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Lesson objectives:
- give an idea of the axioms of geometry unknown to students, repeat the axioms already known to them;
- introduce the axiom of parallel lines;
- introduce the concept of consequences from axioms and theorems;
- show how the axiom of parallel lines and its consequences are used when solving problems;
- education of patriotism and pride in one’s homeland using the example of the great Russian mathematician N.I. Lobachevsky.
Equipment: computer, projector.
PROGRESS OF THE LESSON
1. Checking previous homework
2. Repetition of the axioms of planimetry already known to students
Teacher: In the famous work of Euclid “Elements” (III century BC), the basic geometric information known at that time was systematized. The main thing is that in the “Principles” an axiomatic approach to the construction of geometry was developed, which consists in the fact that first the basic provisions that do not require proof (axioms) are formulated, and then, on their basis, other statements (theorems) are proven through reasoning. Some of the axioms proposed by Euclid are still used in geometry courses.
The word “axiom” itself comes from the Greek “axios”, which means “valuable, worthy”. A complete list of planimetry axioms adopted in our geometry course is given in the appendices at the end of the textbook on pages 344-348. You will consider these axioms at home yourself.
We have already considered some of these axioms. Remember and formulate these axioms.
Students:
1) There are at least three points that do not lie on the same line.
2) A straight line passes through any two points, and only one.
3) Of the three points on a straight line, one and only one lies between the other two.
4) Each point O of a line divides it into two parts (two rays) so that any two points of the same ray lie on the same side of point O, and any two points of different rays lie on opposite sides of point O.
5) Each line a divides the plane into two parts (two half-planes) so that any two points of the same half-plane lie on the same side of line a, and any two points of different half-planes lie on opposite sides of line a.
6) If, during overlap, the ends of two segments are combined, then the segments themselves are combined.
7) On any ray, from its beginning, you can lay off a segment equal to the given one, and, moreover, only one.
8) From any ray into a given half-plane it is possible to plot an angle equal to a given undeveloped angle, and, moreover, only one.
Teacher: What lines are called parallel in a plane?
Students: Two lines in a plane are called parallel if they do not intersect.
Teacher: Formulate the signs of parallelism of lines.
Students:
1) If, when two lines intersect with a transversal, the lying angles are equal, then the lines are parallel.
2) If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.
3) If, when two straight lines intersect with a transversal, the sum of one-sided angles is equal to 180˚, then the straight lines are parallel.
3. New topic. Axiom of parallel lines
Teacher: Let's solve the problem: “Through a point M that does not lie on line a, draw a line parallel to line a.”
The plan for solving the problem is discussed by the whole class. One of the students writes the solution on the board (without writing in their notebooks).
Teacher: The question arises: is it possible to draw another line through point M, parallel to line a?
This question has a long history. Euclid’s Elements contains the fifth postulate: “And if a straight line falling on two straight lines forms interior angles on one side that are less than two right angles, then the extensions of these straight lines will meet indefinitely on the side where the angles are less than two right angles.” Proclus in the 5th century AD reformulated Euclid’s postulate in a simpler and clearer way: “Through a point not lying on a given line, only one line parallel to the given line passes.” This is the axiom of parallel lines. From this it is clear that the problem considered above has a unique solution.
Many mathematicians attempted to prove the fifth postulate, since its formulation was too reminiscent of a theorem. All these attempts failed every time. And only in the 19th century. it was finally clarified that Euclid's fifth postulate cannot be proven; it itself is an axiom.
The great Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856) played a huge role in solving this issue.
4. Watch a presentation about N.I. Lobachevsky
5. Consolidation of what has been learned. Problem solving
Given ∆ABC. How many lines parallel to side AB can be drawn through vertex C?
Solution.
According to the axiom of parallel lines, a single straight line can be drawn.
Four straight lines are drawn through a point not lying on line p. How many of these lines intersect line p? Consider all possible cases.
Solution.
3 straight 4 straight
Answer: 3 or 4 straight.
Corollaries from the axiom of parallel lines.
Statements that are derived directly from axioms or theorems are called corollaries. Let us consider the consequences of the axiom of parallel lines.
Corollary 1˚. If a line intersects one of two parallel lines, then it also intersects the other.
Corollary 2˚. If two lines are parallel to a third line, then they are parallel. (Students are asked to prove it themselves).
The drawing is the same.
Given: a || b, c || b
Prove: a || With
Evidence o (method “by contradiction”):
Let the lines a and c not be parallel. Then they intersect at some point M. Two different straight lines (a and c) parallel to straight line b pass through the point M. This contradicts the parallel axiom. This means our assumption is not correct. But it is true that a || With. Etc.
The second corollary from the axiom of parallel lines is essentially another sign of the parallelism of lines on a plane.
Problem solving: No. 217 (oral), 218 (oral), 198, 200, 213.
№ 217 (orally)
Lines a and b are parallel to line c. Prove that any line intersecting line a also intersects line b.
Solution.
If a || b and b || c, then a || s (corollary 2˚).
If an arbitrary line d ∩ a, then d ∩ b (Corollary 1˚).
№ 218 (orally)
Lines a and b intersect. Is it possible to draw a line that intersects line a and is parallel to line b? Justify your answer.
Solution.
Let's take point A b on line a. Through point A there is only one line parallel to line b (parallel axiom). The constructed line will intersect line a, since it has a common point A with it.
