points that do not lie on the same line? a) intersect; b) nothing can be said; c) do not intersect; d) match; e) have three common points.
2. Which of the following statements is correct? a) If two points of a circle lie in a plane, then the whole circle lies in this plane; b) a straight line lying in the plane of a triangle intersects two of its sides; c) any two planes have only one common point; d) a plane passes through two points, and moreover, only one; e) a line lies in the plane of a given triangle if it intersects two lines containing the sides of the triangle.
3. Can two different planes have only two common points? a) Never; b) I can, but under additional conditions; c) always have; d) the question cannot be answered; d) another answer.
4. Points K, L, M lie on one straight line, point N does not lie on it. One plane is drawn through every three points. How many different planes did this result in? a) 1; b) 2; at 3; d) 4; e) infinitely many.
5. Choose the correct statement. a) A plane passes through any three points, and moreover, only one; b) if two points of a line lie in a plane, then all points of the line lie in this plane; c) if two planes have a common point, then they do not intersect; d) through a line and a point lying on it, a plane passes, and moreover, only one; e) A plane cannot be drawn through two intersecting lines.
6. Name the common line of the planes PBM and MAB. a) PM b) AB; c) PB; d) BM; d) cannot be determined.
7. Lines a and b intersect at point M. Line c not passing through point M intersects lines a and b. What can be said about the mutual position of lines a, b and c? a) All lines lie in different planes; b) lines a and b lie in the same plane; c) all lines lie in the same plane; d) nothing can be said e) line c coincides with one of the lines: either with a or with b.
8. Lines a and b intersect at point O. A € a, B € b, Y € AB. Choose the correct statement. a) Points O and Y do not lie in the same plane; b) lines OY and a are parallel; c) lines a, b and point Y lie in the same plane; d) points O and Y coincide; e) points Y and A coincide.
Option 2.1. What can be said about the relative position of two planes that have three common points that do not lie on one straight line?
a) intersect; b) nothing can be said; c) do not intersect; d) match; e) have three common points.
2. Which of the following statements is correct?
a) If two points of a circle lie in a plane, then the whole circle lies in this plane; b) a straight line lying in the plane of a triangle intersects two of its sides; c) any two planes have only one common point; d) a plane passes through two points, and moreover, only one; e) a line lies in the plane of a given triangle if it intersects two lines containing the sides of the triangle.
3. Can two different planes have only two common points?
a) Never; b) I can, but under additional conditions; c) always have; d) the question cannot be answered; d) another answer.
4. Points K, L, M lie on one straight line, point N does not lie on it. One plane is drawn through every three points. How many different planes did this result in?
a) 1; b) 2; at 3; d) 4; e) infinitely many.
5. Choose the correct statement.
a) A plane passes through any three points, and moreover, only one; b) if two points of a line lie in a plane, then all points of the line lie in this plane; c) if two planes have a common point, then they do not intersect; d) through a line and a point lying on it, a plane passes, and moreover, only one; e) A plane cannot be drawn through two intersecting lines.
6. Name the common line of the planes PBM and MAB.
a) PM b) AB; c) PB; d) BM; d) cannot be determined.
7. Which of the listed planes does the straight line RM intersect (Fig. 1)?
a) DD1C; b) D1PM; c) B1PM; d) ABC; e) CDA.
B1 C1
8. Two planes intersect in a straight line c. The point M lies in only one of the planes. What can be said about the relative position of the point M and the line c?
a) No conclusion can be drawn; b) the line c passes through the point M; c) the point M lies on the line c; d) line c does not pass through point M; d) another answer.
9. Lines a and b intersect at point M. Line c not passing through point M intersects lines a and b. What can be said about the mutual position of lines a, b and c?
a) All lines lie in different planes; b) lines a and b lie in the same plane; c) all lines lie in the same plane; d) nothing can be said e) line c coincides with one of the lines: either with a or with b.
10. Lines a and b intersect at point O. A € a, B € b, Y € AB. Choose the correct statement.
a) Points O and Y do not lie in the same plane; b) lines OY and a are parallel; c) lines a, b and point Y lie in the same plane; d) points O and Y coincide; e) points Y and A coincide.
rays AB and AC are perpendicular to its edge? 2. Is it true that the linear angle BAC dihedral angle if the rays AB and AC lie on the faces of the dihedral angle? 3. Is it true that the angle BAC is the linear angle of a dihedral angle if the rays AB and AC are perpendicular to its edge, and the points E and C lie on the faces of the angle? 4. The linear angle of a dihedral angle is 80 degrees. Is there a line in one of the faces of the angle that is perpendicular to the other face? 5. Angle ABC - a linear angle of a dihedral angle with an alpha edge. Is the line alpha perpendicular to the plane ABC? Is it true that all lines perpendicular to a given plane and intersecting a given line lie in the same plane?
