Straight line. Parallel lines

Which lie in the same plane and either coincide or do not intersect. In some school definitions coincident lines are not considered parallel; such a definition is not considered here.

Properties

1. Parallelism is a binary equivalence relation, therefore it divides the entire set of lines into classes of lines parallel to each other.
2. Through any point you can draw exactly one straight line parallel to the given one. This is a distinctive property of Euclidean geometry; in other geometries the number 1 is replaced by others (in Lobachevsky geometry there are at least two such lines)
3. 2 parallel lines in space lie in the same plane.
4. When 2 parallel lines intersect, a third one, called secant:
   1. The secant necessarily intersects both lines.
   2. When intersecting, 8 angles are formed, some characteristic pairs of which have special names and properties:
      1. Lying crosswise the angles are equal.
      2. Relevant the angles are equal.
      3. Unilateral the angles add up to 180°.

In Lobachevsky geometry

In Lobachevsky geometry in the plane through a point The expression cannot be parsed ( lexical error): Coutside this line $AB$

There are an infinite number of straight lines that do not intersect AB. Of these, parallel to AB only two are named.

Straight CE called an equilateral (parallel) line AB in the direction from A To B, If:

1. points B And E lie on one side of a straight line AC ;
2. straight CE does not intersect a line AB, but every ray passing inside an angle ACE, crosses the ray AB .

A straight line is defined similarly AB in the direction from B To A .

All other lines that do not intersect this one are called ultraparallel or divergent.

See also

Wikimedia Foundation. 2010.

• Crossing lines
• Nesterikhin, Yuri Efremovich

See what “Parallel lines” are in other dictionaries:

PARALLEL DIRECT- PARALLEL LINES, non-intersecting lines lying in the same plane... Modern encyclopedia

PARALLEL DIRECT Big Encyclopedic Dictionary

Parallel lines- PARALLEL LINES, non-intersecting lines lying in the same plane. ... Illustrated Encyclopedic Dictionary

Parallel lines- in Euclidean geometry, straight lines that lie in the same plane and do not intersect. In absolute geometry (See Absolute geometry), through a point that does not lie on a given line, at least one straight line passes through a point that does not intersect the given one. IN… … Great Soviet Encyclopedia

parallel lines- non-intersecting lines lying in the same plane. * * * PARALLEL LINES PARALLEL LINES, non-intersecting lines lying in the same plane... Encyclopedic Dictionary

PARALLEL DIRECT- in Euclidean geometry, straight lines lie in the same plane and do not intersect. In absolute geometry, through a point not lying on a given line there passes at least one line that does not intersect the given one. In Euclidean geometry there is only one... ... Mathematical Encyclopedia

PARALLEL DIRECT- non-intersecting lines lying in the same plane... Natural science. Encyclopedic Dictionary

Parallel worlds in fiction- This article may contain original research. Add links to sources, otherwise it may be set for deletion. More information may be on the talk page. This... Wikipedia

Parallel worlds - Parallel world(in fiction) a reality that somehow exists simultaneously with ours, but independently of it. This autonomous reality can have various sizes: from a small geographical area to an entire universe. In parallel... Wikipedia

Parallel- lines Straight lines are called P. if neither they nor their extensions intersect each other. The news from one of these lines is at the same distance from the other. However, it is customary to say: two P. straight lines intersect at infinity. Such… … Encyclopedia of Brockhaus and Efron

Books

• Set of tables. Mathematics. 6th grade. 12 tables + methodology, . The tables are printed on thick printed cardboard measuring 680 x 980 mm. The kit includes a brochure with methodological recommendations for the teacher. Educational album of 12 sheets. Divisibility…

They do not intersect, no matter how long they are continued. The parallelism of straight lines in writing is denoted as follows: AB|| WITHE

The possibility of the existence of such lines is proved by the theorem.

Theorem.

Through any point taken outside a given line, one can draw a point parallel to this line.

Let AB this straight line and WITH some point taken outside it. It is required to prove that through WITH you can draw a straight line parallelAB. Let's lower it to AB from point WITH perpendicularWITHD and then we will conduct WITHE^ WITHD, which is possible. Straight C.E. parallel AB.

