Straight line. Parallel lines. Basic concepts. Signs of parallelism of two lines. Properties of parallel lines The first sign of parallelism of two lines

The parallelism of two lines can be proved on the basis of the theorem, according to which, two perpendiculars drawn with respect to one line will be parallel. There are certain signs of parallel lines - there are three of them, and we will consider all of them more specifically.

The first sign of parallelism

Lines are parallel if, at the intersection of their third line, the formed internal angles lying across are equal.

Suppose, at the intersection of lines AB and CD with a straight line EF, angles /1 and /2 were formed. They are equal, since the straight line EF runs at the same slope with respect to the other two straight lines. At the intersection of the lines, we put the points Ki L - we have a segment of the secant EF. We find its middle and put a point O (Fig. 189).

On the line AB we drop the perpendicular from the point O. Let's call it OM. We continue the perpendicular until it intersects with the line CD. As a result, the original line AB is strictly perpendicular to MN, which means that CD _ | _ MN, but this statement requires proof. As a result of drawing the perpendicular and the line of intersection, we have formed two triangles. One of them is MINE, the second is NOK. Let's consider them in more detail. signs of parallel lines grade 7

These triangles are equal, because, in accordance with the conditions of the theorem, /1 = /2, and in accordance with the construction of triangles, the side OK = the side OL. Angle MOL =/NOK since these are vertical angles. It follows from this that the side and two angles adjacent to it of one of the triangles are respectively equal to the side and two angles adjacent to it of the other of the triangles. Thus, the triangle MOL \u003d triangle NOK, and hence the angle LMO \u003d angle KNO, but we know that / LMO is a right one, which means that the corresponding angle KNO is also right. That is, we managed to prove that both the line AB and the line CD are perpendicular to the line MN. That is, AB and CD are parallel to each other. This is what we needed to prove. Let us consider the remaining signs of parallel lines (class 7), which differ from the first sign in the way of proof.

The second sign of parallelism

According to the second sign of parallelism of lines, we need to prove that the angles obtained in the process of intersection of parallel lines AB and CD by line EF will be equal. Thus, the signs of parallelism of two lines, both the first and the second, are based on the equality of the angles obtained when they are crossed by the third line. We assume that /3 = /2, and the angle 1 = /3, since it is vertical to it. Thus, and /2 will be equal to angle 1, however, it should be taken into account that both angle 1 and angle 2 are internal, cross-lying angles. Therefore, it remains for us to apply our knowledge, namely, that two segments will be parallel if, at their intersection with a third line, the formed, cross-lying angles will be equal. Thus, we found out that AB || CD.

We managed to prove that under the condition that two perpendiculars are parallel to one straight line, according to the corresponding theorem, the sign of parallel lines is obvious.

The third sign of parallelism

There is also a third criterion for parallelism, which is proved by means of the sum of one-sided interior angles. Such a proof of the sign of parallelism of lines allows us to conclude that two lines will be parallel if, when they intersect with a third line, the sum of the obtained one-sided internal angles will be equal to 2d. See figure 192.

This article is about parallel lines and about parallel lines. First, the definition of parallel lines in the plane and in space is given, notation is introduced, examples and graphic illustrations of parallel lines are given. Further, the signs and conditions of parallelism of straight lines are analyzed. In conclusion, solutions are shown for typical problems of proving the parallelism of straight lines, which are given by some equations of a straight line in a rectangular coordinate system on a plane and in three-dimensional space.

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Parallel lines - basic information.

Definition.

Two lines in a plane are called parallel if they do not have common points.

Definition.

Two lines in three dimensions are called parallel if they lie in the same plane and have no common points.

Note that the "if they lie in the same plane" clause in the definition of parallel lines in space is very important. Let's clarify this point: two straight lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but are skew.

Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel lines. The straight lines along which the plane of the wall of the house intersects the planes of the ceiling and floor are parallel. Railroad tracks on level ground can also be thought of as parallel lines.

The symbol "" is used to denote parallel lines. That is, if the lines a and b are parallel, then you can briefly write a b.

Note that if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.

Let us voice a statement that plays an important role in the study of parallel lines in the plane: through a point not lying on a given line, there passes the only line parallel to the given one. This statement is accepted as a fact (it cannot be proved on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem can be easily proved using the axiom of parallel lines given above (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the bibliography).