Axioms of stereometry.
A1. Through any three points that do not lie on a given line, a plane passes, and moreover, only one;
Sl.1. Through a line and a point not lying on it passes a plane, and moreover, only one;
Sl.2. Through two intersecting lines passes a plane, and moreover, only one;
Sl.3. A plane passes through two parallel lines, and moreover, only one.
A2. If two points of a line lie in a plane, then all points of the line lie in this plane;
A3. If two planes have a common point, then they have a common straight line on which all the common points of these planes lie.
The main figures of stereometry- points (A, B, C…), straight (a, b, c…), plane ( …) , polyhedra and bodies of revolution.
Under cutting plane volumetric figure we will understand the plane, on both sides of which there are points of this figure.
Per measure of distance between a point, a line and a plane we will take the length of their common perpendicular.
2. Mutual arrangement of lines in space.
In space, two straight lines can be parallel, intersect or intersect.
| 1A | Def. Parallel straight lines in space are straight lines that lie in the same plane and do not intersect. According to the 3. A plane passes through two parallel lines, and moreover, only one. | |
| 1B | T 1 (on transitivity). Two lines parallel to a third are parallel to each other. | |
| 2A | According to word 2. After two intersecting straight lines pass through a plane, and moreover, only one | |
| 3A | Def. The two lines are called interbreeding if they do not lie in the same plane. | |
| T 2 (A sign of intersecting lines). If one of the two lines lies in a certain plane, and the other line intersects this plane at a point that does not belong to the first line, then such lines are skew. | ||
| 3B | Def. Angle between skew lines is the angle between intersecting lines parallel to them. | |
| 3B | Def. A common perpendicular of two intersecting lines is a segment that has ends on these lines and is perpendicular to them (distance between skew lines). |
- Mutual arrangement of lines and planes in space.
In space, a straight line and a plane can be parallel, intersect or straight can lie entirely in a plane.
| 1A | Def. Straight called parallel plane, if it is parallel to any line lying in this plane. | |
| 1B | T 3 (A sign of parallelism of a straight line and a plane). A line not lying in a plane is parallel to a plane if it is parallel to some line lying in that plane. | |
| 2A | Def. Direct called perpendicular to the plane , if it is perpendicular to any intersecting lines lying in this plane. | |
| 2B | T 4 (a sign of perpendicularity of a straight line and a plane) If a line intersecting with a plane is perpendicular to any two intersecting lines lying in this plane, then it is also perpendicular to any third line lying in this plane. | |
| 2B | T 5 (about two parallel lines perpendicular to the third). If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to that plane. | |
| 2G | Def. The angle between a line and a plane is the angle between a given line and its projection onto the plane. | |
| 2D | Def. Any other straight line, different from the perpendicular and intersecting the plane, is called oblique to this plane (fig. see below). Def. Projection oblique onto a plane called the segment connecting the base of the perpendicular and the oblique. T 6 (about the length of the perpendicular and oblique). 1) The perpendicular drawn to the plane is shorter than the inclined one to this plane; 2) Equal oblique correspond to equal projections; 3) Of the two inclined ones, the one whose projection is larger is larger. | |
| 2E | T 7 (about three perpendiculars). A straight line drawn on a plane through the base of an inclined projection perpendicular to it is also perpendicular to the most inclined one. T 8 (reverse). A straight line drawn on a plane through the base of an inclined plane and perpendicular to it is also perpendicular to the projection of the inclined plane onto this plane. | |
| 3A | According to axiom 2. If two points of a straight line lie in a plane, then all points of a straight line lie in this plane |
- Mutual arrangement of planes in space.
