By studying the properties of geometric figures, we proved a number of theorems. In doing so, we relied, as a rule, on previously proven theorems. What are the proofs of the very first theorems of geometry based on? The answer to this question is this: certain statements about the properties of geometric figures are accepted as starting points, on the basis of which further theorems are proved and, in general, all geometry is constructed. Such initial positions are called axioms.

Some axioms were formulated back in the first chapter (although they were not called axioms there). For example, it is an axiom that

Many other axioms, although not particularly emphasized, were actually used in our reasoning. Thus, we compared two segments by superimposing one segment on another. The possibility of such an overlap follows from the following axiom:

Comparison of two angles is based on a similar axiom:

All these axioms are clearly obvious and beyond doubt. The word “axiom” itself comes from the Greek “axios”, which means “valuable, worthy”. We provide a complete list of planimetry axioms adopted in our geometry course at the end of the textbook.

This approach to the construction of geometry, when the initial positions - axioms - are first formulated, and then other statements are proven on their basis through logical reasoning, originated in ancient times and was outlined in the famous work “Principles” by the ancient Greek scientist Euclid. Some of Euclid's axioms (some of them he called postulates) and are now used in geometry courses, and the geometry itself, presented in the “Elements”, is called Euclidean geometry. In the next paragraph we will get acquainted with one of the most famous axioms of geometry.

Axiom of parallel lines

Consider an arbitrary straight line a and a point M that does not lie on it (Fig. 110, a). Let us prove that through the point M it is possible to draw a line parallel to the line a. To do this, draw two straight lines through point M: first straight line c perpendicular to straight line a, and then straight line b perpendicular to straight line c (Fig. 110, (b). Since straight lines a and b are perpendicular to straight line c, they are parallel.

Rice. 110

So, through point M there passes a line b parallel to line a. The following question arises: is it possible to draw another line through point M, parallel to straight line a?

It seems to us that if straight line b is “turned” even by a very small angle around point M, then it will intersect straight line a (line b" in Figure 110.6). In other words, it seems to us that it is impossible to draw another straight line through point M ( different from b), parallel to line a. Is it possible to prove this statement?

This question has a long history. Euclid’s “Elements” contains a postulate (Euclid’s fifth postulate), from which it follows that through a point not lying on a given line, only one straight line can be drawn parallel to the given one. Many mathematicians, starting from ancient times, have attempted to prove Euclid's fifth postulate, that is, to derive it from other axioms. However, these attempts were unsuccessful every time. And only in the last century it was finally clarified that the statement about the uniqueness of a line passing through a given point parallel to a given line cannot be proven on the basis of the remaining axioms of Euclid, but is itself an axiom.

The great Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856) played a huge role in solving this difficult issue.

So, as another starting point we accept axiom of parallel lines.

Statements that are derived directly from axioms or theorems are called consequences. For example, statements 1 and 2 (see p. 35) are consequences of the theorem on the bisector of an isosceles triangle.

Let's consider some corollaries from the axiom of parallel lines.

Indeed, let straight lines a and b be parallel and straight line c intersect straight line a at point M (Fig. 111, a). Let us prove that line c also intersects line b. If line c did not intersect line b, then two lines (lines a and c) parallel to line b would pass through point M (Fig. 111, b). But this contradicts the axiom of parallel lines, and, therefore, line c intersects line b.

Rice. 111

Indeed, let straight lines a and b be parallel to straight line c (Fig. 112, a). Let us prove that a || b. Let us assume that lines a and b are not parallel, that is, they intersect at some point M (Fig. 112.6). Then two lines pass through point M (lines a and b), parallel to line c.

Rice. 112

But this contradicts the axiom of parallel lines. Therefore, our assumption is incorrect, which means that lines a and b are parallel.

Theorems on angles formed by two parallel lines and a transversal

Every theorem has two parts: condition And conclusion. The condition of the theorem is what is given, and the conclusion is what needs to be proven.

Let us consider, for example, a theorem expressing the criterion for the parallelism of two straight lines: if, when two straight lines intersect with a transversal, the lying angles are equal, then the straight lines are parallel.

