On a plane, straight lines are called parallel if they have no common points, that is, they do not intersect. To indicate parallelism, use the special icon || (parallel lines a || b).

For straight lines lying in space, the requirement of the absence of common points is not enough - for them to be parallel in space, they must belong to the same plane (otherwise they will be crossing).

There is no need to go far for examples of parallel straight lines, they accompany us everywhere, in the 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 parallelism of two lines and a third line parallel to one of the first two, it will be parallel to the second.

Parallel lines on the plane are connected by a statement that cannot be proved using the axioms of planimetry. It is taken as a fact, as an axiom: for any point on the plane that does not lie on a straight line, there is a single straight 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 straight line, there is a single straight line that passes through it parallel to a given one, is easily proved using the already known axiom of parallelism on the plane.

Parallel Line Properties

• If any of the parallel two straight lines is parallel to the third, then they are mutually parallel.

This property is possessed by parallel lines both on the plane and in space.
As an example, consider its justification in stereometry.

Let us assume parallelism of straight lines b and with straight line a.

The case when all straight lines lie in the same plane will be left 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 betta and gamma planes are different and mark a certain point B on the line b from the betta plane, then the plane drawn through the point B and the line c must intersect the betta plane in a straight line (we denote it by 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, the line b1 must belong to the betta plane and, at the same time, have no points in common with a, therefore, according to the axiom of parallelism, it coincides with b.
We got the line b1 coinciding with the straight line b, which belongs to the same plane with the straight line c and does not intersect it, that is, b and c are parallel

• Through a point that does not lie on a given straight line, only one single straight line can pass parallel to the given one.

• Lying on a plane perpendicular to the third, two straight lines are parallel.

• Provided that the plane intersects one of the parallel two straight lines, the second straight line intersects the same plane.

• The corresponding and criss-crossing internal angles formed by the intersection of two parallel lines of the third are equal, the sum of the resulting internal one-sided angles is 180 °.

The converse statements are also true, which can be taken as signs of parallelism of two straight lines.

Parallelism condition for straight lines

The properties and features formulated above are the conditions for the parallelism of straight lines, and they can be fully proved by the methods of geometry. In other words, to prove the parallelism of two existing straight lines, it is sufficient to prove their parallelism to the third straight line or the equality of the angles, whether corresponding or crosswise, etc.

For the proof, the method is mainly used "by contradiction", that is, with the assumption that the straight lines are not parallel. Proceeding from this assumption, it is easy to show that in this case the specified conditions are violated, for example, cross-lying internal angles turn out to be unequal, which proves the incorrectness of the assumption made.

They do not intersect, no matter how long they continue. 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 this line, you can draw parallel to this line.

Let be AB this straight line and WITH some point taken outside of it. It is required to prove that through WITH you can draw a straight line parallelAB... Let's drop on AB from point WITH perpendicularWITHD and then we run WITHE^ WITHD, what is possible. Straight CE parallel AB.

For the proof, assume the opposite, i.e., that CE intersects AB at some point M... Then from the point M to straight WITHD we would have two different perpendiculars MD and MC, which is impossible. Means, CE cannot intersect with AB, i.e. WITHE parallel AB.

Consequence.

Two perpendiculars (CEandDB) to one straight line (СD) are parallel.

Axiom of parallel lines.

Through the same point, you cannot draw two different straight lines parallel to the same straight line.

So if the straight line WITHD drawn through point WITH parallel to the straight line AB, then any other straight line WITHE drawn through the same point WITH, cannot be parallel AB, i.e. she continued will cross with AB.

The proof of this not quite 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(SV), then it intersects with the other ( AB), because otherwise through the same point WITH would pass two different straight lines 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, assuming that A and B intersect at some point M, then two different straight lines would pass through this point, parallel WITH, which is impossible.

Theorem.

If straight line perpendicular to one of the parallel lines, then it is perpendicular to the other parallel.

Let be AB || WITHD and EF ^ AB. It is required to prove that EF ^ WITHD.

PerpendicularEF intersecting with AB, will certainly cross and WITHD... Let the intersection point be H.

Suppose now that WITHD not perpendicular to EH... Then some other straight line, for example HK, will be perpendicular to EH and, therefore, through the same point H there will be two straight parallel AB: one WITHD, by condition, and the other HK as proved earlier. Since this is impossible, it cannot be assumed that SV was not perpendicular to EH.

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 proof is a method of measuring a ruler. 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 the same, these 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 across the intersecting line are called intersecting. Those that lie on one side are called one-sided. Using a protractor, measure the two intersecting inner corners. If they are equal, then the lines will be parallel. If in doubt, measure the one-sided interior angles and add the resulting values. The straight lines will be parallel if the sum of the one-sided inner angles is equal to 180º.