Lines a and b are perpendicular to line p, line c intersects line a. Does line c intersect line b?
Given:ар, bр, с ∩ а
Find: does c intersect line b?
Solution: if ap and bp, then a || b (theorem).
If c ∩ a and a || b, then c ∩ b (Corollary 1˚).
Answer: c ∩ b.
In the textbook picture AD || p and PQ || B.C. Prove that line p intersects lines AB, AE, AC, BC, PQ.
In the textbook picture, CE = ED, BE = EF and KE = AD. Prove that KE || Sun.
6. Summing up
1) What is the main merit of Euclid?
2) What is called an axiom?
3) What axioms do we know?
4) Which Russian scientist built a coherent theory of non-Euclidean geometry?
5) What is called a consequence in the mathematical sense of the word?
6) What consequences did we learn today?
7. Homework:
§2, paragraph 27, 28, appendix on the axioms of geometry pp. 344-348, questions 7-11 pp. 68, No. 199, 214.
No. 199: Line p is parallel to side AB of triangle ABC. Prove that lines BC and AC intersect line r.
No. 214: A line passing through the middle of the bisector AD of triangle ABC and perpendicular to AD intersects side AC at point M. Prove that MD¦AB.
Literature:
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry, 7-9: Textbook for educational institutions. − M.: Education, 2003.
- Atanasyan L.S., Butuzov V.F., Glazkov Yu.A., Nekrasov V.B., Yudina I.I. Studying geometry in grades 7, 8, 9: Methodological recommendations for the textbook. Book for teachers. − M.: Education, 2003.
- Dorofeeva A.V. Pages of history in mathematics lessons: A book for teachers. − M.: Education, 2007.
- Wikipedia.
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 screen displays the text of the first corollary of the axiom, which states 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.
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)
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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. ... are 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)
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 the 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 sign of parallelism of two lines: if when two lines intersect with a transversal, the lying angles are equal, then the 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 through point M there are two straight lines (straight lines a and MR) parallel to straight 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 with a transversal MN, 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 straight 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 straight 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).
1 0 . ∠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°.
2 0 . ∠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 r.
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, lines a, b and c are intersected by 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 the 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.
§ 1 Axiom of parallel lines
Let's find out which statements are called axioms, give examples of axioms, formulate the axiom of parallel lines and consider some of its consequences.
When studying geometric figures and their properties, the need arises to prove various statements - theorems. When proving them, they often rely on previously proven theorems. The question arises: what are the proofs of the very first theorems based on? In geometry, some initial assumptions are accepted, and on their basis the following theorems are proved. Such initial provisions are called axioms. The axiom is accepted without proof. The word axiom comes from the Greek word "axios", which means "valuable, worthy."
We are already familiar with some axioms. For example, an axiom is the statement: through any two points there passes a straight line, and only one.
When comparing two segments and two angles, we superimposed one segment on the other, and superimposed the angle on the other angle. The possibility of such an imposition follows from the following axioms:
· on any ray from its beginning it is possible to plot a segment equal to the given one, and only one;
· from any ray in a given direction you can put off an angle equal to a given undeveloped angle, and, moreover, only one.
Geometry is an ancient science. For almost two millennia, geometry was studied according to the famous work “Elements” by the ancient Greek scientist Euclid. Euclid first formulated the starting points - postulates, and then, based on them, through logical reasoning he proved other statements. The geometry presented in the Principia is called Euclidean geometry. In the scientist’s manuscripts there is a statement called the fifth postulate, around which controversy flared up for a very long time. Many mathematicians have attempted to prove Euclid's fifth postulate, i.e. derive it from other axioms, but each time the proofs were incomplete or reached a dead end. Only in the 19th century was it finally clarified that the fifth postulate cannot be proven on the basis of the remaining axioms of Euclid, and is itself an axiom. The Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856) played a huge role in solving this issue. So, the fifth postulate is the axiom of parallel lines.
Axiom: through a point not lying on a given line there passes only one line parallel to the given one.
§ 2 Corollaries from the axiom of parallel lines
Statements that are derived directly from axioms or theorems are called corollaries. Let's consider some corollaries from the axiom of parallel lines.
Corollary 1. If a line intersects one of two parallel lines, then it also intersects the other.
Given: lines a and b are parallel, line c intersects line a at point A.
Prove: line c intersects line b.
Proof: if line c did not intersect line b, then two lines a and c would pass through point A, parallel to line b. But this contradicts the axiom of parallel lines: through a point not lying on a given line, only one line parallel to the given line passes. This means that line c intersects line b.
Corollary 2. If two lines are parallel to a third line, then they are parallel.
Given: lines a and b are parallel to line c. (a||c, b||c)
Prove: line a is parallel to line b.
Proof: let’s assume that lines a and b are not parallel, i.e. intersect at some point A. Then two lines a and b pass through point A, parallel to line c. But according to the axiom of parallel lines, through a point not lying on a given line, only one straight line passes through it, parallel to the given one. This means that our assumption is incorrect, therefore, lines a and b are parallel.
List of used literature:
- Geometry. Grades 7-9: textbook. for general education organizations / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al. - M.: Education, 2013. - 383 p.: ill.
- Gavrilova N.F. Lesson developments in geometry grade 7. - M.: “VAKO”, 2004, 288 p. - (To help the school teacher).
- Belitskaya O.V. Geometry. 7th grade. Part 1. Tests. – Saratov: Lyceum, 2014. – 64 p.
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