Corollaries from the axiom of parallel lines definition. Properties of parallel lines
First, let's look at the difference between the concepts of sign, property and axiom.
Definition 1
Sign they call a certain fact by which the truth of a judgment about an object of interest can be determined.
Example 1
Lines are parallel if their transversal forms equal crosswise angles.
Definition 2
Property is formulated in the case when there is confidence in the fairness of the judgment.
Example 2
When parallel lines are parallel, their transversal forms equal crosswise angles.
Definition 3
Axiom they call a statement that does not require proof and is accepted as truth without it.
Each science has axioms on which subsequent judgments and their proofs are based.
Axiom of parallel lines
Sometimes the axiom of parallel lines is accepted as one of the properties of parallel lines, but at the same time other geometric proofs are based on its validity.
Theorem 1
Through a point that does not lie on a given line, only one straight line can be drawn on the plane, which will be parallel to the given one.
The axiom does not require proof.
Properties of parallel lines
Theorem 2
Property1. Transitivity property of parallel lines:
When one of two parallel lines is parallel to the third, then the second line will be parallel to it.
Properties require proof.
Proof:
Let there be two parallel lines $a$ and $b$. Line $c$ is parallel to line $a$. Let us check whether in this case the straight line $c$ will also be parallel to the straight line $b$.
To prove this, we will use the opposite proposition:
Let us imagine that it is possible that line $c$ is parallel to one of the lines, for example, line $a$, and intersects the other line, line $b$, at some point $K$.
We obtain a contradiction according to the axiom of parallel lines. This results in a situation in which two lines intersect at one point, moreover, parallel to the same line $a$. This situation is impossible; therefore, the lines $b$ and $c$ cannot intersect.
Thus, it has been proven that if one of two parallel lines is parallel to the third line, then the second line is parallel to the third line.
Theorem 3
Property 2.
If one of two parallel lines is intersected by a third, then the second line will also be intersected by it.
Proof:
Let there be two parallel lines $a$ and $b$. Also, let there be some line $c$ that intersects one of the parallel lines, for example, line $a$. It is necessary to show that line $c$ also intersects the second line, line $b$.
Let's construct a proof by contradiction.
Let's imagine that line $c$ does not intersect line $b$. Then two lines $a$ and $c$ pass through the point $K$, which do not intersect the line $b$, i.e., they are parallel to it. But this situation contradicts the axiom of parallel lines. This means that the assumption was incorrect and line $c$ will intersect line $b$.
The theorem has been proven.
Properties of corners, which form two parallel lines and a secant: opposite angles are equal, corresponding angles are equal, * the sum of one-sided angles is $180^(\circ)$.
Example 3
Given two parallel lines and a third line perpendicular to one of them. Prove that this line is perpendicular to another of the parallel lines.
Proof.
Let us have straight lines $a \parallel b$ and $c \perp a$.
Since line $c$ intersects line $a$, then, according to the property of parallel lines, it will also intersect line $b$.
The secant $c$, intersecting the parallel lines $a$ and $b$, forms equal internal angles lying crosswise with them.
Because $c \perp a$, then the angles will be $90^(\circ)$.
Therefore, $c \perp b$.
The proof is complete.
We also used other axioms, although we did not particularly highlight them. So, we compared two segments using superposition. The possibility of such an overlap follows from the axiom “On any ray from its beginning, you can lay off a segment equal to the given one, and only one”
These axioms are beyond doubt and with their help other statements are proven. This method originated a long time ago and was outlined in the work “Elements” by the scientist Euclid. Some of Euclid's axioms - postulates are now used in geometry, and the geometry itself, set out in the "Elements", is called Euclidean geometry.
Theorems on angles formed by two parallels and a transversal. The condition is what is given. The conclusion is what needs to be proven. A theorem inverse to a given one is a theorem in which the condition is the conclusion of the given theorem, and the conclusion is the condition of the given theorem.
Comment. If a certain theorem is proven, then the converse statement does not follow. Moreover, the converse is not always true. For example, “vertical angles are equal.” The opposite statement: “if the angles are equal, then they are vertical” is, of course, false.
Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government authorities in the territory of the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
§ 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.
Images used:
Fig.1-2
For example, the task is given to draw two parallel lines, and so that through a given point M at least one of the straight lines passed. Thus, through a given point M draw mutually perpendicular lines MN And CD . And through the point N let's draw a second straight line AB , it must be perpendicular to the line MN .
Let's conclude: straight AB perpendicular to the line MN and straight CD is also perpendicular to the line MN , and since these lines are parallel to one line, then, as a consequence, the line CD parallel AB . So, through the point M there is a straight line CD , which is parallel to the line AB . Let's find out: is it possible to draw another straight line through the point? M so that it is parallel to the line AB ?
This statement is the answer to our question: through a point on the plane that does not lie on a given line, you can draw only one straight line, which will be parallel to the given line. Such a rejection in a different formulation without evidence was accepted in ancient times by the scientist Euclid. It is known that such statements, accepted without proof, are called axioms.
The above statement is called the parallel lines axiom. This axiom of Euclid is of great importance for the proof of many theorems.
Let's consider the converse theorem. If a straight line intersects parallel lines, then the angles lying crosswise at parallel lines are correspondingly equal.
Rice.
3
Proof: suppose that AC And ВD are parallel lines, then the line AB is their secant line. We need to prove that РСАВ =Р АВD .
We need to draw a straight line like this AC1 , to РС1АВ=РАВD . In accordance with the axiom of parallel lines AC1||ВD , in the condition we have AC||ВD . And this means that through this point A two lines pass through, and they are parallel to the line ВD . This results in a contradiction to the axiom of parallel lines, which means that the straight line AC1 carried out incorrectly.
It will be correct if РСАВ=РАВD . Let us conclude: in the case when a given straight line is perpendicular to one of the parallel lines, then it will be perpendicular to the second line.
It turns out if (MN)^(CD) And (CD)||(AB) , That Р1=Р2=90о . And this means: (MN)^(AB) (Fig. 1) .
Let's prove the theorem: if two lines are parallel to the third, then they will be parallel to the second.
Rice. 4
Let it be straight a parallel to the line With and straight b also parallel to the line With (Fig. 4 a) . We need to prove that a||b .
Let's assume that straight lines a And b are not parallel, but they intersect at a point M (Fig. 4 b) . And this means that two straight lines a And b , which are parallel to the line with pass through one point, and this is a complete contradiction to the axiom of parallel lines. So ours are direct a And b parallel.