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem is easily proved using the axiom of parallel lines given above.

Parallelism of lines - signs and conditions of parallelism.

A sign of parallel lines is a sufficient condition for parallel lines, that is, such a condition, the fulfillment of which guarantees parallel lines. In other words, the fulfillment of this condition is sufficient to state the fact that the lines are parallel.

There are also necessary and sufficient conditions for parallel lines in the plane and in three-dimensional space.

Let us explain the meaning of the phrase "necessary and sufficient condition for parallel lines".

We have already dealt with the sufficient condition for parallel lines. And what is the "necessary condition for parallel lines"? By the name "necessary" it is clear that the fulfillment of this condition is necessary for the lines to be parallel. In other words, if the necessary condition for parallel lines is not satisfied, then the lines are not parallel. Thus, necessary and sufficient condition for lines to be parallel is a condition, the fulfillment of which is both necessary and sufficient for parallel lines. That is, on the one hand, this is a sign of parallel lines, and on the other hand, this is a property that parallel lines have.

Before stating the necessary and sufficient condition for lines to be parallel, it is useful to recall a few auxiliary definitions.

secant line is a line that intersects each of the two given non-coincident lines.

At the intersection of two lines of a secant, eight non-deployed ones are formed. The so-called lying crosswise, corresponding And one-sided corners. Let's show them on the drawing.

Theorem.

If two straight lines on a plane are crossed by a secant, then for their parallelism it is necessary and sufficient that the crosswise lying angles are equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphical illustration of this necessary and sufficient condition for parallel lines in the plane.

You can find proofs of these conditions for parallel lines in geometry textbooks for grades 7-9.

Note that these conditions can also be used in three-dimensional space - the main thing is that the two lines and the secant lie in the same plane.

Here are a few more theorems that are often used in proving the parallelism of lines.

Theorem.

If two lines in a plane are parallel to a third line, then they are parallel. The proof of this feature follows from the axiom of parallel lines.

There is a similar condition for parallel lines in three-dimensional space.

Theorem.

If two lines in space are parallel to a third line, then they are parallel. The proof of this feature is considered in the geometry lessons in grade 10.

Let us illustrate the voiced theorems.

Let us give one more theorem that allows us to prove the parallelism of lines in the plane.

Theorem.

If two lines in a plane are perpendicular to a third line, then they are parallel.

There is a similar theorem for lines in space.

Theorem.

If two lines in three-dimensional space are perpendicular to the same plane, then they are parallel.

Let us draw pictures corresponding to these theorems.

All the theorems formulated above, signs and necessary and sufficient conditions are perfectly suitable for proving the parallelism of straight lines by methods of geometry. That is, to prove the parallelism of two given lines, it is necessary to show that they are parallel to the third line, or to show the equality of cross-lying angles, etc. Many of these problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the method of coordinates to prove the parallelism of lines in a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are given in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this section of the article, we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations that determine these lines, and we will also give detailed solutions to typical problems.

Let's start with the condition of parallelism of two lines on the plane in the rectangular coordinate system Oxy . His proof is based on the definition of the directing vector of the line and the definition of the normal vector of the line on the plane.

Theorem.

For two non-coincident lines to be parallel in a plane, it is necessary and sufficient that the direction vectors of these lines are collinear, or the normal vectors of these lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the second line.

Obviously, the condition of parallelism of two lines in the plane reduces to (direction vectors of lines or normal vectors of lines) or to (direction vector of one line and normal vector of the second line). Thus, if and are the direction vectors of the lines a and b, and And are the normal vectors of lines a and b, respectively, then the necessary and sufficient condition for parallel lines a and b can be written as , or , or , where t is some real number. In turn, the coordinates of the directing and (or) normal vectors of the straight lines a and b are found from the known equations of the straight lines.

In particular, if the line a in the rectangular coordinate system Oxy on the plane defines the general equation of the line of the form , and the straight line b - , then the normal vectors of these lines have coordinates and respectively, and the condition of parallelism of lines a and b will be written as .

