of space limited by these half-planes.
Two planes are called perpendicular if the dihedral angle between them is 90 degrees.
§ If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.
§ If from a point belonging to one of two perpendicular planes a perpendicular is drawn to the other plane, then this perpendicular lies entirely in the first plane.
§ If in one of two perpendicular planes a perpendicular is drawn to their line of intersection, then this perpendicular will be perpendicular to the second plane.
Two intersecting planes form four dihedral angles with a common edge: pairs of vertical angles are equal, and the sum of two adjacent angles is 180°. If one of the four angles is right, then the other three are also equal and right. Two planes are called perpendicular if the angle between them is right.
Theorem. If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.
Let and be two planes such that they pass through a straight line AB, perpendicular to and intersecting with it at point A (Fig. 49). Let's prove that _|_ . The planes and intersect along some straight line AC, and AB _|_ AC, because AB _|_ . Let us draw a straight line AD in the plane, perpendicular to the straight line AC.
Then angle BAD is the linear angle of the dihedral angle formed by and . But< ВАD - 90° (ибо AB _|_ ), а тогда, по определению, _|_ . Теорема доказана.
22. A polyhedron is a body whose surface consists of a finite number of flat polygons.
1. any of the polygons that make up the polyhedron can be reached to any of them by moving to the one adjacent to it, and from this, in turn, to the one adjacent to it, etc.
These polygons are called edges, their sides - ribs, and their vertices are peaks polyhedron. The simplest examples of polyhedra are convex polyhedra, that is, the boundary of a limited subset of Euclidean space that is the intersection of a finite number of half-spaces.
The above definition of a polyhedron takes on different meanings depending on how to define a polygon, for which the following two options are possible:
§ Flat closed broken lines (even if they are self-intersecting);
§ Parts of the plane bounded by broken lines.
In the first case, we get the concept of a stellated polyhedron. In the second, a polyhedron is a surface composed of polygonal pieces. If this surface does not intersect itself, then it is the complete surface of some geometric body, which is also called a polyhedron. This gives rise to the third definition of a polyhedron, as the geometric body itself
Straight prism
The prism is called straight, if its lateral edges are perpendicular to the bases.
The prism is called inclined, if its lateral edges are not perpendicular to the bases.
A straight prism has rectangular faces.
The prism is called correct, if its bases are regular polygons.
The lateral surface area of the prism is called the sum of the areas of the lateral faces.
Full prism surface equal to the sum of the lateral surface and the areas of the bases
Prism elements:
Points - called vertices
The segments are called side edges
Polygons and - are called bases. The planes themselves are also called bases and
24. Parallelepiped(from Greek παράλλος - parallel and Greek επιπεδον - plane) - a prism, the base of which is a parallelogram, or (equivalently) a polyhedron, which has six faces and each of them is a parallelogram.
§ The parallelepiped is symmetrical about the middle of its diagonal.
§ Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.
§ Opposite faces of a parallelepiped are parallel and equal.
§ The square of the diagonal length of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.
Surface area of a cuboid equal to twice the sum of the areas of the three faces of this parallelepiped:
| 1. | S= 2(S a+Sb+S c)= 2(ab+bc+ac) |
25 .Pyramid and its elements
Consider a plane, a polygon lying in it, and a point S not lying in it. Let's connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. 
The polygon is called the base, and point S is the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. An alternative name for a triangular pyramid is tetrahedron. The height of a pyramid is the perpendicular descending from its top to the plane of the base. 
A pyramid is called regular if it is a regular polygon and the base of the height of the pyramid (the base of the perpendicular) is its center.
The program is designed to calculate the lateral surface area of a regular pyramid.
A pyramid is a polyhedron with a base in the form of a polygon, and the remaining faces are triangles with a common vertex. 
Formula for calculating the lateral surface area of a regular pyramid:
where p is the perimeter of the base (polygon ABCDE),
a - apothem (OS);
An apothem is the height of the side face of a regular pyramid, which is drawn from its vertex.