In this theorem, the condition is the first part of the statement: “when two lines intersect crosswise, the lying angles are equal” (this is given), and the conclusion is the second part: “the lines are parallel” (this needs to be proven).

The converse of this theorem, is a theorem in which the condition is the conclusion of the theorem, and the conclusion is the condition of the theorem. Let us prove the theorems converse to the three theorems in paragraph 25.

Theorem

Proof

Let parallel lines a and b be intersected by the secant MN. Let us prove that the angles lying crosswise, for example 1 and 2, are equal (Fig. 113).

Rice. 113

Let's assume that angles 1 and 2 are not equal. Let us subtract from the ray MN an angle PMN equal to angle 2, so that ∠PMN and ∠2 are crosswise angles at the intersection of lines MR and b by the secant MN. By construction, these crossed angles are equal, so MR || b. We found that two lines pass through point M (lines a and MP), parallel to line b. But this contradicts the axiom of parallel lines. This means that our assumption is incorrect and ∠1 = ∠2. The theorem has been proven.

Comment

In proving this theorem, we used a method of reasoning called by proof by contradiction.

We assumed that when parallel lines a and b intersect the secant MN crosswise, the lying angles 1 and 2 are not equal, i.e. we assumed the opposite of what needs to be proven. Based on this assumption, through reasoning we came to a contradiction with the axiom of parallel lines. This means that our assumption is incorrect and therefore ∠1 = ∠2.

This way of reasoning is often used in mathematics. We used it earlier, for example, in paragraph 12 when proving that two lines perpendicular to a third do not intersect. We used the same method in paragraph 28 to prove corollaries 1 0 and 2 0 from the axiom of parallel lines.

Consequence

Indeed, let a || b, c ⊥ a, i.e. ∠1 = 90° (Fig. 114). Line c intersects line a, so it also intersects line b. When parallel lines a and b intersect with a transversal c, equal crosswise angles are formed: ∠1=∠2. Since ∠1 = 90°, then ∠2 = 90°, i.e., c ⊥ b, which is what needed to be proved.

Rice. 114

Theorem

Proof

Let parallel lines a and b be intersected by a secant c. Let us prove that the corresponding angles, for example 1 and 2, are equal (see Fig. 102). Since a || b, then the crosswise angles 1 and 3 are equal.

Angles 2 and 3 are equal as vertical. From the equalities ∠1 = ∠3 and ∠2 = ∠3 it follows that ∠1 = ∠2. The theorem has been proven.

Theorem

Proof

Let parallel lines a and b be intersected by a secant c (see Fig. 102). Let us prove, for example, that ∠1 + ∠4 = 180°. Since a || b, then the corresponding angles 1 and 2 are equal. Angles 2 and 4 are adjacent, so ∠2 + ∠4 = 180°. From the equalities ∠1 = ∠2 and ∠2 + ∠4 = 180° it follows that ∠1 + ∠4 = 180°. The theorem has been proven.

Comment

If a certain theorem is proven, then the converse statement does not follow. Moreover, the converse is not always true. Let's give a simple example. We know that if the angles are vertical, then they are equal. The converse statement: “if the angles are equal, then they are vertical” is, of course, false.

Angles with respectively parallel or perpendicular sides

Let us prove the theorem about angles with correspondingly parallel sides.

Theorem

Proof

Let ∠AOB and ∠A 1 O 1 B 1 be the given angles and OA || O 1 A 1 , OB || About 1 In 1. If angle AOB is developed, then angle A 1 O 1 B 1 is also developed (explain why), so these angles are equal. Let ∠AOB be an undeveloped angle. Possible cases of the location of the angles AOB and A 1 O 1 B 1 are shown in Figure 115, a and b. The line O 1 B 1 intersects the line O 1 A 1 and, therefore, intersects the line OA parallel to it at some point M. The parallel lines OB and O 1 B 1 are intersected by the secant OM, therefore one of the angles formed at the intersection of the lines O 1 B 1 and OA (angle 1 in Figure 115), is equal to angle AOB (like crosswise angles). Parallel lines OA and O 1 A 1 are intersected by the secant O 1 M, therefore either ∠1 = ∠A 1 O 1 B 1 (Fig. 115, a), or ∠1 + ∠A 1 O 1 B 1 = 180° (Fig. 115, b). From the equality ∠1 = ∠AOB and the last two equalities it follows that either ∠AOB = ∠A 1 O 1 B 1 (see Fig. 115, a), or ∠AOB + ∠A 1 O 1 B 1 = 180° ( see Fig. 115, b). The theorem has been proven.

