Definition of parallelism. Parallel lines

In this article we will talk about parallel lines, give definitions, and outline the signs and conditions of parallelism. For clarity theoretical material We will use illustrations and solutions to typical examples.

Yandex.RTB R-A-339285-1 Definition 1

Parallel lines on a plane– two straight lines on a plane that have no common points.

Definition 2

Parallel lines in three-dimensional space – two straight lines in three-dimensional space, lying in the same plane and having no common points.

It is necessary to note that to determine parallel lines in space, the clarification “lying in the same plane” is extremely important: 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.

To indicate parallel lines, it is common to use the symbol ∥. That is, if the given lines a and b are parallel, this condition should be briefly written as follows: a ‖ b. Verbal parallelism of lines is indicated in the following way: lines a and b are parallel, or line a is parallel to line b, or line b is parallel to line a.

Let us formulate a statement that plays important role in the topic being studied.

Axiom

Through a point not belonging to a given line there passes the only straight line parallel to the given one. This statement cannot be proven on the basis of the known axioms of planimetry.

In case we're talking about about space, the theorem is true:

Theorem 1

Through any point in space that does not belong to a given line, there will be a single straight line parallel to the given one.

This theorem is easy to prove on the basis of the above axiom (geometry program for grades 10 - 11).

There is a sign of parallelism sufficient condition, during which the parallelism of the lines is guaranteed. In other words, the fulfillment of this condition is sufficient to confirm the fact of parallelism.

In particular, there are necessary and sufficient conditions for the parallelism of lines on the plane and in space. Let us explain: necessary means the condition the fulfillment of which is necessary for parallel lines; if it is not fulfilled, the lines are not parallel.

To summarize, a necessary and sufficient condition for the parallelism of lines is a condition the observance of which is necessary and sufficient for the lines to be parallel to each other. On the one hand, this is a sign of parallelism, on the other hand, it is a property inherent in parallel lines.

Before giving the exact formulation of a necessary and sufficient condition, let us recall a few additional concepts.

Definition 3

Secant line– a straight line intersecting each of two given non-coinciding straight lines.

Intersecting two straight lines, a transversal forms eight undeveloped angles. To formulate a necessary and sufficient condition, we will use such types of angles as crossed, corresponding and one-sided. Let's demonstrate them in the illustration:

Theorem 2

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

Let us illustrate graphically the necessary and sufficient condition for the parallelism of lines on a plane:

The proof of these conditions is present in the geometry program for grades 7 - 9.

In general, these conditions also apply to three-dimensional space, provided that two lines and a secant belong to the same plane.

Let us indicate a few more theorems that are often used to prove the fact that lines are parallel.

Theorem 3

On a plane, two lines parallel to a third are parallel to each other. This feature is proved on the basis of the parallelism axiom indicated above.

Theorem 4

In three-dimensional space, two lines parallel to a third are parallel to each other.

The proof of a sign is studied in the 10th grade geometry curriculum.

Let us give an illustration of these theorems:

Let us indicate one more pair of theorems that prove the parallelism of lines.

Theorem 5

On a plane, two lines perpendicular to a third are parallel to each other.

Let us formulate a similar thing for three-dimensional space.

Theorem 6

In three-dimensional space, two lines perpendicular to a third are parallel to each other.

Let's illustrate:

All the above theorems, signs and conditions make it possible to conveniently prove the parallelism of lines using the methods of geometry. That is, to prove the parallelism of lines, one can show that the corresponding angles are equal, or demonstrate the fact that two given lines are perpendicular to the third, etc. But note that it is often more convenient to use the coordinate method to prove the parallelism of lines on a plane or in three-dimensional space.

Parallelism of lines in a rectangular coordinate system

In a given rectangular system coordinates, a straight line is determined by the equation of a straight line on the plane of one of possible types. Likewise, a straight line defined in a rectangular coordinate system in three-dimensional space corresponds to some equations for a straight line in space.

Let us write down the necessary and sufficient conditions for the parallelism of lines in a rectangular coordinate system depending on the type of equation describing the given lines.

Let's start with the condition of parallelism of lines on a plane. It is based on the definitions of the direction vector of a line and the normal vector of a line on a plane.

Theorem 7

For two non-coinciding lines to be parallel on a plane, it is necessary and sufficient that the direction vectors of the given lines are collinear, or the normal vectors of the given lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the other line.

It becomes obvious that the condition of parallelism of lines on a plane is based on the condition of collinearity of vectors or the condition of perpendicularity of two vectors. That is, if a → = (a x , a y) and b → = (b x , b y) are direction vectors of lines a and b ;

and n b → = (n b x , n b y) are normal vectors of lines a and b, then we write the above necessary and sufficient condition as follows: a → = t · b → ⇔ a x = t · b x a y = t · b y or n a → = t · n b → ⇔ n a x = t · n b x n a y = t · n b y or a → , n b → = 0 ⇔ a x · n b x + a y · n b y = 0 , where t is some real number. The coordinates of the guides or straight vectors are determined by the given equations of the straight lines. Let's look at the main examples.

