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In a plane, straight lines are called parallel if they do not have common points, that is, they do not intersect. To indicate parallelism, use a special icon || (parallel lines a || b).
For lines lying in space, the requirement that there are no common points is not enough - for them to be parallel in space, they must belong to the same plane (otherwise they will intersect).
You don’t have to go far for examples of parallel lines; they accompany us everywhere, in a room - these are the lines of intersection of the wall with the ceiling and floor, on a notebook sheet - opposite edges, etc.
It is quite obvious that, having two lines parallel and a third line parallel to one of the first two, it will also be parallel to the second.
Parallel lines on a plane are related by a statement that cannot be proven using the axioms of planimetry. It is accepted as a fact, as an axiom: for any point on the plane that does not lie on a line, there is a unique line that passes through it parallel to the given one. Every sixth grader knows this axiom.
Its spatial generalization, that is, the statement that for any point in space that does not lie on a line, there is a unique line that passes through it parallel to the given one, is easily proven using the already known axiom of parallelism on the plane.
Properties of parallel lines
- If any of two parallel lines is parallel to the third, then they are mutually parallel.
Parallel lines both on the plane and in space have this property.
As an example, consider its justification in stereometry.
Let us assume that lines b and line a are parallel.
We will leave the case when all straight lines lie in the same plane to planimetry.
Suppose a and b belong to the beta plane, and gamma is the plane to which a and c belong (by the definition of parallelism in space, straight lines must belong to the same plane).
If we assume that the beta and gamma planes are different and mark a certain point B on line b from the beta plane, then the plane drawn through point B and line c must intersect the beta plane in a straight line (let’s denote it b1).
If the resulting straight line b1 intersected the gamma plane, then, on the one hand, the intersection point would have to lie on a, since b1 belongs to the beta plane, and on the other hand, it should also belong to c, since b1 belongs to the third plane.
But parallel lines a and c should not intersect.
Thus, line b1 must belong to the betta plane and at the same time not have common points with a, therefore, according to the parallelism axiom, it coincides with b.
We have obtained a line b1 coinciding with line b, which belongs to the same plane with line c and does not intersect it, that is, b and c are parallel
- Through a point that does not lie on a given line, only one single straight line can pass parallel to the given line.
- Two lines lying on a plane perpendicular to the third are parallel.
- If the plane intersects one of two parallel lines, the second line also intersects the same plane.
- Corresponding and cross-lying internal angles formed by the intersection of two parallel straight lines with a third are equal, the sum of the internal one-sided angles formed is 180°.
The converse statements are also true, which can be taken as signs of the parallelism of two straight lines.
Condition for parallel lines
The properties and characteristics formulated above represent the conditions for the parallelism of lines, and they can be proven using the methods of geometry. In other words, to prove the parallelism of two existing lines, it is enough to prove their parallelism to a third line or the equality of the angles, whether corresponding or crosswise, etc.
For proof, they mainly use the “by contradiction” method, that is, with the assumption that the lines are not parallel. Based on this assumption, it can be easily shown that in this case the given conditions, for example, intersecting internal angles turn out to be unequal, which proves the incorrectness of the assumption made.
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. The conclusion shows solutions characteristic tasks to prove the parallelism of lines that are given by some line equations in rectangular system coordinates on the 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's voice the statement that plays important role when studying parallel lines on a plane: through a point not lying on a given line, there passes a single 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 sufficient condition parallelism of lines, that is, a condition the fulfillment of which guarantees parallelism of lines. 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 corresponding angles were equal, or the sum of one-sided angles was 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 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 straight lines a and b are found by known equations straight
In particular, if straight line a in the rectangular coordinate system Oxy on the plane defines a general straight line equation of the form 
, and straight line b - 
, then the normal vectors of these lines have coordinates and, respectively, and the condition for parallelism of lines a and b will be written as .
