The video course "Get an A" includes all the topics you need to successful delivery USE in mathematics for 60-65 points. Completely all tasks 1-13 profile exam mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
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Theorem
If straight, not belonging to the plane, is parallel to some line in this plane, then it is also parallel to the plane itself.
Proof
Let α be a plane, a a line not lying in it, and a1 a line in the plane α parallel to line a. Let us draw the plane α1 through the lines a and a1. The planes α and α1 intersect along the line a1. If the line a intersected the plane α, then the point of intersection would belong to the line a1. But this is impossible, since the lines a and a1 are parallel. Therefore, the line a does not intersect the plane α, and hence is parallel to the plane α. The theorem has been proven.
18. PLANES
If two parallel planes intersect with a third, then the 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 lines lie in the same plane - the secant plane. They do not intersect, since the parallel planes containing them do not intersect.
So the lines are parallel, which is what we wanted to prove.
Properties
§ If the plane α is parallel to each of two intersecting lines lying in the other plane β, then these planes are parallel
§ If two parallel planes are intersected by a third, then the lines of their intersection are parallel
§ Through a point outside a given plane, it is possible to draw a plane parallel to a given one, and moreover, only one
§ Segments of parallel lines bounded by two parallel planes are equal
§ Two angles with respectively parallel and equally directed sides are equal and lie in parallel planes
19.
If two lines lie in the same plane, the angle between them is easy to measure - for example, using a protractor. And how to measure angle between line and plane?
Let the line intersect the plane, and not at a right angle, but at some other angle. Such a line is called oblique.
Let us drop a perpendicular from some point inclined to our plane. Connect the base of the perpendicular to the point of intersection of the inclined and the plane. We got projection of an oblique plane.
The angle between a line and a plane is the angle between a line and its projection onto a given plane..
Please note - we choose an acute angle as the angle between the 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 is a point. Obviously, in this case the angle between the line and the plane is 90°.
A line is perpendicular to a plane if it is perpendicular to any line in that plane..
This is the definition. But how to work with him? How to check that a given line is perpendicular to all lines lying in the plane? After all, there are an infinite number of them.
In practice, it is applied 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 that plane.
21. Dihedral angle- spatial geometric figure, formed by two half-planes emanating from one straight line, as well as a part of space bounded by these half-planes.
Two planes are said to be 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 the two perpendicular planes, draw a perpendicular to another plane, then this perpendicular lies completely in the first plane.
§ If in one of two perpendicular planes we draw a perpendicular 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 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 those planes are perpendicular.
Let and be two planes such that it passes through the line AB, perpendicular to and intersecting with it at point A (Fig. 49). Let's prove that _|_ . The planes and intersect along some line AC, and AB _|_ AC, because AB _|_ . Let us draw a line AD in the plane, perpendicular to the line AC.
Then angle BAD is a linear angle dihedral angle, educated 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, you can get to any of them by going to the one adjacent to it, and from this, in turn, to the one adjacent to it, etc.
These polygons are called faces, their sides - ribs, and their vertices are peaks polyhedron. The simplest examples of polyhedra are convex polyhedra, that is, the boundary of a bounded subset of the Euclidean space, which is the intersection of a finite number of half-spaces.
The above definition of a polyhedron takes on a different meaning depending on how the polygon is defined, 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 star polyhedron. In the second, a polyhedron is a surface composed of polygonal pieces. If this surface does not intersect itself, then it is the full surface of some geometric body, which is also called a polyhedron. Hence the third definition of the polyhedron arises, as the geometric body itself.
straight prism
The prism is called straight if it side ribs perpendicular to the bases.
The prism is called oblique if its side edges are not perpendicular to the bases.
A straight prism has faces that are rectangles.
The prism is called correct if its bases are regular polygons.
The area of the lateral surface of the prism is called the sum of the areas of the side faces.
Full surface of the prism equal to the sum of the lateral surface and the areas of the bases
Prism elements:
Points - called vertices
The segments are called lateral edges
Polygons and - are called bases. The planes themselves are also called bases.
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 midpoint of its diagonal.
§ Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided by it in half; in particular, all the diagonals of the parallelepiped intersect at one point and bisect it.
§ Opposite faces of a parallelepiped are parallel and equal.
§ Square of diagonal length cuboid is equal to the sum squares of its three dimensions.
Surface area of a cuboid is 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. Connect S to all vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called lateral edges. 
