To question 1. Give a definition of parallel lines. What two line segments are called parallel? given by the author Sasha Nizhevyasov the best answer is which on the plane will never intersect

Answer from adaptability[guru]
Parallel lines are lines that lie in the same plane and either coincide or do not intersect.

Answer from Naumenko[guru]
segments. belonging to parallel lines. are parallel.
straight lines on the plane called. parallel. if they do not intersect or coincide.

Answer from Neurologist[newbie]
Two lines that lie in the same plane and have no common point are called parallel.

Answer from Throw[master]

Answer from Varvara Lamekina[newbie]
two lines in a plane are said to be parallel if they do not intersect)

Answer from Maxim Ivanov[newbie]
Which do not intersect on the plane.

Answer from Sem2805[active]
two lines in a plane are called parallel if they do not intersect (Grade 7)

Answer from Sasha Klyuchnikov[newbie]
Parallel lines in Euclidean geometry, lines that 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 line. In Euclidean geometry, there is only one such line. This fact is equivalent to Euclid's fifth postulate (about parallel). In Lobachevsky geometry (see Lobachevsky geometry) in the plane through the point C (see figure) outside the given line AB there are an infinite set of lines that do not intersect AB. Of these, only two are called parallel to AB. Line CE is called parallel to line AB in the direction from A to B if: 1) points B and E lie on the same side of line AC; 2) line CE does not intersect line AB; any ray passing inside angle ACE intersects ray AB. The straight line CF parallel to AB in the direction from B to A is defined similarly.

Answer from Anatoly Mishin[newbie]
Two lines in space are called parallel if they lie in the same plane and do not intersect.

Answer from Ўliya[active]
Parallel lines are lines that do not intersect

Answer from said charakov[newbie]
Parallel are two straight lines that lie in the same plane and have no common points.
Through a point, only one line can be drawn parallel to a given plane.

Answer from Olga Nemtyreva[newbie]
Parallel lines are lines that lie in the same plane and either coincide or do not intersect. ..Lobachevsky geometry) in the plane through the point C (see Fig.) outside the given line AB there passes an infinite set of lines that do not intersect AB. Of these, only two are called parallel to AB.

Answer from Oksana Tyshchenko[newbie]
Parallel lines are two lines in a plane that do not intersect. Two line segments are called parallel if they lie on parallel lines.

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

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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 dimensions are called parallel if they lie in the same plane and have no common points.

Note that the "if they lie in the same plane" clause in the definition of parallel lines in space is very important. Let's clarify this point: two straight lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but are skew.

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 tracks on level ground can also be thought of as parallel lines.

The symbol "" is used to denote parallel lines. That is, if the lines a and b are parallel, then you can briefly write a b.

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

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem is easily proved using the axiom of parallel lines given above.

Parallelism of lines - signs and conditions of parallelism.

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

There are also necessary and sufficient conditions for parallel lines in the 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 the "necessary condition for parallel lines"? By the name "necessary" it is clear that the fulfillment of this condition is necessary for the lines to be parallel. In other words, if the necessary condition for parallel lines is not satisfied, then the lines are not parallel. In this way, necessary and sufficient condition for lines to be parallel 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 parallel lines, and on the other hand, this is a property that parallel lines have.

Before stating the necessary and sufficient condition for lines to be parallel, it is useful to recall a few auxiliary definitions.

secant line is a line that intersects each of the two given non-coincident lines.

At the intersection of two lines of a secant, eight non-deployed ones are formed. The so-called lying crosswise, corresponding and one-sided corners. Let's show them on the drawing.

Theorem.

If two lines on a plane are intersected by a secant, then for their parallelism it is necessary and sufficient that the crosswise lying angles are equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphical illustration of this necessary and sufficient condition for parallel lines in the plane.

You can find proofs of these conditions for parallel 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 in proving 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 feature 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 feature is considered in the geometry lessons in grade 10.

Let us illustrate the voiced theorems.

Let us give one more theorem that allows us to prove the parallelism of lines in the 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 formulated above, signs and necessary and sufficient conditions are perfectly suitable for proving the parallelism of straight lines by methods of geometry. That is, to prove the parallelism of two given lines, it is necessary to show that they are parallel to the third line, or to show the equality of cross-lying angles, etc. Many of these problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the method of coordinates to prove the parallelism of lines in a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are given in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this section of the article, we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations that determine these lines, and we will also give detailed solutions to typical problems.

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

Theorem.

For two non-coincident 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 in the plane reduces 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 the direction vectors of the lines a and b, and and are the normal vectors of lines a and b, respectively, then the necessary and sufficient condition for parallel lines a and b can be written as , or , or , where t is some real number. In turn, the coordinates of the directing and (or) normal vectors of the straight lines a and b are found from the known equations of the straight lines.

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

If the straight line a corresponds to the equation of the straight line with the slope coefficient of the form . Therefore, if straight lines on a plane in a rectangular coordinate system are parallel and can be given by equations of straight lines with slope coefficients, then the slope coefficients of the lines will be equal. And vice versa: if non-coinciding straight lines on a plane in a rectangular coordinate system can be given by the equations of a straight line with equal slope coefficients, then such straight lines are parallel.