If line a corresponds to the equation of a line with an angular coefficient of the form , and line b - , then the normal vectors of these lines have coordinates and , and the condition for parallelism of these lines takes the form 
. Consequently, if straight lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of straight lines with angular coefficients, then slopes straight 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 straight line in segments in the form general equation direct: 
. Now we can see that is the normal vector of the line , a is the normal vector of the line. These vectors are not collinear, since there is no such real number t for which the equality ( 
). Consequently, the necessary and sufficient condition for the parallelism of lines on a plane is not satisfied, therefore, the given lines are not parallel.
Answer:
No, the lines are not parallel.
Example.
Are straight lines and parallel?
Solution.
Let's give canonical equation straight line to the equation of a straight line with an angular coefficient: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the angular coefficients of the lines are equal, therefore, the original lines are parallel.
In the section on the question how to prove that lines are parallel???? given by the author Alyonka Yakovleva the best answer is Properties of parallel lines
Theorem
Two lines parallel to a third are parallel.
Proof.
Let lines a and b be parallel to line c. Let's assume that lines a and b are not parallel. Then they intersect at some point C. It turns out that through point C there are two straight lines parallel to straight line c. But this contradicts the axiom “Through a point not lying on a given line, you can draw on the plane at most one straight line parallel to the given one.” The theorem has been proven.
Theorem
If two parallel lines are intersected by a third line, then the intersecting interior angles are equal.
Proof.
Let there be parallel lines a and b that are intersected by a secant line c. Line c intersects line a at point A and line b at point B. Let us draw line a1 through point A so that lines a1 and b with transversal c form equal internal angles lying crosswise. According to the criterion of parallelism of lines, lines a1 and b are parallel. And since only one line parallel to b can be drawn through point A, then a and a1 coincide.
This means that the internal crosswise angles formed by the lines a and b are equal. The theorem has been proven.
Based on the theorem it is proved:
If two parallel lines are intersected by a third line, then the corresponding angles are equal.
If two parallel lines are intersected by a third line, then the sum of the interior one-sided angles is 180º
Instructions
Before starting the proof, make sure that the lines lie in the same plane and can be drawn on it. Most in a simple way The proof is the ruler measurement method. To do this, use a ruler to measure the distance between the straight lines in several places as far apart as possible. If the distance remains unchanged, the given lines are parallel. But this method is not accurate enough, so it is better to use other methods.
Draw a third line so that it intersects both parallel lines. It forms four outer and four inner corners with them. Consider the interior corners. Those that lie through the secant line are called cross-lying. Those that lie on one side are called unilateral. Using a protractor, measure the two internal intersecting angles. If they are equal to each other, then the lines will be parallel. If in doubt, measure one-sided internal angles and add the resulting values. The lines will be parallel if the sum of the one-sided lines internal corners will be equal to 180º.
If you don't have a protractor, use a 90º square. Use it to construct a perpendicular to one of the lines. After this, continue this perpendicular so that it intersects another line. Using the same square, check at what angle this perpendicular intersects it. If this angle is also 90º, then the lines are parallel to each other.
In the event that the lines are given in Cartesian system coordinates, find their direction or normal vectors. If these vectors, respectively, are collinear with each other, then the lines are parallel. Reduce the equation of lines to a general form and find the coordinates of the normal vector of each line. Its coordinates are equal to the coefficients A and B. If the ratio of the corresponding coordinates of the normal vectors is the same, they are collinear and the lines are parallel.
For example, straight lines are given by the equations 4x-2y+1=0 and x/1=(y-4)/2. The first equation is general view, the second – canonical. Bring the second equation to its general form. Use the proportion conversion rule for this, the result will be 2x=y-4. After reduction to the general form, you get 2x-y+4=0. Since the general equation for any line is written Ax+By+C=0, then for the first line: A=4, B=2, and for the second line A=2, B=1. For the first direct coordinate of the normal vector (4;2), and for the second – (2;1). Find the ratio of the corresponding coordinates of the normal vectors 4/2=2 and 2/1=2. These numbers are equal, which means the vectors are collinear. Since the vectors are collinear, the lines are parallel.