The polygon is called the base, and the point S is called 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. Alternative title triangular pyramid – tetrahedron. The height of a pyramid is the perpendicular drawn from its apex to the base plane. 
A pyramid is called correct if 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 correct pyramid.
The pyramid is a polyhedron with a base in the form of a polygon, and the remaining faces are triangles with a common vertex. 
The formula for calculating the lateral surface area of a regular pyramid is:
where p is the perimeter of the base (polygon ABCDE),
a - apothem (OS);
The apothem is the height of the side face of a regular pyramid, which is drawn from its top.
To find the lateral surface area of a regular pyramid, enter the pyramid perimeter and apothem values, 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 a part complete pyramid enclosed between the base and a section parallel to it. | |
| The cross section is called upper base of a truncated pyramid, and the base of the full pyramid is bottom base truncated pyramid. (The bases are similar.) Side faces truncated pyramid - trapezoid. In a 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 (the segment cut off from the height of the full pyramid). |  |
| Square full surface truncated pyramid is equal to the sum of the areas of its faces. | |
| The volume of the truncated pyramid ( S and s- base area, H- height) |  |
Body of rotation called a body formed as a result of the rotation of a line around a straight line.
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 axis of the cylinder. [ 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 sphere lies neither in the middle of the axis of the cylinder. [ 3 ]
Volume of any cylinder is equal to the product base area to height:
| 1. | V=π r 2 h |
Full area surface of the cylinder is equal to the sum of the lateral surface of the cylinder and double square base of the cylinder.
The formula for calculating the total surface area of a cylinder is:
27. A round cone can be obtained by rotation right triangle around one of its legs, so the round cone is also called the cone of revolution. See also Volume of a round cone
Total surface area of a circular cone is equal to the sum of the areas of the lateral surface of the cone and its base. The base of a 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 obtained by drawing a section parallel to the base of a cone. The body bounded by this section, the base and the side surface of the cone is called a truncated cone. See also Volume of a truncated cone
Total surface area of a truncated cone is equal to the sum of the areas of the lateral surface of the 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. Ball - geometric body bounded by a surface all of whose points are on equal distance from the center. This distance is called the radius of the sphere.
Sphere(Greek σφαῖρα - ball) - a closed surface, geometric place 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 (half axes, radii) are equal. A sphere is the surface of a ball.
The area of the spherical surface of the 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 the pyramid, the base of which has the same area as the surface of the ball, and the height is the radius of the ball
The volume of a sphere is one and a half times less than the volume of a cylinder circumscribed around it.
ball elements
Ball Segment The cutting plane splits the ball into two ball segments. H- segment height, 0< H < 2 R,
r- segment base radius,  Ball segment volume  The area of the spherical surface of the spherical segment |  |
| Spherical layer A spherical layer is a part of a sphere enclosed between two parallel sections. Distance ( H) between sections is called layer height, and the sections themselves - layer bases. Spherical surface area( volume) of the spherical layer can be found as the difference in areas spherical surfaces(volumes) of spherical segments. |  |
1. Multiplying a vector by a number(Fig. 56).
Vector product BUT per number λ called vector AT, whose modulus is equal to the product of the modulus of the vector BUT per modulo number λ :
The direction does not change if λ > 0 ; changes to the opposite if λ < 0 . If a λ = −1, then the vector
called a vector, opposite vector BUT, and is denoted
2. Vector addition. To find the sum of two vectors BUT and AT vector
Then the sum will be a vector, the beginning of which coincides with the beginning of the first, and the end - with the end of the second. This vector addition rule is called the “triangle rule” (Fig. 57). it is necessary to depict the summand 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 from a change in the places of the terms."
Let us indicate one more rule for adding vectors - the “parallelogram rule”. If we combine the beginnings of the summand vectors and build 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). In order to find 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 coincides with the beginning of the first, and the end With- with the end of the latter (Fig. 59).
3. Subtraction of vectors. The subtraction operation is reduced to the two previous operations: the difference of 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: it is necessary to combine the beginnings of the vectors BUT and AT, then their difference will be the vector
Drawn from the end of the vector AT towards the end of the vector BUT(Fig. 60).
In what follows, we will talk about the displacement vector material point, that is, a vector connecting the initial and final positions of the point. Agree that the introduced rules of action on vectors are quite obvious for displacement vectors.
