Like two parallel lines. Straight line

In this article we will talk about parallel lines, give definitions, and outline the signs and conditions of parallelism. To make the theoretical material clearer, 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. Verbally, parallelism of lines is denoted as follows: 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 an important role in the topic under study.

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 the case when we are talking 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).

The parallelism criterion is a sufficient condition, the fulfillment of which guarantees parallelism of lines. 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 are also applicable for three-dimensional space, despite the fact 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 geometry methods. 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 coordinate system, a straight line is determined by the equation of a straight line on a plane of one of the 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 for 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.

Line a in a rectangular coordinate system is determined by the general equation of the 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 the angular coefficients of the 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) . The 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. given the equations of 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 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 the direction vectors of the lines a and b, respectively, then in order for them to be parallel, the existence of such a real number t is necessary, 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 by the canonical equations of one line in space and the parametric equations of 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 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..

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. The opposite edges of the 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. Railroad 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. What is a “necessary condition for parallel 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 the lines to be parallel 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 straight 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 criterion 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 lines, and we will also provide detailed solutions to characteristic problems.

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 lines a and b are found using the known equations of lines.

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 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 lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of lines with angular coefficients, then the angular coefficients of the 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 line in segments in the form of a general equation of a line: . 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 us reduce the canonical equation of a 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.

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 Unable to parse expression (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:

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, intersects 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.

Like two parallel lines. Straight line

Parallelism of lines in a rectangular coordinate system

Parallel lines - basic information.

Parallelism of lines - signs and conditions of parallelism.

Parallelism of lines in a rectangular coordinate system.

Properties

In Lobachevsky geometry

see also

Nesterikhin, Yuri Efremovich

Books