To find the lateral surface area of a regular pyramid, enter the values for the pyramid's perimeter and apothem, then click the "CALCULATE" button. The program will determine the lateral surface area of a regular pyramid, the value of which can be placed on the clipboard.
Truncated pyramid
| A truncated pyramid is the part of a complete pyramid enclosed between the base and a section parallel to it. | |
| The section is called the upper base of a truncated pyramid, and the base of a complete pyramid is bottom base truncated pyramid. (The bases are similar.) The side faces of the truncated pyramid are trapezoids. In the truncated pyramid 3 n ribs, 2 n peaks, n+ 2 faces, n(n- 3) diagonals. The distance between the upper and lower bases is the height of the truncated pyramid (a segment cut off from the height of the full pyramid). |  |
| The total surface area of a truncated pyramid is equal to the sum of the areas of its faces. | |
| Volume of a truncated pyramid ( S And s- base area, N- height) |  |
Body of rotation A body formed as a result of the rotation of a line around a straight line is called.
A right circular cylinder is inscribed in a sphere if the circles of its bases lie on the sphere. The bases of the cylinder are small circles of the ball, the center of the ball coincides with the middle of the cylinder axis. [ 2 ]
A right circular cylinder is inscribed in a sphere if the circles of its bases lie on the sphere. Obviously, the center of the ball lies in the middle of the cylinder axis. [ 3 ]
Volume of any cylinder equal to the product of the area of the base and the height:
| 1. | V=π r 2 h |
The total surface area of a cylinder is equal to the sum of the side surface of the cylinder and twice the area of the base of the cylinder.
Formula for calculating the total surface area of a cylinder:
27. A circular cone can be obtained by rotating a right triangle around one of its legs, which is why a circular cone is also called a cone of revolution. See also Volume of a circular cone
Total surface area of a circular cone equal to the sum of the areas of the lateral surface of the cone and its base. The base of the cone is a circle and its area is calculated using the formula for the area of a circle:
| 2. | S=π r l+π r 2 =π r(r+l) |
28. Frustum it will work if you draw a section in the cone parallel to the base. The body bounded by this section, the base and the lateral surface of the cone is called a truncated cone. See also Volume of a truncated cone
Total surface area of a truncated cone equal to the sum of the areas of the lateral surface of a truncated cone and its bases. The bases of a truncated cone are circles and their area is calculated using the formula for the area of a circle: S= π (r 1 2 + (r 1 + r 2)l+ r 2 2)
29. A ball is a geometric body bounded by a surface, all points of which are at equal distances from the center. This distance is called the radius of the ball.
Sphere(Greek σφαῖρα - ball) - a closed surface, a geometric locus of points in space equidistant from a given point, called the center of the sphere. A sphere is a special case of an ellipsoid, in which all three axes (semi-axes, radii) are equal. A sphere is the surface of a ball.
The area of the spherical surface of a spherical segment (spherical sector) and the spherical layer depends only on their height and the radius of the ball and is equal to the circumference of the great circle of the ball multiplied by the height
Ball volume equal to the volume of a pyramid whose base has the same area as the surface of the ball, and the height is the radius of the ball
The volume of the sphere is one and a half times less than the volume of the cylinder circumscribed around it.
Ball elements
Ball segment The cutting plane splits the ball into two ball segments. N- segment height, 0< N < 2 R,
r- radius of the segment base,  Ball segment volume  Spherical surface area of a ball segment |  |
| Spherical layer A spherical layer is the part of a sphere enclosed between two parallel sections. Distance ( N) between sections is called layer height, and the sections themselves - bases of the layer. Spherical surface area( volume) of the spherical layer can be found as the difference between the areas of the spherical surfaces (volumes) of the spherical segments. |  |
1. Multiplying a vector by a number(Fig. 56).
Product of a vector A per number λ called a vector IN, the modulus of which is equal to the product of the modulus of the vector A per modulus of number λ :
The direction does not change if λ > 0 ; changes to the opposite if λ < 0 . If λ = −1, then the vector
Called the vector opposite to the vector A, and is denoted
2. Vector addition. To find the sum of two vectors A And IN vector
Then the sum will be a vector whose beginning coincides with the beginning of the first, and the end with the end of the second. This rule for adding vectors is called the “triangle rule” (Fig. 57). it is necessary to depict the component vectors so that the beginning of the second vector coincides with the end of the first.