Rice. 115

Let us now prove the theorem about angles with correspondingly perpendicular sides.

Theorem

Proof

Let ∠AOB and ∠A 1 O 1 B 1 be given angles, OA ⊥ O 1 A 1 , OB ⊥ O 1 B 1 . If angle AOB is reversed or straight, then angle A 1 O 1 B 1 is reversed or straight (explain why), so these angles are equal. Let ∠AOB< 180°, О ∉ О 1 А 1 , О ∉ О 1 В 1 (случаи О ∈ O 1 А 1 , О ∈ О 1 В 1 рассмотрите самостоятельно).

Two cases are possible (Fig. 116).

10 . ∠AOB< 90° (см. рис. 116, а). Проведём луч ОС так, чтобы прямые ОА и ОС были взаимно перпендикулярными, а точки В и С лежали по разные стороны от прямой О А. Далее, проведём луч OD так, чтобы прямые ОВ и OD были взаимно перпендикулярными, а точки С и D лежали по одну сторону от прямой О А. Поскольку ∠AOB = 90° - ∠AOD и ∠COD = 90° - ∠AOD, то ∠AOB = ∠COD. Стороны угла COD соответственно параллельны сторонам угла А 1 О 1 В 1 (объясните почему), поэтому либо ∠COD = ∠A 1 O 1 B 1 , либо ∠COD + ∠A 1 O 1 B 1 = 180°. Следовательно, либо ∠AOB = ∠A 1 O 1 B 1 , либо ∠AOB + ∠A 1 O 1 B 1 = 180°.

20 . ∠AOB > 90° (see Fig. 116, b). Let us draw the ray OS so that the angle AOS is adjacent to the angle AOB. Angle AOC is acute, and its sides are correspondingly perpendicular to the sides of angle A 1 O 1 B 1 . Therefore, either ∠AOC + ∠A 1 O 1 B 1 = 180°, or ∠AOC = ∠A 1 O 1 B 1 . In the first case, ∠AOB = ∠A 1 O 1 B 1, in the second case, ∠AOB + ∠A 1 O 1 B 1 = 180°. The theorem has been proven.

Tasks

196. Given a triangle ABC. How many lines parallel to side AB can be drawn through vertex C?

197. Four straight lines are drawn through a point not lying on the line p. How many of these lines intersect line p? Consider all possible cases.

198. Lines a and b are perpendicular to line p, line c intersects line a. Does line c intersect line b?

199. Line p is parallel to side AB of triangle ABC. Prove that lines BC and AC intersect line p.

200. In Figure 117 AD || p and PQ || Sun. Prove that line p intersects lines AB, AE, AC, BC and PQ.

Rice. 117

201. The sum of crosswise angles when two parallel lines intersect with a transversal is equal to 210°. Find these angles.

202. In Figure 118, straight lines a, b and c are intersected by straight line d, ∠1 = 42°, ∠2 = 140°, ∠3 = 138°. Which of the lines a, b and c are parallel?

Rice. 118

203. Find all the angles formed when two parallel lines a and b intersect with a transversal c, if:

a) one of the angles is 150°;
b) one of the angles is 70° greater than the other.

204. The ends of the segment AB lie on parallel lines a and b. The straight line passing through the middle O of this segment intersects lines a and b at points C and D. Prove that CO = OD.

205. Using the data in Figure 119, find ∠1.

Rice. 119

206. ∠ABC = 70°, and ABCD = 110°. Can direct AB and CD be:

a) parallel;
b) intersecting?