4. Dot product of vectors. result dot product two vectors BUT and AT is the number c equal to the product of the modules of the vectors and the cosine of the angle α between
The scalar product of vectors is very widely used in physics. In the future, we will often have to deal with such an operation.
The article considers the concepts of parallelism of a straight line and a plane. The main definitions will be considered and examples will be given. Consider the sign of parallelism of a straight line to a plane with necessary and sufficient conditions for parallelism, we will solve examples of tasks in detail.
Yandex.RTB R-A-339285-1 Definition 1
Line and plane are called parallel if they don't have common points, that is, they do not intersect.
Parallelism is indicated by "∥". If in the task by condition the line a and the plane α are parallel, then the notation is a ∥ α . Consider the figure below.
It is believed that the line a, parallel to the plane α and the plane α, parallel to the line a, are equivalent, that is, the line and the plane are parallel to each other in any case.
Parallelism of a straight line and a plane - a sign and conditions of parallelism
It is not always obvious that a line and a plane are parallel. Often this needs to be proven. Nessesary to use sufficient condition, which will guarantee parallelism. Such a sign is called the sign of parallelism of a line and a plane. It is recommended to study the definition of parallel lines first.
Theorem 1
If a given line a, not lying in the plane α, is parallel to the line b, which belongs to the plane α, then the line a is parallel to the plane α.
Consider the theorem used to establish the parallelism of a straight line with a plane.
Theorem 2
If one of two parallel lines is parallel to a plane, then the other line lies in or is parallel to that plane.
A detailed proof is considered in the textbook of grades 10 - 11 on geometry. A necessary and sufficient condition for the parallelism of a straight line with a plane is possible if there is a definition of the directing vector of the straight line and the normal vector of the plane.
Theorem 3
For parallelism of the line a, which does not belong to the plane α, and the given plane, a necessary and sufficient condition is the perpendicularity of the directing vector to the line with normal vector given plane.
The condition is applicable when it is necessary to prove parallelism in rectangular system coordinates three-dimensional space. Let's look at the detailed proof.
Proof
Suppose the line a in the coordinate system O x y is given by the canonical equations of the line in space, which have the form x - x 1 a x \u003d y - y 1 a y \u003d z - z 1 a z or parametric equations line in space x = x 1 + a x λ y = y 1 + a y λ z = z 1 + a z λ , plane α with general equations of the plane A x + B y + C z + D = 0 .
Hence a → = (a x, a y, a z) is a directing vector with coordinates of the straight line a, n → = (A, B, C) is the normal vector of the given plane alpha.
To prove the perpendicularity of n → = (A , B , C) and a → = (a x , a y , a z) , you need to use the concept of dot product. That is, with the product a → , n → = a x · A + a y · B + a z · C, the result must be equal to zero from the condition of perpendicularity of the vectors.
This means that the necessary and sufficient condition for the parallelism of the line and the plane is written as follows: a → , n → = a x · A + a y · B + a z · C . Hence a → = (a x , a y , a z) is the direction vector of the line a with coordinates, and n → = (A , B , C) is the normal vector of the plane α .
Example 1
Determine if the line x = 1 + 2 λ y = - 2 + 3 λ z = 2 - 4 λ is parallel to the plane x + 6 y + 5 z + 4 = 0 .
Decision
We get that the provided line does not belong to the plane, since the coordinates of the line M (1 , - 2 , 2) do not fit. When substituting, we get that 1 + 6 (- 2) + 5 2 + 4 = 0 ⇔ 3 = 0 .
It is necessary to check for the feasibility of the necessary and sufficient condition for the parallelism of a straight line and a plane. We get that the coordinates of the directing vector of the line x = 1 + 2 λ y = - 2 + 3 λ z = 2 - 4 λ have the values a → = (2 , 3 , - 4) .
The normal vector for the x + 6 y + 5 z + 4 = 0 plane is n → = (1 , 6 , 5) . Let's proceed to the calculation of the scalar product of the vectors a → and n → . We get that a → , n → = 2 1 + 3 6 + (- 4) 5 = 0 .
Hence, the perpendicularity of the vectors a → and n → is obvious. It follows that the line and the plane are parallel.
Answer: line and plane are parallel.
Example 2
Determine the parallelism of the line A B in the coordinate plane O y z when the coordinates are given A (2, 3, 0) , B (4, - 1, - 7) .
Decision
By the condition, it can be seen that the point A (2, 3, 0) does not lie on the O x axis, since the value of x is not equal to 0.