In space, planes can be parallel or cross.
| 1A | Def. Two plane called parallel if they do not intersect. | |
| T 9 (sign of parallel planes). If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel. | ||
| 1B | T 10 If two parallel planes are intersected by a third plane, then the direct intersections are parallel (property of parallel planes 1). | |
| 1B | T 11 Segments of parallel lines enclosed between parallel planes are equal (property of parallel planes 2). | |
| 2A | By axiom 3 . If two planes have a common point, then they have a common line on which all common points of these planes lie ( planes intersect in a straight line). | |
| 2B | T 12 (a sign of perpendicularity of planes). If a plane passes through a line perpendicular to another plane, then these planes are perpendicular. | |
| 2B | Def. dihedral angle a figure formed by two half-planes emanating from one straight line is called. A plane perpendicular to an edge of a dihedral angle intersects its faces along two rays. The angle formed by these rays is called linear angle of a dihedral angle. Per dihedral angle measure the measure of the corresponding linear angle is taken. |
I5 Whatever the three points that do not lie on the same line, there is at most one plane passing through these points.
I6 If two points A and B of a line lie in the plane a, then each point of the line a lies in the plane a. (In this case we will say that the line a lies in the plane a or that the plane a passes through the line a.
I7 If two planes a and b have a common point A, then they have at least one more common point B.
I8 There are at least four points that do not lie in the same plane.
Already from these 8 axioms, several theorems of elementary geometry can be deduced, which are clearly obvious and, therefore, are not proved in the school geometry course and even sometimes, for logical reasons, are included in the axioms of a particular school course
For example:
1. Two lines have at most one common point.
2. If two planes have a common point, then they have a common line on which all common points of these two planes lie
Proof: (for show off):
By I 7 $ B, which also belongs to a and b, because A, B "a, then according to I 6 AB "b. So the line AB is common to two planes.
3. Through a line and a point not lying on it, as well as through two intersecting lines, one and only one plane passes.
4. There are three points on each plane that do not lie on one straight line.
COMMENT: With these axioms, you can prove a few theorems, and most of them are so simple. In particular, it cannot be proved from these axioms that the set geometric elements endlessly.
GROUP II Axioms of order.
If three points are given on a straight line, then one of them can be located to the other two in the relation "to lie between", which satisfies the following axioms:
II1 If B lies between A and C, then A, B, C are distinct points of the same line, and B lies between C and A.
II2 Whatever two points A and B are, there is at least one point C on line AB such that B lies between A and C.
II3 Among any three points of a line, there is at most one point lying between two others.
According to Hilbert, a pair of points A and B is understood over a segment AB(BA). Points A and B are called the ends of the segment, and any point lying between points A and B is called an interior point of the segment AB(BA).
COMMENT: But from II 1-II 3 it does not yet follow that every segment has interior points, but from II 2, z that the segment has exterior points.
II4 (Pasch's axiom) Let A, B, C be three points that do not lie on the same straight line, and let A be a straight line in the plane ABC that does not pass through any of the points A, B, C. Then if the line a passes through the point of the segment AB, then it also passes through the point of the segment AC or BC.
Sl.1: Whatever the points A and C, there is at least one point D on the line AC lying between A and C.
Doc-in: I 3 Þ$ i.e. not lying on the line AC
Sl.2. If C lies on the segment AD and B between A and C, then B lies between A and D, and C lies between B and D.Now we can prove two statements
DC3 Assertion II 4 also holds if the points A, B and C lie on the same straight line.
And the most interesting.
Sl.4 . Between any two points of a line there is an infinite number of other points on it (self-sufficient).
However, it cannot be established that the set of points of the line is uncountable. .
The axioms of groups I and II allow us to introduce such important concepts as half-plane, ray, half-space and angle. Let's prove the theorem first.
Th1. The line a lying in the plane a divides the set of points of this plane that do not lie on the line a into two non-empty subsets so that if points A and B belong to the same subset, then the segment AB has no common points with the line a; if these points belong to different subsets, then the segment AB has a common point with the line a.
Idea: a relation is introduced, namely, t. A and B Ï a are in relation to Δ if the segment AB has no common points with the line a or these points coincide. Then the sets of equivalence classes with respect to Δ were considered. It is proved that there are only two of them using simple arguments.
ODA1 Each of the subsets of points defined by the previous theorem is called a half-plane with boundary a.
Similarly, we can introduce the concepts of a ray and a half-space.
Ray- h, and the straight line is .
ODA2 An angle is a pair of rays h and k emanating from the same point O and not lying on the same straight line. so O is called the vertex of the angle, and the rays h and k are called the sides of the angle. Denoted in the usual way: Ðhk.