207. Answer the questions in Problem 206 if ∠ABC = 65° and ∠BCD = 105°.

208. The difference between two one-sided angles when two parallel lines intersect with a transversal is 50°. Find these angles.

209. In figure 120 a || b, c || d, ∠4 = 45°. Find angles 1, 2 and 3.

Rice. 120

210. Two bodies P 1 and P 2 are suspended at the ends of a thread thrown over blocks A and B (Fig. 121). The third body P 3 is suspended from the same thread at point C and balances the bodies P 1 and P 2. (In this case, AP 1 || BP 2 || CP 3 .) Prove that ∠ACB = ∠CAP 1 + ∠CBP 2 .

Rice. 121

211. Two parallel lines are intersected by a transversal. Prove that: a) bisectors of opposite angles are parallel; b) bisectors of one-sided angles are perpendicular.

212. Straight lines containing altitudes AA 1 and BB 1 of triangle ABC intersect at point H, angle B is obtuse, ∠C = 20°. Find angle ABB.

Answers to problems

196. One straight line.

197. Three or four.

201. 105°, 105°.

203. b) Four angles are 55°, four other angles are 125°.

206. a) Yes; b) yes.

207. a) No; b) yes.

208. 115° and 65°.

209. ∠1 = 135°, ∠2 = 45°, ∠3=135°.

210. Instruction. Consider the continuation of the beam CP 3.

German physicist Albert Einstein, using mathematical methods, developed the theory of relativity, revolutionizing physics in the 20th century.

It is believed that the foundations of modern mathematics were laid by Euclid in his 13-volume work "Elements" around 300 BC. e. Unlike his predecessors, Euclid explains geometry here not with the help of countless drawings, but purely logically. First, he describes a number of facts that he considers true and immutable. These facts are called postulates. One of these postulates of Euclid, for example, says: “From each point one can draw one straight line to any other point.” Then, based on these postulates, he deduces everything else. Thus, Euclid for the first time demonstrated modern mathematical thinking: based on certain assumptions, once made and not subject to revision, he proved many other statements.

For centuries, there have been disputes over Euclid's fifth postulate, the so-called axiom of parallel lines: through any point P lying outside the line g, only one straight line can be drawn that does not intersect g. Such a line is called parallel to the line g passing through point P. Many scientists sought not only to accept this position, but to derive it from the first four. But this turned out to be impossible. Mathematicians began to create geometry that was based on the first four axioms of Euclid and rejected the fifth. What at first looked like a mathematical game, at the beginning of the 20th century. turned out to be in demand. Albert Einstein saw these models of geometry as the basis for his general theory of relativity.

Completed by a student of grade 7 "G" MBOU "OK "Lyceum No. 3" Gavrilov Dmitry

Axiom
Comes from the Greek “axios”, which means “valuable, worthy”. A position accepted without logical proof due to immediate persuasiveness is the true starting position of the theory. (Soviet encyclopedic dictionary)

Download:

Preview:

To use presentation previews, create a Google account and log in to it: https://accounts.google.com

Slide captions:

Axiom of parallel lines Completed by a student of grade 7 "G" MBOU "OK "Lyceum No. 3" Gavrilov Dmitry 2015-2016 academic year (teacher Konareva T.N.)

Known definitions and facts. Finish the sentence. 1. Line x is called a transversal in relation to lines a and b if... 2. When two straight lines intersect, a transversal forms... undeveloped angles. 3. If lines AB and C D are intersected by line B D, then line B D is called... 4. If points B and D lie in different half-planes relative to the secant AC, then angles BAC and DCA are called... 5. If points B and D lie in one half-plane relative to the secant AC, then the angles BAC and DCA are called... 6. If the interior angles of one pair are equal, then the interior angles of the other pair are equal... D C A C B D A B

Checking the task. 1 . ...if it intersects them at two points 2. 8 3. ... secant 4. ... lying crosswise 5. ... one-sided 6. ... equal

Match a) a b m 1) a | | b, since internal crosswise angles are equal b) 2) a | | b, since the corresponding angles are equal c) a b 3) a | | b, since the sum of internal one-sided angles is equal to 180° 50 º 130 º 45 º 45 º m a b m a 150 º 150º

About the axioms of geometry

Axiom Comes from the Greek "axios", which means "valuable, worthy." A position accepted without logical proof due to immediate persuasiveness is the true initial position of the theory. Soviet encyclopedic dictionary

A straight line passes through any two points, and only one. How many straight lines can be drawn through any two points lying on a plane?