Straight a in a rectangular coordinate system is defined general equation straight line: A 1 x + B 1 y + C 1 = 0; straight line b - A 2 x + B 2 y + C 2 = 0. Then the normal vectors of the given lines will have coordinates (A 1, B 1) and (A 2, B 2), respectively. We write the parallelism condition as follows:

A 1 = t A 2 B 1 = t B 2

Line a is described by the equation of a line with a slope of the form y = k 1 x + b 1 . Straight line b - y = k 2 x + b 2. Then the normal vectors of the given lines will have coordinates (k 1, - 1) and (k 2, - 1), respectively, and we will write the parallelism condition as follows:

k 1 = t k 2 - 1 = t (- 1) ⇔ k 1 = t k 2 t = 1 ⇔ k 1 = k 2

Thus, if parallel lines on a plane in a rectangular coordinate system are given by equations with angular coefficients, then slopes given lines will be equal. And the opposite statement is true: if non-coinciding lines on a plane in a rectangular coordinate system are determined by the equations of a line with identical angular coefficients, then these given lines are parallel.

Lines a and b in a rectangular coordinate system are specified by the canonical equations of a line on a plane: x - x 1 a x = y - y 1 a y and x - x 2 b x = y - y 2 b y or by parametric equations of a line on a plane: x = x 1 + λ · a x y = y 1 + λ · a y and x = x 2 + λ · b x y = y 2 + λ · b y .

Then the direction vectors of the given lines will be: a x, a y and b x, b y, respectively, and we will write the parallelism condition as follows:

a x = t b x a y = t b y

Let's look at examples.

Example 1

Two lines are given: 2 x - 3 y + 1 = 0 and x 1 2 + y 5 = 1. It is necessary to determine whether they are parallel.

Solution

Let us write the equation of a straight line in segments in the form of a general equation:

x 1 2 + y 5 = 1 ⇔ 2 x + 1 5 y - 1 = 0

We see that n a → = (2, - 3) is the normal vector of the line 2 x - 3 y + 1 = 0, and n b → = 2, 1 5 is the normal vector of the line x 1 2 + y 5 = 1.

The resulting vectors are not collinear, because there is no such value of tat which the equality will be true:

2 = t 2 - 3 = t 1 5 ⇔ t = 1 - 3 = t 1 5 ⇔ t = 1 - 3 = 1 5

Thus, the necessary and sufficient condition for the parallelism of lines on a plane is not satisfied, which means the given lines are not parallel.

Answer: the given lines are not parallel.

Example 2

The lines y = 2 x + 1 and x 1 = y - 4 2 are given. Are they parallel?

Solution

Let's transform the canonical equation of the straight line x 1 = y - 4 2 to the equation of the straight line with the slope:

x 1 = y - 4 2 ⇔ 1 · (y - 4) = 2 x ⇔ y = 2 x + 4

We see that the equations of the lines y = 2 x + 1 and y = 2 x + 4 are not the same (if it were otherwise, the lines would be coincident) and the angular coefficients of the lines are equal, which means the given lines are parallel.

Let's try to solve the problem differently. First, let's check whether the given lines coincide. We use any point on the line y = 2 x + 1, for example, (0, 1), the coordinates of this point do not correspond to the equation of the line x 1 = y - 4 2, which means the lines do not coincide.

The next step is to determine whether the condition of parallelism of the given lines is satisfied.

The normal vector of the line y = 2 x + 1 is the vector n a → = (2 , - 1) , and the direction vector of the second given line is b → = (1 , 2) . Scalar product of these vectors is equal to zero:

n a → , b → = 2 1 + (- 1) 2 = 0

Thus, the vectors are perpendicular: this demonstrates to us the fulfillment of the necessary and sufficient condition for the parallelism of the original lines. Those. the given lines are parallel.

Answer: these lines are parallel.

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

Theorem 8

For two non-coinciding lines in three-dimensional space to be parallel, it is necessary and sufficient that the direction vectors of these lines be collinear.

Those. at given equations of straight lines in three-dimensional space, the answer to the question: are they parallel or not, is found by determining the coordinates of the direction vectors of the given straight lines, as well as checking the condition of their collinearity. In other words, if a → = (a x , a y , a z) and b → = (b x , b y , b z) are direction vectors of straight lines a and b respectively, then in order for them to be parallel, the existence of such real number t so that the equality holds:

a → = t b → ⇔ a x = t b x a y = t b y a z = t b z

Example 3

The lines x 1 = y - 2 0 = z + 1 - 3 and x = 2 + 2 λ y = 1 z = - 3 - 6 λ are given. It is necessary to prove the parallelism of these lines.

Solution

The conditions of the problem are given canonical equations one straight line in space and parametric equations another line in space. Guide vectors a → and b → the given lines have coordinates: (1, 0, - 3) and (2, 0, - 6).

1 = t · 2 0 = t · 0 - 3 = t · - 6 ⇔ t = 1 2 , then a → = 1 2 · b → .

Consequently, the necessary and sufficient condition for the parallelism of lines in space is satisfied.

Answer: the parallelism of the given lines is proven.

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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, what is possible. Straight C.E. parallel AB.

To prove this, let us assume the opposite, i.e., that C.E. intersects 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 in 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 between 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 the straight lines have 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 reference

WITH Greek language The concept of “parallelos” is translated as “nearby” or “held next to each other.” This term was used in the ancient school of Pythagoras even before parallel lines were 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 sign 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 that in ordinary life we are surrounded by a huge number of 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 straight lines will look like two converging lines.

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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

Parallelism is a binary equivalence relation, therefore it divides the entire set of lines into classes of lines parallel to each other.
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)
2 parallel lines in space lie in the same plane.
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 $AB$

Passes infinite set 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:

points B And E lie on one side of a straight line AC ;
straight CE does not intersect the 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.