It is easy to prove that for vectors “the sum does not change when the terms are changed.”
Let us indicate one more rule for adding vectors - the “parallelogram rule”. If you combine the beginnings of the vector-summands and construct a parallelogram on them, then the sum will be a vector that coincides with the diagonal of this parallelogram (Fig. 58).
It is clear that addition according to the “parallelogram rule” leads to the same result as according to the “triangle rule”.
The “triangle rule” is easy to generalize (to the case of several terms). To find the sum of vectors
It is necessary to combine the beginning of the second vector with the end of the first, the beginning of the third with the end of the second, etc. Then the beginning of the vector WITH will coincide with the beginning of the first, and the end WITH− with the end of the latter (Fig. 59).
3. Vector subtraction. The subtraction operation is reduced to the two previous operations: the difference between two vectors is the sum of the first with the vector opposite to the second:
You can also formulate the “triangle rule” for subtracting vectors: you need to combine the origins of the vectors A And IN, then their difference will be the vector
Drawn from the end of the vector IN towards the end of the vector A(Fig. 60).
In the future we will talk about the displacement vector material point, that is, the vector connecting the initial and final positions of the point. Agree that the introduced rules for acting on vectors are quite obvious for displacement vectors.
4. Dot product of vectors. The result of the scalar product of two vectors A And IN is the number c equal to the product of the vector moduli and the cosine of the angle α between
The scalar product of vectors operation is very widely used in physics. In the future, we will have to deal with such an operation quite often.
The definition of parallel lines and their properties in space are the same as on the plane (see paragraph 11).
At the same time, another case of arrangement of lines in space is possible - crossing lines. Lines that do not intersect and do not lie in the same plane are called intersecting.
Figure 121 shows a layout of a living room. You see that the lines to which the segments AB and BC belong are intersecting.
The angle between intersecting lines is the angle between intersecting parallel lines. This angle does not depend on which intersecting lines are taken.
The degree measure of the angle between parallel lines is considered equal to zero.
The common perpendicular of two skew lines is a segment with ends on these lines, which is perpendicular to each of them. It can be proven that two skew lines have a common perpendicular, and only one. It is the common perpendicular of parallel planes passing through these lines.
The distance between intersecting lines is the length of their common perpendicular. It is equal to the distance between parallel planes passing through these lines.
Thus, to find the distance between intersecting straight lines a and b (Fig. 122), you need to draw parallel planes a and b through each of these straight lines. The distance between these planes will be the distance between the crossing lines a and b. In Figure 122 this distance is, for example, distance AB.
Example. Lines a and b are parallel, and lines c and d intersect. Can each of the lines a and intersect both lines?
Solution. Lines a and b lie in the same plane, and therefore any line intersecting each of them lies in the same plane. Consequently, if each of the lines a and b intersected both lines c and d, then the lines would lie in the same plane as the lines a and b, but this cannot happen, since the lines intersect.
42. Parallelism of a straight line and a plane.
A straight line and a plane are called parallel if they do not intersect, that is, they do not have common points. If straight line a is parallel to plane a, then write: .
Figure 123 shows a straight line a, parallel to the plane a.
If a line that does not belong to a plane is parallel to some line in this plane, then it is parallel to the plane itself (a sign of parallelism between a line and a plane).
This theorem allows specific situation prove that a line and a plane are parallel. Figure 124 shows a straight line b parallel to a straight line a lying in plane a, i.e. straight b is parallel to plane a, i.e.