On any ray, from its beginning, one can lay off a segment equal to the given one, and, moreover, only one. How many segments of a given length can be laid off from the beginning of the ray?

From any ray in a given direction it is possible to plot an angle equal to a given undeveloped angle, and only one. How many angles equal to a given one can be plotted from a given ray to a given half-plane?

axioms theorems logical reasoning famous essay “Principia” Euclidean geometry Logical construction of geometry

Axiom of parallel lines

M a Let us prove that through the point M it is possible to draw a line parallel to the line a c b a ┴ c b ┴ c a II c

Is it possible to draw another line through point M parallel to line a? a M in 1 Is it possible to prove this?

Many mathematicians, since ancient times, have tried to prove this statement, and in Euclid’s Elements this statement is called the fifth postulate. Attempts to prove Euclid's fifth postulate were unsuccessful, and only in the 19th century it was finally clarified that the statement about the uniqueness of a line passing through a given point parallel to a given line cannot be proven on the basis of the rest of Euclid's axioms, but is itself an axiom. The Russian mathematician Nikolai Ivanovich Lobachevsky played a huge role in solving this issue.

Fifth postulate of Euclid 1792-1856 Nikolai Ivanovich

“Through a point not lying on a given line, only one line parallel to the given line passes.” “Through a point not lying on a given line, one can draw a line parallel to the given one.” Which of these statements is an axiom? How are the above statements different?

Through a point not lying on a given line there passes only one line parallel to the given one. Statements that are derived from axioms or theorems are called corollaries. Corollary 1. If a line intersects one of two parallel lines, then it also intersects the other. a II b , c b ⇒ c a Axiom of parallelism and consequences from it. a A Corollary 2. If two lines are parallel to a third line, then they are parallel. a II c, b II c a II b a b c c b

Consolidation of knowledge. Test Mark correct statements with a “+” sign and erroneous statements with a “-” sign. Option 1 1. An axiom is a mathematical statement about the properties of geometric figures that requires proof. 2. A straight line passes through any two points. 3. On any ray, from the beginning, you can plot segments equal to the given one, and as many as you like. 4. Through a point not lying on a given line, only one line parallel to the given line passes. 5. If two lines are parallel to a third, then they are parallel to each other. Option 2 1. An axiom is a mathematical statement about the properties of geometric figures, accepted without proof. 2. A straight line passes through any two points, and only one. 3. Through a point not lying on a given line, only two lines parallel to the given line pass through. 4. If a line intersects one of two parallel lines, then it is perpendicular to the other line. 5. If a line intersects one of two parallel lines, then it also intersects the other.

Test answers Option 1 1. “-” 2. “-” 3. “-” 4. “+” 5. “+” Option 2 “+” “+” “-” “-” “+”

“Geometry is full of adventure because behind every problem lies an adventure of thought. Solving a problem means experiencing an adventure.” (V. Proizvolov)

The video lesson “Axiom of parallel lines” involves a detailed consideration of an important axiom of geometry - the axiom of parallel lines, its features, consequences from this axiom, which are widely used in the practice of solving geometric problems. The purpose of this video lesson is to make it easier to memorize the axiom and its consequences, to form an idea of ​​its features and application in solving problems.

Presenting the material in the form of a video lesson opens up new opportunities for the teacher. The delivery of a standard block of educational material to students is automated. At the same time, the quality of the presentation of the material improves, since it is enriched with a visual representation and animation effects that bring the constructions closer to the real ones carried out on the board. Historical information is presented with drawings and photos, arousing interest in the topic being studied. Video also frees up the teacher to deepen individual work during teaching.