If the straight line a corresponds to the equation of the straight line with the slope coefficient of the form . Therefore, if straight lines on a plane in a rectangular coordinate system are parallel and can be given by equations of straight lines with slope coefficients, then the slope coefficients of the lines will be equal. And vice versa: if non-coinciding straight lines on a plane in a rectangular coordinate system can be given by the equations of a straight line with equal slope coefficients, then such straight lines are parallel.

If the line a and the line b in a rectangular coordinate system define the canonical equations of the line on the plane of the form And , or parametric equations of a straight line on a plane of the form And respectively, then the direction vectors of these lines have coordinates and , and the parallelism condition for lines a and b is written as .

Let's take a look at a few examples.

Example.

Are the lines parallel? And ?

Solution.

We rewrite the equation of a straight line in segments in the form of a general equation of a straight line: . Now we can see that is the normal vector of the straight line , and is the normal vector of the straight line. These vectors are not collinear, since there is no real number t for which the equality ( ). Consequently, the necessary and sufficient condition for the parallelism of lines on the plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are lines and parallels?

Solution.

We bring the canonical equation of a straight line to the equation of a straight line with a slope: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the slopes of the lines are equal, therefore, the original lines are parallel.

The second solution.

First, let's show that the original lines do not coincide: take any point of the line, for example, (0, 1) , the coordinates of this point do not satisfy the equation of the line, therefore, the lines do not coincide. Now let's check the fulfillment of the condition of parallelism of these lines. The normal vector of the line is the vector , and the direction vector of the line is the vector . Let's calculate and : . Consequently, the vectors and are perpendicular, which means that the necessary and sufficient condition for the parallelism of the given lines is satisfied. So the lines are parallel.

Answer:

The given lines are parallel.

To prove the parallelism of lines in a rectangular coordinate system in three-dimensional space, the following necessary and sufficient condition is used.

Theorem.

For non-coincident lines to be parallel in three-dimensional space, it is necessary and sufficient that their direction vectors be collinear.

Thus, if the equations of lines in a rectangular coordinate system in three-dimensional space are known and you need to answer the question whether these lines are parallel or not, then you need to find the coordinates of the direction vectors of these lines and check the fulfillment of the condition of collinearity of the direction vectors. In other words, if And - direction vectors of straight lines a given lines have coordinates and . Because , That . Thus, the necessary and sufficient condition for two lines to be parallel in space is satisfied. This proves the parallelism of the lines And .

Bibliography.

• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 - 9: a textbook for educational institutions.
• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
• Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
• Bugrov Ya.S., Nikolsky S.M. Higher Mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
• Ilyin V.A., Poznyak E.G. Analytic geometry.

Class: 2

The purpose of the lesson:

• form the concept of parallelism of 2 lines, consider the first sign of parallel lines;
• develop the ability to apply the sign in solving problems.

Tasks:

1. Educational: repetition and consolidation of the studied material, the formation of the concept of parallelism of 2 lines, proof of the 1st sign of parallelism of 2 lines.
2. Educational: to cultivate the ability to accurately keep notes in a notebook and follow the rules for constructing drawings.
3. Developmental tasks: development of logical thinking, memory, attention.

Lesson equipment:

• multimedia projector;
• screen, presentations;
• drawing tools.

During the classes

I. Organizational moment.

Greetings, checking readiness for the lesson.

II. Preparation for active UPD.

Stage 1.

In the first lesson of geometry, we considered the relative position of 2 lines on the plane.

Question. How many common points can two lines have?
Answer. Two lines can either have one common point, or not have more than one common point.

Question. How will the 2 lines be located relative to each other if they have one common point?
Answer. If lines have one common point, then they intersect

Question. How are the 2 lines located relative to each other if they do not have common points?
Answer. In this case, the lines do not intersect.

Stage 2.

In the last lesson, you were given the task of making a presentation where we meet with non-intersecting lines in our life and in nature. Now we will look at these presentations and choose the best of them. (The jury included students who, due to low intelligence, find it difficult to create their own presentations.)

Viewing presentations made by students: "Parallelism of lines in nature and life", and choosing the best of them.

III. Active UPD (explanation of new material).

Stage 1.

Picture 1

Definition. Two lines in a plane that do not intersect are called parallel.

This table shows various cases of arranging 2 parallel lines on a plane.