If you don't have a protractor, use a 90º square. Use it to draw a perpendicular to one of the lines. After that, 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 equal to 90º, then the straight lines are parallel to each other.

In the event that the straight lines are given in the Cartesian coordinate system, find their direction or normal vectors. If these vectors, respectively, are collinear with each other, then the straight lines are parallel. Bring the equation of the straight lines to a general form and find the coordinates of the normal vector of each of the straight lines. Its coordinates are equal to the coefficients A and B. In the event that the ratio of the corresponding coordinates of the normal vectors is the same, they are collinear, and the straight 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, the second is canonical. Generalize the second equation. Use the rule of conversion of proportions for this, as a result you get 2x = y-4. After reducing to a general view, get 2x-y + 4 = 0. Since the general equation for any straight line is written Ax + Vy + C = 0, then for the first straight line: A = 4, B = 2, and for the second straight line A = 2, B = 1. For the first line, the coordinates of the normal vector are (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 straight lines are parallel.

Parallel lines concept

Definition 1

Parallel lines- lines that lie in the same plane do not coincide and do not have common points.

If the lines have a common point, then they intersect.

If all points are straight match up, then we essentially have one straight line.

If the straight lines lie in different planes, then the conditions for their parallelism are somewhat greater.

When considering straight lines on one plane, the following definition can be given:

Definition 2

Two straight lines in the plane are called parallel if they do not overlap.

In mathematics, parallel lines are usually denoted using the parallel sign "$ \ parallel $". For example, the fact that line $ c $ is parallel to line $ d $ is denoted as follows:

$ c \ parallel d $.

The concept of parallel lines is often considered.

Definition 3

The two segments are called parallel if they lie on parallel lines.

For example, in the figure, the segments $ AB $ and $ CD $ are parallel, since they belong to parallel lines:

$ AB \ parallel CD $.

At the same time, the segments $ MN $ and $ AB $ or $ MN $ and $ CD $ are not parallel. This fact can be written using symbols as follows:

$ MN ∦ AB $ and $ MN ∦ CD $.

Similarly, the parallelism of a straight line and a segment, a straight line and a ray, a segment and a ray, or two rays is determined.

Historical reference

From the Greek language, the concept of "parallelos" is translated "walking side by side" or "held next to each other." This term was used in the ancient school of Pythagoras even before parallel lines were defined. According to historical facts by Euclid in the $ III $ c. BC. in his works, the meaning of the concept of parallel lines was nevertheless revealed.

In ancient times, the sign for denoting parallel lines had a different kind of what we use in modern mathematics. For example, the ancient Greek mathematician Pappus in $ III $ c. AD parallelism was denoted with an equal sign. Those. the fact that the line $ l $ is parallel to the line $ m $ was previously denoted "$ l = m $". Later, the familiar “$ \ parallel $” sign began to be used to denote the parallelism of straight lines, and the equal sign began to be used to denote the equality of numbers and expressions.

Parallel lines in life

Often we do not notice that in everyday life we ​​are surrounded by a huge number of parallel lines. For example, in a music book and a songbook with scores, the staff is made using parallel lines. Same parallel lines are also found in musical instruments (for example, harp strings, guitar strings, piano keys, etc.).

Electric wires that run along streets and roads also run parallel. The rails of the metro lines and railways are arranged in parallel.

In addition to everyday life, parallel lines can be found in painting, in architecture, in the construction of buildings.

Parallel lines in architecture

In the presented images, architectural structures contain parallel straight lines. The use of parallelism of straight lines in construction helps to increase the service life of such structures and gives them extraordinary beauty, attractiveness and grandeur. Power lines are also deliberately run in parallel to avoid crossing or touching them, which would lead to short circuits, interruptions and no electricity. So that the train can move freely, the rails are also made in parallel lines.

In painting, parallel lines are depicted as converging into one line or close to that. This technique is called perspective, which follows from the illusion of sight. If you look into the distance for a long time, then the parallel lines will look like two converging lines.

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

Page navigation.

Parallel lines - basic information.

Definition.

Two straight lines on a plane are called parallel if they have no common points.

Definition.

Two straight lines in three-dimensional space are called parallel if they lie in the same plane and have no common points.

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 straight 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 straight lines. The straight lines along which the plane of the house wall intersects the planes of the ceiling and floor are parallel. Railroad tracks on level ground can also be viewed as parallel straight lines.

To denote parallel lines use the symbol "". That is, if straight lines a and b are parallel, then we can briefly write a b.

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 straight lines on a plane: through a point that does not lie on a given straight line, there is a single straight line parallel to a given one. This statement is taken as a fact (it cannot be proved on the basis of the well-known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the following theorem is true: through any point in space that does not lie on a given straight line, there is a single straight 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 for grades 10-11, which is indicated at the end of the article in the bibliography).