First, this video demonstrates the name of the topic. Consideration of an axiom begins with the construction of its model. The screen shows a line a and a point M lying outside it. Next, we describe the proof of the statement that through a given point M it is possible to construct a line parallel to the given one. A line c is drawn perpendicular to line a, then line b is drawn perpendicular to line c at point M. Based on the statement about the parallelism of two lines perpendicular to the third, we note that line b is parallel to the original line a. Taking this into account, we indicate that at point M a straight line is drawn parallel to this one. However, it is still necessary to check whether it is possible to draw another parallel line through M. The screen shows that any rotation of straight line b at point M will lead to the construction of a straight line that will intersect straight line a. However, is it possible to prove the impossibility of drawing another straight line?

The question of proving the impossibility of drawing another line parallel to this one has a long history. Students are offered a short excursion into the history of the issue. It is noted that in Euclid’s work “Elements” this statement is given in the form of the fifth postulate. Attempts by scientists to prove the statement were unsuccessful. For many centuries, mathematicians have been interested in this problem. However, only in the last century it was finally proven that this statement is unprovable in Euclidean geometry. It is an axiom. Students are introduced to one of the famous mathematicians who made significant contributions to mathematical science - Nikolai Ivanovich Lobachevsky. It was he who played an important role in the final resolution of the issue. Therefore, the statement discussed in this lesson is an axiom that lies in the foundation of science along with other axioms.

Next, we propose to consider the consequences of this axiom. To do this, it is necessary to clarify the concept of “consequence”. The screen displays the definition of corollaries as statements that are derived directly from theorems or axioms. This definition can be offered to students to write in their notebooks. The concept of consequences is demonstrated using an example that has already been discussed in video lesson 18 “Properties of an isosceles triangle.” A theorem about the properties of an isosceles triangle is displayed on the screen. It is recalled that after the proof of this theorem, no less important consequences from it were considered. So, if the main theorem stated that the bisector of an isosceles triangle is a median and an altitude, then the corollaries had a similar content, stating that the altitude of an isosceles triangle is a bisector and a median, and also the median of an isosceles triangle is both a bisector and an altitude.

Having clarified the concept of consequences, we directly consider the consequences arising from this axiom of parallel lines. The text of the first corollary of the axiom is displayed on the screen, stating that the intersection of a line with one of the parallel lines means its intersection with the second parallel line. The figure below the text of the corollary shows a straight line b and a parallel straight line a. The second line intersects line c at point M, which belongs to line a. A proof is given of the statement that line c will also intersect line b. The proof is made by contradiction, using the axiom of parallel lines. If we assume that line c does not intersect b, this means that through this point we can draw another line parallel to the indicated one. But this is impossible, given the parallel lines axiom. Therefore, c also intersects line b. The investigation has been proven.

Next, we consider the second corollary of this axiom. The screen displays the text of a corollary stating that if two lines are parallel to a third, then we can assert that they are parallel to each other. In the figure demonstrating this statement, straight lines a, b, c are constructed. In this case, the line c, as parallel to both lines, is highlighted in blue. It is proposed to prove this statement. During the proof, it is assumed that lines a and b parallel to lines c are not parallel to each other. This means that they have an intersection point. This means that both lines passing through the point M are parallel to this one, which contradicts the axiom of parallel lines. This corollary is correct.

The video lesson “Axiom of Parallel Lines” can make it easier for the teacher to explain to students the features of the axiom, the proof of its consequences, and make it easier for students to memorize the material in a regular lesson. Also, this video material can be used for distance learning and is recommended for self-study.

Euclid's Axiom of Parallelism

Euclid's Axiom of Parallelism, or fifth postulate- one of the axioms underlying classical planimetry. First given in Euclid's Elements:

Euclid distinguishes concepts postulate And axiom without explaining their differences; in different manuscripts of Euclid’s Elements, the division of statements into axioms and postulates is different, just as their order does not coincide. In the classic edition of Heyberg's Principia, the stated statement is the fifth postulate.