For the O x z plane, the vector with coordinates i → = (1 , 0 , 0) is considered to be a normal vector of this plane. Denote the direction vector of the straight line A B as A B → . Now, using the coordinates of the beginning and end, we calculate the coordinates of the vector A B . We get that A B → = (2 , - 4 , - 7) . It is necessary to check for the feasibility of the necessary and sufficient conditions for the vectors A B → = (2 , - 4 , - 7) and i → = (1 , 0 , 0) to determine their perpendicularity.
Let's write A B → , i → = 2 1 + (- 4) 0 + (- 7) 0 = 2 ≠ 0 .
It follows from this that the line A B c coordinate plane O y z are not parallel.
Answer: are not parallel.
Not always the specified condition contributes easy definition proof of the parallelism of a line and a plane. There is a need to check whether the line a belongs to the plane α . There is one more sufficient condition by means of which the parallelism is proved.
For a given straight line a using the equation of two intersecting planes A 1 x + B 1 y + C 1 z + D 1 \u003d 0 A 2 x + B 2 y + C 2 z + D 2 \u003d 0, by the plane α - general equation plane A x + B y + C z + D = 0 .
Theorem 4
A necessary and sufficient condition for the parallelism of the line a and the plane α is the absence of solutions to the system linear equations, having the form A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 A x + B y + C z + D = 0 .
Proof
It follows from the definition that the line a with the plane α should not have common points, that is, they should not intersect, only in this case they will be considered parallel. This means that the coordinate system O x y z should not have points belonging to it and satisfying all the equations:
A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 , as well as the equation of the plane A x + B y + C z + D = 0 .
Therefore, a system of equations that has the form A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 A x + B y + C z + D = 0 , is called inconsistent.
The opposite is true: if there are no solutions to the system A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 A x + B y + C z + D = 0 there are no points in O x y z that satisfy all given equations simultaneously. We get that there is no such point with coordinates that could immediately be solutions of all equations A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 and equations A x + B y + C z + D = 0 . This means that we have a parallel line and a plane, since their intersection points are absent.
The system of equations A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 A x + B y + C z + D = 0 has no solution, when the rank of the main matrix is less than the rank of the extended one. This is verified by the Kronecker-Capelli theorem for solving linear equations. You can apply the Gauss method to determine its incompatibility.
Example 3
Prove that the line x - 1 = y + 2 - 1 = z 3 is parallel to the plane 6 x - 5 y + 1 3 z - 2 3 = 0 .
Decision
For solutions this example should move from canonical equation direct to the form of the equation of two intersecting planes. Let's write it like this:
x - 1 = y + 2 - 1 = z 3 ⇔ - 1 x = - 1 (y + 2) 3 x = - 1 z 3 (y + 2) = - 1 z ⇔ x - y - 2 = 0 3 x + z = 0
To prove the parallelism of the given line x - y - 2 = 0 3 x + z = 0 with the plane 6 x - 5 y + 1 3 z - 2 3 = 0 , it is necessary to transform the equations into a system of equations x - y - 2 = 0 3 x + z = 0 6 x - 5 y + 1 3 z - 2 3 = 0 .
We see that it is not solvable, so we will resort to the Gauss method.
Having written the equations, we get that 1 - 1 0 2 3 0 1 0 6 - 5 1 3 2 3 ~ 1 - 1 0 2 0 3 1 - 6 0 1 1 3 - 11 1 3 ~ 1 - 1 0 2 0 3 1 - 6 0 0 0 - 9 1 3 .
From this we conclude that the system of equations is inconsistent, since the line and the plane do not intersect, that is, they do not have common points.
We conclude that the line x - 1 \u003d y + 2 - 1 \u003d z 3 and the plane 6 x - 5 y + 1 3 z - 2 3 \u003d 0 are parallel, since the necessary and sufficient condition for parallelism of the plane with a given line was met.
Answer: line and plane are parallel.
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Some consequences of the axioms
Theorem 1:
Through a line and a point not lying on it passes a plane, and moreover, only one.
Given: M ₵ a
Prove: 1) There is α: a∈ α , М ∈ b ∈ α
2)
α is the only
Proof:
1) On a straight line and select points P and Q. Then we have 3 points - R, Q, M that do not lie on the same line.
2) According to axiom A1, a plane passes through three points that do not lie on one straight line, and moreover, only one, i.e. plane α, which contains the line a and the point M, exist.