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

Parallelism of lines - signs and conditions of parallelism.

Parallelism of straight lines is an sufficient condition parallelism of straight lines, that is, such a condition, the fulfillment of which guarantees the parallelism of straight lines. In other words, the fulfillment of this condition is sufficient to state the fact of parallelism of straight lines.

There are also necessary and sufficient conditions for the parallelism of straight lines on the plane and in three-dimensional space.

Let us clarify the meaning of the phrase "a necessary and sufficient condition for the parallelism of straight lines."

We have already figured out the sufficient condition for the parallelism of straight lines. But what is the "necessary condition for the parallelism of straight lines"? By the name "necessary" it is clear that the fulfillment of this condition is necessary for the parallelism of straight lines. In other words, if the necessary condition for the parallelism of lines is not satisfied, then the lines are not parallel. Thus, necessary and sufficient condition for parallelism of straight lines Is a condition, the fulfillment of which is both necessary and sufficient for the parallelism of straight lines. That is, on the one hand, this is a sign of parallelism of straight lines, and on the other hand, it is a property that parallel straight lines have.

Before formulating a necessary and sufficient condition for the parallelism of straight lines, it is advisable to recall several auxiliary definitions.

Secant line Is a line that intersects each of the two specified non-coincident lines.

When two straight secants intersect, eight undeveloped ones are formed. The so-called criss-crossing, corresponding and one-sided corners... Let's show them in the drawing.

Theorem.

If two straight lines on a plane are intersected by a secant, then for their parallelism it is necessary and sufficient that the cross-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 graphic illustration of this necessary and sufficient condition for the parallelism of straight lines on a plane.

Proofs of these conditions of parallelism of straight lines can be found in geometry textbooks for grades 7-9.

Note that these conditions can be used in three-dimensional space - the main thing is that the two lines and the secant lie in the same plane.

Here are some more theorems that are often used to prove the parallelism of straight lines.

Theorem.

If two straight lines in the plane are parallel to the third straight line, then they are parallel. The proof of this criterion follows from the parallel line axiom.

There is a similar condition for the parallelism of straight lines in three-dimensional space.

Theorem.

If two lines in space are parallel to the third line, then they are parallel. The proof of this sign is considered in geometry lessons in grade 10.

Let us illustrate the stated theorems.

Let us present one more theorem that allows us to prove the parallelism of straight lines in the plane.

Theorem.

If two straight lines in the plane are perpendicular to the third straight line, then they are parallel.

There is a similar theorem for lines in space.

Theorem.

If two straight 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 perfect for proving the parallelism of straight lines by the methods of geometry. That is, in order 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 intersecting 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 method of coordinates to prove the parallelism of straight lines on a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of straight lines, which are given in a rectangular coordinate system.

Parallelism of straight lines in a rectangular coordinate system.

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

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

Theorem.

For the parallelism of two non-coinciding straight lines on the plane, it is necessary and sufficient that the direction vectors of these straight lines are collinear, or the normal vectors of these straight lines are collinear, or the direction vector of one straight line is perpendicular to the normal vector of the second straight line.

Obviously, the condition of parallelism of two straight lines on a plane is reduced to (direction vectors of straight lines or normal vectors of straight lines) or to (direction vector of one straight line and normal vector of the second straight line). Thus, if and are direction vectors of straight lines a and b, and and are normal vectors of lines a and b, respectively, then the necessary and sufficient condition for parallelism of lines a and b can be written as , or , or, where t is some real number. In turn, the coordinates of the guides and (or) normal vectors of the straight lines a and b are found from the well-known equations of the straight lines.

In particular, if the straight line a in the rectangular coordinate system Oxy on the plane is defined by the general equation of the straight line of the form , and 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 the straight line a corresponds to the equation of a straight line with a slope of the form, and to a straight line b -, then the normal vectors of these straight lines have coordinates and, and the condition for the parallelism of these straight lines takes the form ... Therefore, if the straight lines on the plane in a rectangular coordinate system are parallel and can be specified by equations of straight lines with slope coefficients, then the slope coefficients of the straight lines will be equal. And vice versa: if mismatched straight lines on a plane in a rectangular coordinate system can be specified by equations of a straight line with equal slope coefficients, then such straight lines are parallel.

If straight line a and straight line b in a rectangular coordinate system are determined by the canonical equations of a straight line in the plane of the form and , or parametric equations of a straight line on the 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 the solutions of several examples.

Example.

Are the lines parallel and ?

Solution.

We rewrite the equation of a straight line in segments in the form general equation straight: ... Now you can see that is the normal vector of the line , a is the normal vector of a straight 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 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 straight and parallel?

Solution.

Let us bring the canonical equation of the straight line to the equation of the straight line with the 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.