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

Let's take a look at a few examples.

Example.

Are the lines parallel? and ?

Solution.

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

Answer:

No, the lines are not parallel.

Example.

Are lines and parallels?

Solution.

We bring the canonical equation of a straight line to the equation of a straight line with a slope: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the slopes of the lines are equal, therefore, the original lines are parallel.

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

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

Parallel lines in the plane are two straight lines in the plane that do not have common points.

Definition 2

Parallel lines in 3D space- two straight lines in three-dimensional space that lie in the same plane and do not have common points.

It should be noted that in order 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 denote 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, the parallelism of lines is indicated 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 that does not belong to a given line, there is only one line parallel to the given line. This statement cannot be proved on the basis of the known axioms of planimetry.

In the case when it comes to space, the theorem is true:

Theorem 1

Through any point in space that does not belong to a given line, there will be only one 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 sign of parallelism is a sufficient condition under which parallel lines are 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 in 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 satisfied, the lines are not parallel.

Summarizing, a necessary and sufficient condition for the parallelism of lines is such 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, a property inherent in parallel lines.

Before giving a precise formulation of the necessary and sufficient conditions, we recall a few more additional concepts.

Definition 3

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

Intersecting two straight lines, the secant forms eight non-expanded angles. To formulate the necessary and sufficient condition, we will use such types of angles as cross-lying, corresponding, and one-sided. Let's demonstrate them in the illustration:

Theorem 2

If two lines on a plane intersect a secant, then for the given lines to be parallel it is necessary and sufficient that the crosswise lying angles be equal, or the corresponding angles be equal, or the sum of one-sided angles be equal to 180 degrees.

Let us graphically illustrate the necessary and sufficient condition for parallel lines on the 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, provided that the two lines and the secant belong to the same plane.

Let us point out a few more theorems that are often used in proving the fact that lines are parallel.

Theorem 3

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

Theorem 4

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

The proof of the attribute is studied in the 10th grade geometry program.

We give an illustration of these theorems:

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

Theorem 5

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

Let us formulate a similar one for a 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 by 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, and so on. But we note that it is often more convenient to use the coordinate method to prove the parallelism of lines in 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. Similarly, a straight line given in a rectangular coordinate system in three-dimensional space corresponds to some equations of a straight line in space.

Let us write 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 parallel lines in the plane. It is based on the definitions of the direction vector of the line and the normal vector of the line in the plane.

Theorem 7

In order for two non-coincident lines to be parallel on a plane, it is necessary and sufficient that the direction vectors of the given lines be 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 parallel lines on the plane is based on the condition of collinear vectors or the condition of perpendicularity of two vectors. That is, if a → = (a x , a y) and b → = (b x , b y) are the 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 directing or direct vectors are determined by the given equations of the lines. Let's consider the main examples.

1. The 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 ; 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 condition of parallelism as follows:

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

1. The straight line a is described by the equation of a straight line with a slope of the form y = k 1 x + b 1 . Straight line b - y \u003d 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 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 slope coefficients, then the slope coefficients of the given lines will be equal. And the converse statement is true: if non-coinciding lines on a plane in a rectangular coordinate system are determined by the equations of a line with the same slope coefficients, then these given lines are parallel.

1. The lines a and b in a rectangular coordinate system are given by the canonical equations of the line on the plane: x - x 1 a x = y - y 1 a y and x - x 2 b x = y - y 2 b y or the parametric equations of the line on the 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 write the parallelism condition as follows:

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

Let's look at examples.

Example 1

Given two lines: 2 x - 3 y + 1 = 0 and x 1 2 + y 5 = 1 . You need to determine if they are parallel.

Solution

We 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 t for 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 of parallelism of lines on the plane is not satisfied, which means that the given lines are not parallel.

Answer: given lines are not parallel.

Example 2

Given lines y = 2 x + 1 and x 1 = y - 4 2 . Are they parallel?

Solution

Let's transform the canonical equation of the straight line x 1 \u003d y - 4 2 to the equation of a straight line with a 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 the same) and the slopes of the lines are equal, which means that the given lines are parallel.

Let's try to solve the problem differently. First, we check whether the given lines coincide. We use any point of the line y \u003d 2 x + 1, for example, (0, 1), the coordinates of this point do not correspond to the equation of the line x 1 \u003d y - 4 2, which means that the lines do not coincide.

The next step is to determine the fulfillment of the parallelism condition for the given lines.

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 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 original lines to be parallel. Those. 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-coincident lines in three-dimensional space to be parallel, it is necessary and sufficient that the direction vectors of these lines be collinear.

Those. for given 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 equality holds:

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

Example 3

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

Solution

The conditions of the problem are the canonical equations of one straight line in space and the parametric equations of another straight line in space. Direction vectors a → and b → 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 → .

Therefore, the necessary and sufficient condition for parallel lines in space is satisfied.

Answer: the parallelism of the given lines is proved.

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