A line and a plane are called parallel if they have no common points. If a line not in a given plane is parallel to a line in that plane
1. If a plane passes through a given line parallel to another plane and intersects this plane, then the line of intersection of the planes is parallel to the given line.
2. If one of the two parallel lines is parallel to a given plane, and the other line has a plane common point, then this line lies in the given plane. plane, then it is parallel to the plane itself.
cases relative position straight and plane: a) the line lies in a plane;
b) a line and a plane have only one common point; c) a line and a plane have no common point.
2. Determination of the natural size of a segment of a straight line in general position by the method of a right triangle.
The natural value (n.v.) of a line segment AB in general position is the hypotenuse of a right triangle ABK. In this triangle, the leg AK is parallel to the plane of projections π1 and is equal to the horizontal projection of the segment A"B". The leg BK is equal to the difference between the distances of points A and B from the plane π1.
In the general case, to determine the natural size of a straight line segment, it is necessary to construct the hypotenuse of a right triangle, one leg of which is the horizontal (frontal) projection of the segment, the other leg is a segment equal in magnitude to the algebraic difference of the Z (Y) coordinates of the extreme points of the segment.
Angle α is found from a right-angled triangle - the angle of inclination of a straight line to the horizontal plane of projections.
To determine the angle of inclination of a straight line to the frontal projection plane, it is necessary to perform similar constructions on the frontal projection of the segment.
3. The main lines of the plane (horizontal, frontal).
The horizontal of the plane P is a straight line that lies in this plane and is parallel to the horizontal plane. The horizontal as a straight line parallel to the horizontal plane has a frontal projection ѓ parallel to the x-axis.
The front of the plane P is a straight line that lies in this plane and is parallel to the frontal plane.
The frontal is a straight line parallel to the frontal plane, and its horizontal projection f is parallel to the x-axis.
4. Mutual position of straight lines in space. Determination of visibility by competing points. Two straight lines in space can have a different location: A) intersect (lie in the same plane). A special case of intersection - at a right angle; B) can be parallel (lie in the same plane); C) coincide - a special case of parallelism; D) cross (lie in different planes and do not intersect).
Points whose projections on P1 coincide are called competing with respect to the plane P1, and the points whose projections on P2 coincide are called competing with respect to the plane P2.
Points K and L are competing with respect to the plane P1, since on the plane P1 points K and L are projected into one point: K1 = L1.
Point K is higher than point L, because K2 is higher than point L2, therefore K1 is visible on P1.
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 the line is parallel to the 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 a parallelepiped and passing through the middle of its diagonal, divides 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 name for the 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 rotating a right triangle around one of its legs, which is why a round cone is also called a 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 rl+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. A ball is a geometric body bounded by a surface, all points of which 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 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.