Consider which segments will be parallel.

Figure 2

1) If line a is parallel to b, then segments AB and CD are also parallel.

2) A line segment can be parallel to a straight line. So the segment MN is parallel to the line a.

Figure 3

3) Segment AB is parallel to ray h. Ray h is parallel to beam k.

4) If line a is perpendicular to line c, and line b is perpendicular to line c, then lines a and b are parallel.

Stage 2.

Angles formed by two parallel lines and a transversal.

Figure 4

Two parallel lines intersect a third line at two points. In this case, eight corners are formed, indicated in the figure by numbers.

Some pairs of these angles have special names (see figure 4).

Exists three signs, parallelism of two lines associated with these angles. In this lesson, we'll look at first sign.

Stage 3.

Let us repeat the material needed to prove this feature.

Figure 5

Question. What are the names of the corners shown in Figure 5?
Answer. Angles AOC and COB are called adjacent.

Question. What angles are called adjacent? Give a definition.
Answer. Two angles are called adjacent if they have one side in common and the other two are extensions of each other.

Question. What are the properties of adjacent angles?
Answer. Adjacent angles add up to 180 degrees.
AOC + COB = 180°

Question. What are angles 1 and 2 called?
Answer. Angles 1 and 2 are called vertical.

Question. What are the properties of vertical angles?
Answer. The vertical angles are equal to each other.

Stage 4.

Proof of the first sign of parallelism.

Theorem. If at the intersection of two lines by a transversal, the lying angles are equal, then the lines are parallel.

Figure 6

Given: a and b are straight
AB - secant
1 = 2
Prove: a//b.

1st case.

Figure 7

If 1 and 2 are straight lines, then a is perpendicular to AB, and b is perpendicular to AB, then a//b.

2nd case.

Figure 8

Consider the case when 1 and 2 are not straight lines. We divide the segment AB in half by the point O.

Question. What will be the length of the segments AO and OB?
Answer. Segments AO and OB are equal in length.

1) From the point O we draw a perpendicular to the line a, OH is perpendicular to a.

Question. What will angle 3 be?
Answer. Corner 3 will be right.

2) From point A on the straight line b, we set aside the segment AH 1 = BH with a compass.

3) Let's draw a segment OH 1.

Question. What triangles were formed as a result of the proof?
Answer. Triangle ONV and triangle OH 1 A.

Let's prove that they are equal.

Question. What angles are equal according to the hypothesis of the theorem?
Answer. Angle 1 is equal to angle 2.

Question. Which sides are equal in construction.
Answer. AO = OB and AN 1 = VN

Question. On what basis are triangles congruent?
Answer. Triangles are equal in two sides and the angle between them (the first sign of equality of triangles).

Question. What property do congruent triangles have?
Answer. Equal triangles have equal angles opposite equal sides.

Question. What angles will be equal?
Answer. 5 = 6, 3 = 4.

Question. What are 5 and 6 called?
Answer. These angles are called vertical.

From this it follows that the points: H 1 , O, H lie on one straight line.
Because 3 is straight, and 3 = 4, then 4 is straight.

Question. How are the lines a and b in relation to the line HH 1 if angles 3 and 4 are right?
Answer. Lines a and b are perpendicular to HH 1 .

Question. What can we say about two perpendiculars to one straight line?
Answer. Two perpendiculars of one line are parallel.

So a//b. The theorem has been proven.

Now I will repeat all the proof from the beginning, and you will listen to me carefully and try to understand everything to remember.

IV. Consolidation of new material.

Work in groups with different levels of intelligence, followed by a check on the screen and on the board. 3 students work at the blackboard (one from each group).

№1 (for students with a reduced level of intellectual development).

Given: a and b are straight
c - secant
1 = 37°
7 = 143°
Prove: a//b.

Solution.

7 = 6 (vertical) 6 = 143°
1 + 4 = 180° (adjacent) 4 =180° – 37° = 143°
4 \u003d 6 \u003d 143 °, and they lie crosswise a//b 5 \u003d 48 °, 3 and 5 are cross-lying angles, they are equal to a//b.

Figure 11

V. Summary of the lesson.

The result of the lesson is carried out using figures 1-8.

