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Signs of parallelism of two lines
Theorem 1. If, when two lines intersect with a secant:
crossed angles are equal, or
corresponding angles are equal, or
the sum of one-sided angles is 180°, then
lines are parallel(Fig. 1).
Proof. We limit ourselves to proving case 1.
Let the intersecting lines a and b be crosswise and the angles AB be equal. For example, ∠ 4 = ∠ 6. Let us prove that a || b.
Suppose that lines a and b are not parallel. Then they intersect at some point M and, therefore, one of the angles 4 or 6 will be the external angle of triangle ABM. For definiteness, let ∠ 4 be the external angle of the triangle ABM, and ∠ 6 the internal one. From the theorem about external angle triangle it follows that ∠ 4 is greater than ∠ 6, and this contradicts the condition, which means that lines a and 6 cannot intersect, so they are parallel.
Corollary 1. Two different lines in a plane perpendicular to the same line are parallel(Fig. 2).
Comment. The way we just proved case 1 of Theorem 1 is called the method of proof by contradiction or reduction to absurdity. This method received its first name because at the beginning of the argument an assumption is made that is contrary (opposite) to what needs to be proven. It is called leading to absurdity due to the fact that, reasoning on the basis of the assumption made, we come to an absurd conclusion (to the absurd). Receiving such a conclusion forces us to reject the assumption made at the beginning and accept the one that needed to be proven.
Task 1. Construct a line passing through this point M and parallel to a given line a, not passing through the point M.
Solution. We draw a straight line p through the point M perpendicular to the straight line a (Fig. 3).
Then we draw a line b through point M perpendicular to the line p. Line b is parallel to line a according to the corollary of Theorem 1.
An important conclusion follows from the problem considered:
through a point not lying on a given line, it is always possible to draw a line parallel to the given one.
The main property of parallel lines is as follows.
Axiom of parallel lines. Through a given point that does not lie on a given line, there passes only one line parallel to the given one.
Let us consider some properties of parallel lines that follow from this axiom.
1) If a line intersects one of two parallel lines, then it also intersects the other (Fig. 4).
2) If two different lines are parallel to a third line, then they are parallel (Fig. 5).
The following theorem is also true.
Theorem 2. If two parallel lines are intersected by a transversal, then:
crosswise angles are equal;
corresponding angles are equal;
the sum of one-sided angles is 180°.
Corollary 2. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other(see Fig. 2).
Comment. Theorem 2 is called the inverse of Theorem 1. The conclusion of Theorem 1 is the condition of Theorem 2. And the condition of Theorem 1 is the conclusion of Theorem 2. Not every theorem has an inverse, i.e. if this theorem is true, then the converse theorem may not be true.
Let us explain this using the example of the theorem about vertical corners. This theorem can be formulated as follows: if two angles are vertical, then they are equal. The converse theorem would be: if two angles are equal, then they are vertical. And this, of course, is not true. Two equal angles don't have to be vertical at all.
Example 1. Two parallel lines are crossed by a third. It is known that the difference between two internal one-sided angles is 30°. Find these angles.
Solution. Let Figure 6 meet the condition.
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.
- 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 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 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 canonical equation straight line x 1 = 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 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 by the canonical equations of 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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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. 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 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 straight 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.
Exists similar condition parallelism of 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 the methods of geometry. 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. A bunch of similar tasks 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 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 conversely: if non-coinciding lines on a plane in a rectangular coordinate system can be specified by the 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.
To question 1. Give the definition of parallel lines. Which two segments are called parallel? given by the author Sasha Nizhevyasov the best answer is which will never intersect on a plane
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 a plane are called parallel. if they do not intersect or coincide.
Answer from Neuropathologist[newbie]
Two straight lines lying in the same plane and having no common point, are called parallel
Answer from Add[master]
Answer from Varvara Lamekina[newbie]
two lines in a plane are called parallel if they do not intersect)
Answer from Maxim Ivanov[newbie]
Which will not intersect on a 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 are 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 one. In Euclidean geometry there is only one such line. This fact is equivalent to Euclid's V postulate (on parallels). In Lobachevsky geometry (see Lobachevsky geometry) in the plane through point C (see figure) outside a given straight line AB passes infinite set straight lines not intersecting 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 the 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 Oliya[active]
Parallel lines are lines that do not intersect
Answer from Said Charakov[newbie]
Parallel lines are two lines that lie in the same plane and do not have common points.
Through a point you can only draw one straight line parallel to a given plane.
Answer from Oliya 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 point C (see figure) outside a given line AB there passes an infinite number of straight 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 segments are called parallel if they lie on parallel lines.