Example. Through the top right angle A plane is drawn from a right triangle ABC parallel to the hypotenuse at a distance of 10 cm from it. The projections of the legs onto this plane are 30 and 50 cm. Find the projection of the hypotenuse onto the same plane.
Solution. From right triangles BBVC and (Fig. 125) we find:
From triangle ABC we find:
The projection of the hypotenuse AB onto plane a is equal to . Since AB is parallel to plane a, then So, .
43. Parallel planes.
The two planes are called parallel. if they don't intersect.
Two planes are parallel” if one of them is parallel to two intersecting lines lying in the other plane (a sign of parallelism of two planes).
In Figure 126, plane a is parallel to the intersecting straight lines a and b, lying in the plane, then along these planes are parallel.
Through a point outside a given plane it is possible to draw a plane parallel to the given one, and only one.
If two parallel planes intersect with a third, then the straight lines of intersection are parallel.
Figure 127 shows two parallel planes, and plane y intersects them along straight lines a and b. Then, by Theorem 2.7, we can say that lines a and b are parallel.
The segments of parallel lines contained between two parallel planes are equal.
According to T.2.8, the segments AB and those shown in Figure 128 are equal, since
Let these planes intersect. Let us draw a plane perpendicular to the line of their intersection. It intersects these planes along two straight lines. The angle between these straight lines is called the angle between these planes (Fig. 129). The angle between the planes determined in this way does not depend on the choice of the cutting plane.
Some corollaries from the axioms
Theorem 1:
A plane passes through a straight line and a point not lying on it, and only one.
Given: M ₵ a
Prove: 1) There is α: a∈ α, M ∈ b ∈ α
2)
α is the only one
Proof:
1) On a straight line, and select points P And Q. Then we have 3 points - R, Q, M, which do not lie on the same straight line.
2) According to axiom A1, a plane passes through three points that do not lie on the same line, and only one, i.e. plane α, which contains a straight line a and a point M, exists.
3) Now let's prove thatα the only one. Let us assume that there is a plane β that passes through both the point M and the straight line a, but then this plane passes through the pointsR, Q, M. And after three points P, Q, M, not lying on the same straight line, by virtue of axiom 1, only one plane passes through.
4) This means that this plane coincides with the plane α.Therefore 1) On a straight line, and select points P And Q. Then we have 3 points - P, Q, M, which do not lie on the same straight line.Therefore α is unique.
The theorem has been proven.
1) On line b, take point N, which does not coincide with point M, that is, N ∈ b, N≠M
2) Then we have a point N that does not belong to line a. According to the previous theorem, a plane passes through a straight line and a point not lying on it. Let's call it the plane α. This means that such a plane that passes through the line a and the point N exists.
3) Let us prove the uniqueness of this plane. Let's assume the opposite. Let there be a plane β such that it passes through both line a and line b. But then it also passes through the line a and the point N. But according to the previous theorem, this plane is unique, i.e. plane β coincides with plane α.
4) This means that we have proven the existence of a unique plane passing through two intersecting lines.
The theorem has been proven.
Theorem on parallel lines
Theorem:
Through any point in space not lying on a given line passes a line parallel to the given line.
Given: straight a, M₵ a
Prove:There is only one straight lineb ∥ a, M ∈ b
Proof:
1) Through a straight line a and a point M not lying on it, a unique plane can be drawn (Corollary 1). In the α plane we can draw a straight line b, parallel to a, passing through M.
2) Let's prove that she is the only one. Suppose that there is another line c passing through the point M and parallel to the line a. Let parallel lines a and c lie in the plane β. Then β passes through M and the straight line a. But the plane α passes through the straight line a and the point M.
3) This means that α and β are the same. From the axiom of parallel lines it follows that lines b and c coincide, since in the plane there is a single straight line passing through this point and parallel to a given line.
The theorem has been proven.