The activity of students in the lesson is assessed (each student receives an appropriate emoticon).

Homework: teach - pp. 52-53; solve No. 186 (b, c).

Parallelism is a very useful property in geometry. In real life, parallel sides allow you to create beautiful, symmetrical things that are pleasing to any eye, so geometry has always needed ways to check this parallelism. We will talk about the signs of parallel lines in this article.

Definition for parallelism

Let us single out the definitions that you need to know to prove the signs of parallelism of two lines.

Lines are called parallel if they have no points of intersection. In addition, in solutions, parallel lines usually go in conjunction with a secant line.

A secant line is a line that intersects both parallel lines. In this case, lying, corresponding and one-sided angles are formed crosswise. The pairs of angles 1 and 4 will be lying across; 2 and 3; 8 and 6; 7 and 5. Corresponding will be 7 and 2; 1 and 6; 8 and 4; 3 and 5.

Unilateral 1 and 2; 7 and 6; 8 and 5; 3 and 4.

When properly formatted, it is written: “Cross-lying angles with two parallel lines a and b and a secant c”, because for two parallel lines there can be an infinite number of secants, so you need to specify which secant you mean.

Also, for the proof, we need the theorem on the external angle of a triangle, which states that the external angle of a triangle is equal to the sum of two angles of a triangle that are not adjacent to it.

signs

All signs of parallel lines are tied to the knowledge of the properties of angles and the theorem on the external angle of a triangle.

Feature 1

Two lines are parallel if the intersecting angles are equal.

Consider two lines a and b with a secant c. Crosswise lying angles 1 and 4 are equal. Assume that the lines are not parallel. This means that the lines intersect and there should be an intersection point M. Then a triangle AVM is formed with an external angle of 1. The external angle must be equal to the sum of angles 4 and AVM as non-adjacent to it according to the theorem on the external angle in a triangle. But then it turns out that angle 1 is greater than angle 4, and this contradicts the condition of the problem, which means that the point M does not exist, the lines do not intersect, that is, they are parallel.

Rice. 1. Drawing for proof.

Feature 2

Two lines are parallel if the corresponding secant angles are equal.

Consider two lines a and b with a secant c. The corresponding angles 7 and 2 are equal. Let's pay attention to angle 3. It is vertical for angle 7. Therefore, angles 7 and 3 are equal. So angles 3 and 2 are also equal, since<7=<2 и <7=<3. А угол 3 и угол 2 являются накрест лежащими. Следовательно, прямые параллельны, что и требовалось доказать.

Rice. 2. Drawing for proof.

Feature 3

Two lines are parallel if the sum of one-sided angles is 180 degrees.

Rice. 3. Drawing for proof.

Consider two lines a and b with a secant c. The sum of one-sided angles 1 and 2 is 180 degrees. Let's pay attention to angles 1 and 7. They are adjacent. That is:

$$<1+<7=180$$

$$<1+<2=180$$

Subtract the second from the first expression:

$$(<1+<7)-(<1+<2)=180-180$$

$$(<1+<7)-(<1+<2)=0$$

$$<1+<7-<1-<2=0$$

$$<7-<2=0$$

$<7=<2$ - а они являются соответственными. Значит, прямые параллельны.

What have we learned?

We analyzed in detail what angles are obtained when cutting parallel lines with a third line, identified and described in detail the proof of three signs of parallelism of lines.

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1. The first sign of parallelism.

If, at the intersection of two lines with a third, the interior angles lying across are equal, then these lines are parallel.

Let lines AB and CD be intersected by line EF and ∠1 = ∠2. Let's take the point O - the middle of the segment KL of the secant EF (Fig.).

Let us drop the perpendicular OM from the point O to the line AB and continue it until it intersects with the line CD, AB ⊥ MN. Let us prove that CD ⊥ MN as well.

To do this, consider two triangles: MOE and NOK. These triangles are equal to each other. Indeed: ∠1 = ∠2 by the hypothesis of the theorem; OK = OL - by construction;

∠MOL = ∠NOK as vertical angles. Thus, the side and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle; therefore, ΔMOL = ΔNOK, and hence ∠LMO = ∠KNO,
but ∠LMO is direct, hence ∠KNO is also direct. Thus, the lines AB and CD are perpendicular to the same line MN, therefore, they are parallel, which was to be proved.