To prove this, let us assume the opposite, i.e., that C.E. intersects with AB at some point M. Then from the point M to a straight line WITHD we would have two different perpendiculars MD And MS, which is impossible. Means, C.E. can't cross with AB, i.e. WITHE parallel AB.

Consequence.

Two perpendiculars (CEAndD.B.) to one straight line (CD) are parallel.

Axiom of parallel lines.

Through the same point it is impossible to draw two different lines parallel to the same line.

So, if straight WITHD, drawn through the point WITH parallel to the line AB, then every other line WITHE, drawn through the same point WITH, cannot be parallel AB, i.e. she's on continuation will intersect With AB.

Proving this not entirely obvious truth turns out to be impossible. It is accepted without proof, as a necessary assumption (postulatum).

Consequences.

1. If straight(WITHE) intersects with one of parallel(NE), then it intersects with another ( AB), because otherwise through the same point WITH there would be two different lines passing parallel AB, which is impossible.

2. If each of the two direct (AAndB) are parallel to the same third line ( WITH) , then they parallel among themselves.

Indeed, if we assume that A And B intersect at some point M, then two different lines parallel to this point would pass through WITH, which is impossible.

Theorem.

If line is perpendicular to one of the parallel lines, then it is perpendicular to the other parallel.

Let AB || WITHD And E.F. ^ AB.It is required to prove that E.F. ^ WITHD.

PerpendicularEF, intersecting with AB, will certainly cross and WITHD. Let the intersection point be H.

Let us now assume that WITHD not perpendicular to E.H.. Then some other straight line, for example H.K., will be perpendicular to E.H. and therefore through the same point H there will be two straight parallel AB: one WITHD, by condition, and the other H.K. as previously proven. Since this is impossible, it cannot be assumed that NE was not perpendicular to E.H..

The concept of parallel lines

Definition 1

Parallel lines– straight lines that lie in the same plane do not coincide and do not have common points.

If straight lines have a common point, then they intersect.

If all points are straight match, then we essentially have one straight line.

If the lines lie in different planes, then the conditions for their parallelism are somewhat greater.

When considering straight lines on the same plane, the following definition can be given:

Definition 2

Two lines in a plane are called parallel, if they do not intersect.

In mathematics, parallel lines are usually denoted using the parallelism sign “$\parallel$”. For example, the fact that line $c$ is parallel to line $d$ is denoted as follows:

$c\parallel d$.

The concept of parallel segments is often considered.

Definition 3

The two segments are called parallel, if they lie on parallel lines.

For example, in the figure the segments $AB$ and $CD$ are parallel, because they belong to parallel lines:

$AB \parallel CD$.

At the same time, the segments $MN$ and $AB$ or $MN$ and $CD$ are not parallel. This fact can be written using symbols as follows:

$MN ∦ AB$ and $MN ∦ CD$.

The parallelism of a straight line and a segment, a straight line and a ray, a segment and a ray, or two rays is determined in a similar way.

Historical background

From the Greek, the concept of “parallelos” is translated as “coming next to” or “held next to each other.” This term was used in ancient school Pythagoras before parallel lines were even defined. According to historical facts Euclid in the $III$ century. BC his works nevertheless revealed the meaning of the concept of parallel lines.

In ancient times, the symbol for designating parallel lines had a different appearance from what we use in modern mathematics. For example, the ancient Greek mathematician Pappus in the $III$ century. AD parallelism was indicated using an equal sign. Those. the fact that line $l$ is parallel to line $m$ was previously denoted by “$l=m$”. Later, the familiar “$\parallel$” sign began to be used to denote the parallelism of lines, and the equal sign began to be used to denote the equality of numbers and expressions.

Parallel lines in life

We often do not notice what surrounds us in everyday life. huge number parallel lines. For example, in a music book and a collection of songs with notes, the staff is made using parallel lines. Parallel lines are also found in musical instruments(for example, harp strings, guitar strings, piano keys, etc.).

Electrical wires that are located along streets and roads also run parallel. Metro line rails and railways are located in parallel.

In addition to everyday life, parallel lines can be found in painting, in architecture, and in the construction of buildings.

