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 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 is 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 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 is 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 a given point and parallel to a given line.
The theorem is proven.
Elementary geometry studies the concepts and relationships of objects. Without a clear justification, one cannot navigate the application field. The sign of parallelism between a straight line and a plane is the first step into the geometry of space. Mastery of initial categories will allow you to get closer to the fascinating world of precision, logic, clarity.
Correlation of objects: possible options
Stereometry is a tool for understanding the world. She examines the relationship of objects to each other and teaches how to calculate distances without a ruler. Successful practice requires master basic concepts.
There is a surface a and a line l. There are three cases of object relationships. They are determined by the points of intersection. Easy to remember:
- 0 points - parallel;
- 1 point - mutually intersect;
- infinitely many - the straight line lies in the plane.
It is easy to describe the sign of parallelism of objects. On surface a there is a line with || l, then l || A.
A simple statement requires proof. Let the surface be drawn through the lines: l || c. In Ω a = c. Let l have a common point with a. It should lie on p. This contradicts the condition: l || c. Then l is parallel to plane a. The starting position is correct.
Important! There is at least one line in space || flat surface. This is consistent with the statement of initial geometry (planimetry).
A simple thought: a belongs to more than one point l, which means the straight line l belongs entirely to a.
a || l only in case lack of a single point of intersection.
This is a logical definition of parallelism between a line and a plane.
It is easy to find a practical application of the position. How to prove that one line is parallel to a plane?
It is enough to use the studied feature.
What is useful to know
To solve problems competently, you need to study additional locations of objects. The basis is a sign of parallelism between a straight line and a plane. Its use will make it easier to understand other elements. The geometry of space considers special cases.
Intersections in stereometry
The objects are the same: flat surface a, lines c, l. How are they next to each other? With || l. L intersects a. It’s easy to understand: c will definitely intersect a. This idea is a lemma about the intersection of a plane by parallel lines.
The field of activity is expanding. Surface c is added to the objects under study. She owns l. Nothing changes in the original objects: l || A. Again it's simple: in case of intersection of planes, common line d || l. The concept immediately follows: which two planes are called intersecting. Those that have a common line.
What theorems need to be studied
The main concepts of the relationship of objects lead to a description of the main statements. They require extensive evidence. First: theorems on the parallelism of one line and a plane. Various cases are considered.
- Objects: surfaces P, Q, R, straight lines AB, CD. Condition: P||Q, R intersects them. Naturally, AB||CD.
- Subjects of research: lines AB, CD, A1B1, C1D1. AB intersects CD in one plane, A1B1 intersects C1D1 in another. AB||A1B1, CD||C1D1. Conclusion: surfaces containing pairwise intersecting parallel lines, ||.
A new concept emerges . Crossing lines are not themselves parallel. although they lie in parallel planes. These are C1D1 and AB, A1B1 and CD. This phenomenon is widely used in practical stereometry.
Natural statement: through one of the crossing lines it is real there is only one plane parallel to the indicated plane.
- Then it is easy to arrive at the trace theorem. This is the third statement about the parallelism of a line and a surface. There is a straight line l. She || A. l belongs to. In Ω a = d. The only possible option: d || l.
Important! A straight line and a plane are called || in the absence of common objects - points.
Properties of parallelism and their proofs
It is easy to come to the concept of the arrangement of flat surfaces:
- the empty set of common points (called parallel);
- intersect in a straight line.
They are used in stereometry properties of parallelism. Any spatial picture has surfaces and lines. To successfully solve problems, you need to study the basic theorems:
- Objects under study: a || b; c Ω b = l, c Ω a = m. Conclusion: l ||m. The assumption requires proof. The location of l and m is one of two: intersecting or parallel. But in the second case, the surfaces do not have common points. Then l || m. The statement has been proven. It should be remembered: if a line lies in a plane, then they have more than one intersection point.
- There is a surface a, point A does not belong to a. Then there is only one surface b || a passing through A. Proving the position is simple. Let l Ω m; l, m belong to a. A plane is constructed through each of them and A. She crosses a. There is a line in it passing through A and || A. At point A they intersect. They form a single surface b || a.