3) Now let's prove thatα the only one. Suppose that there is a plane β that passes both through the point M and through the line a, but then this plane through the pointsP, Q, M. And after three points P, Q, M, not lying on one straight line, by virtue of axiom 1, only one plane passes.
4) Hence, this plane coincides with the plane α.Hence 1) On a straight line, but choose points P and Q. Then we have 3 points - P, Q, M, which do not lie on the same line.Hence α is unique.
The theorem has been proven.
1) On the line b, take a point N, which does not coincide with the point M, that is, N ∈ b, N≠M
2) Then we have a point N, which does not belong to the line a. According to the previous theorem, a plane passes through a 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 the line a and the line b. But then it also passes through the line a and the point N. But by the previous theorem, this plane is unique, i.e. the plane β coincides with the plane α.
4) So, we have proved the existence of a unique plane passing through two intersecting lines.
The theorem has been proven.
Parallel lines theorem
Theorem:
Through any point in space that does not lie on a given line, there passes a line parallel to the given line.
Given: straight a, M₵ a
Prove:There is only one directb ∥ a, M ∈ b
Proof:
1) Through the line a and the point M, which does not lie on it, one can draw a single plane (1st corollary). In the plane α one can draw a line b, parallel to a, passing through M.
2) Let's prove that it is the only one. Let us 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 line a. But through the line a and the point M passes the plane α.
3) Hence, α and β coincide. From the axiom of parallel lines it follows that the lines b and c coincide, since there is a unique line in the plane passing through given point and parallel to a given line.
The theorem has been proven.
The definition of parallel lines and their properties in space are the same as in the plane (see item 11).
At the same time, one more case of the arrangement of lines is possible in space - skew lines. Lines that do not intersect and do not lie in the same plane are called intersecting lines.
Figure 121 shows the layout of the living room. You see that the lines to which the segments AB and BC belong are skew.
The angle between intersecting lines is the angle between intersecting lines parallel to them. This angle does not depend on which intersecting lines are taken.
The degree measure of the angle between parallel lines is assumed to be zero.
A common perpendicular of two intersecting lines is a segment with ends on these lines, which is a perpendicular to each of them. It can be proved that two intersecting lines have a common perpendicular, and moreover, only one. It is a common perpendicular of the 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 the intersecting lines a and b (Fig. 122), it is necessary to draw parallel planes a and through each of these lines. The distance between these planes will be the distance between the intersecting lines a and b. In figure 122, this distance is, for example, the 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
Decision. The lines a and b lie in the same plane, and therefore any line intersecting each of them lies in the same plane. Therefore, if each of the lines a, b intersects both lines c and d, then the lines would lie in the same plane with the lines a and b, and this cannot be, since the lines intersect.
42. Parallelism of a straight line and a plane.
A line and a plane are called parallel if they do not intersect, that is, they do not have common points. If the line a is parallel to the plane a, then they 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 also parallel to the plane itself (a sign of parallelism of the line and the 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 the plane a, i.e. along the straight line b parallel to the plane a, i.e.
Example. Through the top right angle From rectangular triangle ABC A plane is drawn parallel to the hypotenuse at a distance of 10 cm from it. The projections of the legs on this plane are 30 and 50 cm. Find the projection of the hypotenuse on the same plane.
Decision. From right triangles BBVC and (Fig. 125) we find:
From triangle ABC we find:
The projection of the hypotenuse AB on the plane a is . Since AB is parallel to the plane a, then So,.
43. Parallel planes.
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 another plane (a sign of parallelism of two planes).
In Figure 126, the plane a is parallel to the intersecting lines a and b lying in the plane, then along these planes are parallel.
Through a point outside a given plane, one can draw a plane parallel to the given one, and moreover, only one.
If two parallel planes intersect with a third, then the lines of intersection are parallel.
Figure 127 shows two parallel planes, and the plane y intersects them along straight lines a and b. Then, by Theorem 2.7, we can assert that the lines a and b are parallel.
Segments of parallel lines enclosed between two parallel planes are equal.
According to T.2.8, the segments AB and shown in Figure 128 are equal, since
Let these planes intersect. Draw a plane perpendicular to the line of their intersection. It intersects these planes along two straight lines. The angle between these lines is called the angle between these planes (Fig. 129). The angle between the planes defined in this way does not depend on the choice of the secant plane.