Note. The intersection of the lines MO and CD can be established by rotating the triangle MOL around the point O by 180°.

2. The second sign of parallelism.

Let's see if the lines AB and CD are parallel if, at the intersection of their third line EF, the corresponding angles are equal.

Let some corresponding angles be equal, for example ∠ 3 = ∠2 (Fig.);

∠3 = ∠1 as vertical angles; so ∠2 will be equal to ∠1. But angles 2 and 1 are internal crosswise angles, and we already know that if at the intersection of two lines by a third, the internal crosswise lying angles are equal, then these lines are parallel. Therefore, AB || CD.

If at the intersection of two lines of the third the corresponding angles are equal, then these two lines are parallel.

The construction of parallel lines with the help of a ruler and a drawing triangle is based on this property. This is done as follows.

Let us attach a triangle to the ruler as shown in Fig. We will move the triangle so that one side of it slides along the ruler, and draw several straight lines along any other side of the triangle. These lines will be parallel.

3. The third sign of parallelism.

Let us know that at the intersection of two lines AB and CD by the third line, the sum of any internal one-sided angles is equal to 2 d(or 180°). Will the lines AB and CD be parallel in this case (Fig.).

Let ∠1 and ∠2 be one-sided interior angles and add up to 2 d.

But ∠3 + ∠2 = 2 d as adjacent angles. Therefore, ∠1 + ∠2 = ∠3+ ∠2.

Hence ∠1 = ∠3, and these interior angles are crosswise. Therefore, AB || CD.

If at the intersection of two lines by a third, the sum of the interior one-sided angles is equal to 2 d (or 180°), then the two lines are parallel.

Signs of parallel lines:

1. If at the intersection of two straight lines by a third, the internal cross lying angles are equal, then these lines are parallel.

2. If at the intersection of two lines of the third, the corresponding angles are equal, then these two lines are parallel.

3. If at the intersection of two lines of the third, the sum of the internal one-sided angles is 180 °, then these two lines are parallel.

4. If two lines are parallel to the third line, then they are parallel to each other.

5. If two lines are perpendicular to the third line, then they are parallel to each other.

Euclid's axiom of parallelism

Task. Through a point M taken outside the line AB, draw a line parallel to the line AB.

Using the proven theorems on the signs of parallelism of lines, this problem can be solved in various ways,

Solution. 1st s o s o b (Fig. 199).

We draw MN⊥AB and through the point M we draw CD⊥MN;

we get CD⊥MN and AB⊥MN.

Based on the theorem ("If two lines are perpendicular to the same line, then they are parallel.") we conclude that СD || AB.

2nd s p o s o b (Fig. 200).

We draw a MK intersecting AB at any angle α, and through the point M we draw a straight line EF, forming an angle EMK with a straight line MK, equal to the angle α. Based on the theorem () we conclude that EF || AB.

Having solved this problem, we can consider it proved that through any point M, taken outside the line AB, it is possible to draw a line parallel to it. The question arises, how many lines parallel to a given line and passing through a given point can exist?

The practice of constructions allows us to assume that there is only one such line, since with a carefully executed drawing, lines drawn in various ways through the same point parallel to the same line merge.

In theory, the answer to this question is given by the so-called axiom of Euclid's parallelism; it is formulated like this:

Through a point taken outside a given line, only one line can be drawn parallel to this line.

In the drawing 201, a straight line SK is drawn through the point O, parallel to the straight line AB.

Any other line passing through the point O will no longer be parallel to the line AB, but will intersect it.

The axiom adopted by Euclid in his Elements, which states that on a plane through a point taken outside a given line, only one line can be drawn parallel to this line, is called Euclid's axiom of parallelism.

For more than two thousand years after Euclid, many mathematicians tried to prove this mathematical proposition, but their attempts were always unsuccessful. Only in 1826, the great Russian scientist, professor of Kazan University Nikolai Ivanovich Lobachevsky proved that, using all other Euclid's axioms, this mathematical proposition cannot be proved, that it really should be taken as an axiom. N. I. Lobachevsky created a new geometry, which, in contrast to the geometry of Euclid, was called the geometry of Lobachevsky.