- There are skew lines l and m. Then there are || surfaces a and b to which l and m belong. The logical thing to do is to choose arbitrary points on l and m. Spend m1 || m, l1 || l. Intersecting lines in pairs || => a || b. The situation has been proven.
Knowledge of the properties of parallelism of one straight line and plane will allow you to skillfully apply them in practice. Simple and logical proofs will help you navigate the fascinating world of stereometry.
Planes: Parallelism Evaluation
The concept is easy to describe. Question: what does it mean that one straight line and a plane are parallel, solved. The study of the initial categories of space geometry led to a more complex statement.
When solving applied problems, the parallelism feature is used. Simple description: let l Ω m, l1 Ω m1, l, m belong to a, l1, m1 – b. In this case l || l1, m || m1. Then a || b.
Without the use of mathematical symbols: planes are called parallel if they are drawn through pairs of intersecting parallel lines.
Stereometry reviews properties of parallel planes. They are described by theorems:
Objects under study: a || b, a Ω c = l, b Ω c = m. Then l || m. The evidence is clear. and Lines lie in the same plane if they || or intersect. The statement about the parallelism of the line and the surface should be applied. Then it becomes obvious: l and m cannot intersect. The only thing left is l || m.
1. Formulate the definition of skew lines. Formulate and prove a theorem expressing the characteristic of skew lines. 2/Prove that if twolines are parallel to the third line, then they are parallel. 3. Construct a section of the parallelepiped ABCDA1B1C1D1 with a plane passing through points A, C and M, where M is the middle of the edge AlDl.
Which of the figures is not the main figure in space? 1) point; 2) segment; 3) straight; 4) plane.2. Directa andb interbreeding. How is the line located?b relative to the plane α, if the line is a ϵ α?
1) crosses; 2) parallel; 3) lies in a plane; 4) crosses.
3. Determine which statement is true:
1) The perpendicular is longer than the inclined one.
2) If two obliques are not equal, then the larger oblique has a smaller projection.
3) A line is perpendicular to a plane if it is perpendicular to the two sides of the triangle lying in this plane.
4) The angle between a parallel line and a plane is 90º.
4. The distance between two parallel planes is 8 cm. A straight segment, the length of which is 17 cm, is located between them so that its ends belong to the planes. Find the projection of this segment onto each of the planes.
1) 15 cm; 2) 9 cm; 3) 25 cm) 4) 12 cm.
5. A perpendicular TE equal to 6 dm is drawn to the MCRT plane. Calculate the distance from point E to the vertex of the rhombus K, if MK = 8 dm, the angle M of the rhombus is 60º.
1) 10 dm; 2) 14 dm; 3) 8 dm; 4) 12 dm.
6. The hypotenuse of a right triangle is 12 cm. Outside the plane of the triangle, a point is given that is 10 cm away from each vertex of the triangle. Find the distance from the point to the plane of the triangle.
1) 4 cm; 2) 16 cm; 3) 8 cm; 4) 10 cm.
7. From a certain point, a perpendicular and an inclined line are drawn to a given plane, the angle between which is 60º. Find the projection of the inclined plane on the given plane if the perpendicular is 5 cm.
1) 5√3 cm; 2) 10 cm; 3) 5 cm; 4) 10√3 cm.
8. Find the lateral surface of a regular triangular pyramid if the side of the base is 2 cm and all dihedral angles at the base are 30º.
1) 2 cm2; 2) 2√3 cm2; 3) √3 cm2; 4) 3√2 cm2.
9. Find the surface area of a rectangular parallelepiped based on its three dimensions, equal to 3 cm, 4 cm, 5 cm.
1) 94 cm2; 2) 47 cm2; 3) 20 cm2; 4) 54 cm2.
plane.
b) if one of two parallel lines intersects a given plane, then the other line also intersects this plane.
c) if two lines are parallel to a third line, then they intersect
d) if a straight line and a plane do not have common points, then the straight line lies in the plane
e) a straight line and a plane are called intersecting if they do not have common points
plane; b) if one of two parallel lines intersects a given plane, then the other line also intersects this plane; c) if two lines are parallel to a third line, then they intersect; d) if the line and the plane do not have common points, then the line lies in plane) a straight line and a plane are said to intersect if they do not have common points.
2. Line c, parallel to line a, intersects plane β. Line b is parallel to line a, then:
