On a plane, 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 method “by contradiction”, 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 specified conditions are violated, for example, the internal angles lying across each other turn out to be unequal, which proves the incorrectness of the assumption made.
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 restrict 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 on the external angle of a 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 a given 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, that is, if a given theorem is true, then the inverse theorem may be false.
Let us explain this using the example of the theorem on vertical angles. 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 do not have to be vertical.
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.
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.
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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-dimensional space are called parallel, if they lie in the same plane and do not have common points.
Please note that the clause “if they lie in the same plane” in the definition of parallel lines in space is very important. Let us clarify this point: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.
Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel lines. The straight lines along which the plane of the wall of the house intersects the planes of the ceiling and floor are parallel. Railroad rails on level ground can also be considered as parallel lines.
To denote parallel lines, use the symbol “”. That is, if lines a and b are parallel, then we can briefly write a b.
Please note: if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.
Let us voice a statement that plays an important role in the study of parallel lines on a plane: through a point not lying on a given line, there passes the only straight line parallel to the given one. This statement is accepted as a fact (it cannot be proven on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.
For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem is easily proven using the above axiom of parallel lines (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the list of references).
For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem can be easily proven using the above parallel line axiom.
Parallelism of lines - signs and conditions of parallelism.
A sign of parallelism of lines is a sufficient condition for the lines to be parallel, that is, a condition the fulfillment of which guarantees the lines to be parallel. In other words, the fulfillment of this condition is sufficient to establish the fact that the lines are parallel.
There are also necessary and sufficient conditions for the parallelism of lines on a plane and in three-dimensional space.
Let us explain the meaning of the phrase “necessary and sufficient condition for parallel lines.”
We have already dealt with the sufficient condition for parallel lines. What is a “necessary condition for parallel lines”? From the name “necessary” it is clear that the fulfillment of this condition is necessary for parallel lines. In other words, if the necessary condition for the lines to be parallel is not met, then the lines are not parallel. Thus, necessary and sufficient condition for parallel lines is a condition the fulfillment of which is both necessary and sufficient for parallel lines. That is, on the one hand, this is a sign of parallelism of lines, and on the other hand, this is a property that parallel lines have.
Before formulating a necessary and sufficient condition for the parallelism of lines, it is advisable to recall several auxiliary definitions.
Secant line is a line that intersects each of two given non-coinciding lines.
When two straight lines intersect with a transversal, eight undeveloped ones are formed. In the formulation of the necessary and sufficient condition for the parallelism of lines, the so-called lying crosswise, corresponding And one-sided angles. Let's show them in the drawing.
Theorem.
If two straight lines in a plane are intersected by a transversal, then for them to be parallel it is necessary and sufficient that the intersecting angles be equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.
Let us show a graphic illustration of this necessary and sufficient condition for the parallelism of lines on a plane.
You can find proofs of these conditions for the parallelism of lines in geometry textbooks for grades 7-9.
Note that these conditions can also be used in three-dimensional space - the main thing is that the two lines and the secant lie in the same plane.
Here are a few more theorems that are often used to prove the parallelism of lines.
Theorem.
If two lines in a plane are parallel to a third line, then they are parallel. The proof of this criterion follows from the axiom of parallel lines.
There is a similar condition for parallel lines in three-dimensional space.
Theorem.
If two lines in space are parallel to a third line, then they are parallel. The proof of this criterion is discussed in geometry lessons in the 10th grade.
Let us illustrate the stated theorems.
Let us present another theorem that allows us to prove the parallelism of lines on a plane.
Theorem.
If two lines in a plane are perpendicular to a third line, then they are parallel.
There is a similar theorem for lines in space.
Theorem.
If two lines in three-dimensional space are perpendicular to the same plane, then they are parallel.
Let us draw pictures corresponding to these theorems.
All the theorems, criteria and necessary and sufficient conditions formulated above are excellent for proving the parallelism of lines using 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. Many similar problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the coordinate method to prove the parallelism of lines on a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are specified in a rectangular coordinate system.
Parallelism of lines in a rectangular coordinate system.
In this paragraph of the article we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations defining these lines, and we will also provide detailed solutions to characteristic problems.
Let's start with the condition of parallelism of two straight lines on a plane in the rectangular coordinate system Oxy. His proof is based on the definition of the direction vector of a line and the definition of the normal vector of a line on a plane.
Theorem.
For two non-coinciding lines to be parallel in a plane, it is necessary and sufficient that the direction vectors of these lines are collinear, or the normal vectors of these lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the second line.
Obviously, the condition of parallelism of two lines on a plane is reduced to (direction vectors of lines or normal vectors of lines) or to (direction vector of one line and normal vector of the second line). Thus, if and are direction vectors of lines a and b, and 
And 
are normal vectors of lines a and b, respectively, then the necessary and sufficient condition for the parallelism of lines a and b will be written as 
, or 
, or , where t is some real number. In turn, the coordinates of the guides and (or) normal vectors of lines a and b are found using the known equations of lines.
In particular, if straight line a in the rectangular coordinate system Oxy on the plane defines a general straight line equation of the form 
, and straight line b - 
, then the normal vectors of these lines have coordinates and, respectively, and the condition for parallelism of lines a and b will be written as .
If line a corresponds to the equation of a line with an angular coefficient of the form , and line b - , then the normal vectors of these lines have coordinates and , and the condition for parallelism of these lines takes the form 
. Consequently, if lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of lines with angular coefficients, then the angular coefficients of the lines will be equal. And 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.
1. If two lines are parallel to a third line, then they are parallel:
If a||c And b||c, That a||b.
2. If two lines are perpendicular to the third line, then they are parallel:
If a⊥c And b⊥c, That a||b.
The remaining signs of parallelism of lines are based on the angles formed when two straight lines intersect with a third.
3. If the sum of internal one-sided angles is 180°, then the lines are parallel:
If ∠1 + ∠2 = 180°, then a||b.
4. If the corresponding angles are equal, then the lines are parallel:
If ∠2 = ∠4, then a||b.
5. If internal crosswise angles are equal, then the lines are parallel:
If ∠1 = ∠3, then a||b.
Properties of parallel lines
Statements inverse to the properties of parallel lines are their properties. They are based on the properties of angles formed by the intersection of two parallel lines with a third line.
1. When two parallel lines intersect a third straight line, the sum of the internal one-sided angles formed by them is equal to 180°:
If a||b, then ∠1 + ∠2 = 180°.
2. When two parallel lines intersect a third line, the corresponding angles formed by them are equal:
If a||b, then ∠2 = ∠4.
3. When two parallel lines intersect a third line, the crosswise angles they form are equal:
If a||b, then ∠1 = ∠3.
The following property is a special case for each previous one:
4. If a line on a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other:
If a||b And c⊥a, That c⊥b.
The fifth property is the axiom of parallel lines:
5. Through a point not lying on a given line, only one line can be drawn parallel to the given line.
They do not intersect, no matter how long they are continued. The parallelism of straight lines in writing is denoted as follows: AB|| WITHE
The possibility of the existence of such lines is proved by the theorem.
Theorem.
Through any point taken outside a given line, one can draw a point parallel to this line.
Let AB this straight line and WITH some point taken outside it. It is required to prove that through WITH you can draw a straight line parallelAB. Let's lower it to AB from point WITH perpendicularWITHD and then we will conduct WITHE^ WITHD, what is possible. Straight C.E. parallel AB.
To prove this, let us assume the opposite, i.e., that C.E. intersects AB at some point M. Then from the point M to a straight line WITHD we would have two different perpendiculars MD And MS, which is impossible. Means, C.E. can't cross with AB, i.e. WITHE parallel AB.
Consequence.
Two perpendiculars (CEAndD.B.) to one straight line (CD) are parallel.
Axiom of parallel lines.
Through the same point it is impossible to draw two different lines parallel to the same line.
So, if straight WITHD, drawn through the point WITH parallel to the line AB, then every other line WITHE, drawn through the same point WITH, cannot be parallel AB, i.e. she's on continuation will intersect With AB.
Proving this not entirely obvious truth turns out to be impossible. It is accepted without proof, as a necessary assumption (postulatum).
Consequences.
1. If straight(WITHE) intersects with one of parallel(NE), then it intersects with another ( AB), because otherwise through the same point WITH there would be two different lines passing parallel AB, which is impossible.
2. If each of the two direct (AAndB) are parallel to the same third line ( WITH) , then they parallel between themselves.
Indeed, if we assume that A And B intersect at some point M, then two different lines parallel to this point would pass through WITH, which is impossible.
Theorem.
If line is perpendicular to one of the parallel lines, then it is perpendicular to the other parallel.
Let AB || WITHD And E.F. ^ AB.It is required to prove that E.F. ^ WITHD.
PerpendicularEF, intersecting with AB, will certainly cross and WITHD. Let the intersection point be H.
Let us now assume that WITHD not perpendicular to E.H.. Then some other straight line, for example H.K., will be perpendicular to E.H. and therefore through the same point H there will be two straight parallel AB: one WITHD, by condition, and the other H.K. as previously proven. Since this is impossible, it cannot be assumed that NE was not perpendicular to E.H..