In modern language, Euclid's text can be reformulated as follows:

If the sum of interior angles with a common side formed by two straight lines when they intersect a third, on one side of the secant, is less than 180°, then these straight lines intersect, and, moreover, on the same side of the secant.

The fifth postulate is extremely different from Euclid's other postulates, which are simple and intuitively obvious (see Euclid's Elements). Therefore, over the course of 2 thousand years, attempts have not stopped to exclude it from the list of axioms and derive it as a theorem. All these attempts ended in failure. "It is probably impossible to find a more exciting and dramatic story in science than the story of Euclid's fifth postulate." Despite the negative result, these searches were not in vain, as they ultimately led to a complete revision of scientific ideas about the geometry of the Universe.

Equivalent formulations of the parallel postulate

Modern sources usually give another formulation of the parallel postulate, equivalent to the V postulate and belonging to Proclus (abroad, it is often called Playfair’s axiom):

In a plane, through a point not lying on a given line, one and only one line can be drawn parallel to the given one.

In this formulation, the words “one and only one” are often replaced with “only one” or “at most one”, since the existence of at least one such parallel immediately follows from Theorems 27 and 28 of Euclid’s Elements.

In general, postulate V has a huge number of equivalent formulations, many of which seem quite obvious. Here are some of them:

§ There is a rectangle ( at least one), that is, a quadrilateral with all right angles.

§ There are triangles that are similar but not equal ( Wallis's axiom, 1693).

§ Any figure can be proportionally enlarged.

§ There is a triangle of arbitrarily large area.

§ A line passing through a point inside an angle intersects at least one of its sides ( Lorentz's axiom, 1791).

§ Through every point inside an acute angle it is always possible to draw a straight line intersecting both its sides.

§ If two straight lines diverge in one direction, then in the other they come closer.

§ Converging lines will intersect sooner or later.

§ Option: a perpendicular and an oblique line to the same line will certainly intersect (Legendre’s axiom).

§ Points equidistant from a given line (on one side of it) form a straight line,

§ If two straight lines begin to approach each other, then it is impossible for them to then begin (in the same direction, without intersecting) to diverge ( Robert Simson's axiom, 1756).

§ The sum of angles is the same for all triangles.

§ There is a triangle whose angles add up to two right angles.

§ Two lines parallel to a third are parallel to each other ( Ostrogradsky's axiom, 1855).

§ A straight line intersecting one of the parallel lines will certainly intersect the other.

§ Through any three points you can draw either a straight line or a circle.

§ Option: for every non-degenerate triangle there is a circumscribed circle ( Farkas Bolyai's axiom).

§ The Pythagorean theorem is true.

Their equivalence means that all of them can be proven if we accept the V postulate, and vice versa, replacing the V postulate with any of these statements, we can prove the original V postulate as a theorem.

If instead of the V postulate we assume that for a point-line pair the V postulate is incorrect, then the resulting system of axioms will describe Lobachevsky’s geometry. It is clear that in Lobachevsky geometry all of the above equivalent statements are false.

The system of axioms of spherical geometry also requires changes to other axioms of Euclid.

The fifth postulate stands out sharply from the others, which are quite obvious; it is more like a complex, non-obvious theorem. Euclid was probably aware of this, and therefore the first 28 sentences in the Elements are proved without his help.

“Euclid certainly must have known various forms of the parallel postulate.” Why did he choose the reduced, complex and cumbersome one? Historians have made various assumptions about the reasons for this choice. V.P. Smilga believed that Euclid, with this formulation, indicated that this part of the theory was incomplete. M. Klein draws attention to the fact that Euclid’s fifth postulate has local character, that is, it describes an event on a limited portion of the plane, while, for example, Proclus’ axiom states the fact of parallelism, which requires consideration of the entire infinite straight line. It should be made clear that ancient mathematicians avoided using actual infinity; for example, Euclid’s second postulate does not assert the infinity of a straight line, but only that “a straight line can be continuously extended.” From the point of view of ancient mathematicians, the above equivalents of the parallel postulate could seem unacceptable: they either refer to actual infinity or the (not yet introduced) concept of measurement, or are also not very obvious.