Parallel lines in architecture

In the presented images, architectural structures contain parallel lines. The use of parallel lines in construction helps to increase the service life of such structures and gives them extraordinary beauty, attractiveness and grandeur. Power lines are also deliberately laid in parallel to avoid crossing or touching them, which would lead to short circuits, outages and loss of electricity. So that the train can move freely, the rails are also made in parallel lines.

In painting, parallel lines are depicted as converging into one line or close to it. This technique is called perspective, which follows from the illusion of vision. If you look into the distance for a long time, parallel lines will look like two converging lines.

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

Page navigation.

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-dimensional space are called parallel, if they lie in the same plane and do not have common points.

Please note that the clause “if they lie in the same plane” in the definition of parallel lines in space is very important. Let us clarify this point: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

Here are some examples of parallel lines. Opposite edges 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. Railway rails on level ground can also be considered as parallel lines.

To denote parallel lines, use the symbol “”. That is, if lines a and b are parallel, then we can briefly write a b.

Please note: 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 on a plane: through a point not lying on a given line, there passes the only straight line parallel to the given one. This statement is accepted as a fact (it cannot be proven 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 valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem is easily proven using the above axiom of parallel lines (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the list of references).

For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem can be easily proven using the above parallel line axiom.

Parallelism of lines - signs and conditions of parallelism.

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

There are also necessary and sufficient conditions for the parallelism of lines on a 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 “ necessary condition parallelism of lines"? From the name “necessary” it is clear that the fulfillment of this condition is necessary for parallel lines. In other words, if the necessary condition for parallel lines is not met, then the lines are not parallel. Thus, necessary and sufficient condition for parallel lines 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 parallelism of lines, and on the other hand, this is a property that parallel lines have.

Before formulating a necessary and sufficient condition for the parallelism of lines, it is advisable to recall several auxiliary definitions.

Secant line is a line that intersects each of two given non-coinciding lines.

When two straight lines intersect with a transversal, eight undeveloped ones are formed. In the formulation of the necessary and sufficient condition for the parallelism of lines, the so-called lying crosswise, corresponding And one-sided angles. Let's show them in the drawing.

Theorem.

If two lines in a plane are intersected by a transversal, then for them to be parallel it is necessary and sufficient that the intersecting angles be equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphic illustration of this necessary and sufficient condition for the parallelism of lines on a plane.

You can find proofs of these conditions for the parallelism of 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 to prove 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 property 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 criterion is discussed in geometry lessons in the 10th grade.

Let us illustrate the stated theorems.

Let us present another theorem that allows us to prove the parallelism of lines on a 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, criteria and necessary and sufficient conditions formulated above are excellent for proving the parallelism of lines using geometry methods. That is, to prove the parallelism of two given lines, you need to show that they are parallel to a third line, or show the equality of crosswise lying angles, etc. Many similar problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the coordinate method to prove the parallelism of lines on a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are specified in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this paragraph of the article we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations defining these straight lines, and we also present detailed solutions characteristic tasks.

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

Theorem.

For two non-coinciding 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 on a plane is reduced 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 direction vectors of lines a and b, and And are normal vectors of lines a and b, respectively, then the necessary and sufficient condition for the parallelism of lines a and b will be written as , or , or , where t is some real number. In turn, the coordinates of the guides and (or) normal vectors of straight lines a and b are found by known equations straight

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

If line a corresponds to the equation of a line with an angular coefficient of the form , and line b - , then the normal vectors of these lines have coordinates and , and the condition for parallelism of these lines takes the form . Consequently, if straight lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of straight lines with angular coefficients, then slope coefficients straight lines will be equal. And vice versa: if non-coinciding lines on a plane in a rectangular coordinate system can be specified by equations of a line with equal angular coefficients, then such lines are parallel.

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

Let's look at solutions to several examples.

Example.

Are the lines parallel? And ?

Solution.

Let us rewrite the equation of a straight line in segments in the form general equation direct: . Now we can see that is the normal vector of the line , a is the normal vector of the 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 a plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are straight lines and parallel?

Solution.

Let's give canonical equation straight line to the equation of a straight line with an angular coefficient: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the angular coefficients of the lines are equal, therefore, the original lines are parallel.

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