The point M is called an internal point of the angle hk if the point M and the ray k lie in the same half-plane with the boundary and the point M and the ray k lie in the same half-plane with the boundary. The set of all interior points is called the interior of the angle.
outer area angle - an infinite set, because all points of the segment with ends on different sides of the angle are internal. For methodological reasons, the following property is often included in axioms.
Property: If a ray starts from a vertex of an angle and passes through at least one interior point of that angle, then it intersects any segment with ends on different sides of the angle. (Self.)
GROUP III. Axioms of congruence (equality)
On the set of segments and angles, a congruence or equality relation is introduced (denoted by “=”), which satisfies the axioms:
III 1 If given a segment AB and a ray emanating from point A / , then $ t.B / belonging to this ray, so that AB=A / B / .
III 2 If A / B / =AB and A // B // =AB, then A / B / =A // B // .
III 3 Let А-В-С, А / -В / -С / , АВ=А / В / and ВС=В / С / , then AC=А / С /
ODA3 If O / is a point, h / is a ray emanating from this point, and l / is a half-plane with boundary , then the triple of objects O / ,h / and l / is called a flag (O / ,h / ,l /).
III 4 Let Ðhk and a flag (O / ,h / ,l /) be given. Then in the half-plane l / there is a unique ray k / emanating from the point O / such that Ðhk = Ðh / k / .
III 5 Let A, B and C be three points that do not lie on the same straight line. If at the same time AB=A / B / , AC=A / C / , ÐB / A / C / = ÐBAC, then RABC = ÐA / B / C / .
1. Point B / B III 1 is the only one on this beam (self.)
2. The relation of congruence of segments is an equivalence relation on the set of segments.
3. In an isosceles triangle, the angles at the bases are equal. (According to III 5).
4. Signs of equality of triangles.
5. An angle congruence relation is an equivalence relation on a set of angles. (Report)
6. An exterior angle of a triangle is greater than every angle of the triangle that is not adjacent to it.
7. In each triangle, a larger angle lies opposite the larger side.
8. Any segment has one and only one midpoint
9. Any angle has one and only one bisector
You can introduce the following concepts:
ODA4 An angle equal to its adjacent angle is called a right angle..
Can define vertical angles, perpendicular and oblique, etc.
It is possible to prove the uniqueness of ^. You can introduce the concepts > and< для отрезков и углов:
ODA5 If segments AB and A / B / and $ t.C are given, so that A / -C-B / and A / C \u003d AB, then A / B / > AB.
ODA6 If two angles Ðhk and Ðh / k / are given, and if a ray l can be drawn through the interior of Ðhk and its vertex such that Ðh / k / = Ðhl, then Ðhk > Ðh / k / .
And the most interesting thing is that with the help of the axioms of groups I-III it is possible to introduce the concept of movement (overlay).
It's done like this:
Let two sets of points p and p / be given. Let us assume that a one-to-one correspondence is established between the points of these sets. Each pair of points M and N of the set p determines the segment MN. Let М / and N / be points of the set p / corresponding to points МN. We will agree to call the segment M / N / corresponding to the segment MN.
ODA7 If $ the correspondence between p and p / is such that the corresponding segments always turn out to be mutually congruent, then sets p and p / are called congruent . It is also said that each of the sets p and p / is obtained movement from another or that one of these sets can be superimposed on another. The corresponding points of the set p and p / are called superimposed.
App1: Points lying on a line, when moving, pass into points also lying on some line.
Utv2 The angle between two segments connecting any point of the set with two other points is congruent to the angle between the corresponding segments of the congruent set.
You can introduce the concept of rotation, shift, composition of movements, etc.
GROUP IV. Axioms of continuity and.
IV 1 (Axiom of Archimedes). Let AB and CD be some segments. Then on the line AB there is a finite set of points А 1 , А 2 , …, А n such that the following conditions are met:
1. A-A 1 -A 2, A 1 -A 2 -A 3, ..., A n -2 -A n -1 -A n
2. AA 1 = A 1 A 2 = … = A n-1 A n = CD
3. A-B-An
IV2 (Cantor's Axiom) Let an infinite sequence of segments А1В1, А2В2,… be given on an arbitrary line a, of which each subsequent one lies inside the previous one and, in addition, for any segment CD there is natural number n such that AnBn< СD. Тогда на прямой а существует т.М, принадлежащая каждому из отрезков данной последовательности.
