Instruction

Before starting the proof, make sure that the lines lie in the same plane and can be drawn on it. The simplest method of proof is the method of measuring with 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, 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 interior corners. Those that lie through the secant line are called cross-lying. Those that lie on one side are called one-sided. Using a protractor, measure the two inner diagonal corners. If they are equal, then the lines will be parallel. If in doubt, measure one-sided interior angles and add up the resulting values. Lines will be parallel if the sum of one-sided interior angles is 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 that, continue this perpendicular in such a way 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 lines are parallel to each other.

In the event that the lines are given in the Cartesian coordinate system, find their guides or normal vectors. If these vectors are, respectively, collinear with each other, then the lines are parallel. Bring the equation of lines to a general form and find the coordinates of the normal vector of each of the 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 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 of general form, the second is canonical. Bring the second equation to a general form. Use the proportion conversion rule for this, and you'll end up with 2x=y-4. After reduction to a general form, get 2x-y + 4 = 0. Since the general equation for any line is written Ax + Vy + 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.

AB And WITHD crossed by the third line MN, then the angles formed in this case receive the following names in pairs:

corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7;

internal cross-lying corners: 3 and 5, 4 and 6;

external cross-lying corners: 1 and 7, 2 and 8;

internal one-sided corners: 3 and 6, 4 and 5;

external one-sided corners: 1 and 8, 2 and 7.

So, ∠ 2 = ∠ 4 and ∠ 8 = ∠ 6, but by the proven ∠ 4 = ∠ 6.

Therefore, ∠ 2 = ∠ 8.

3. Respective angles 2 and 6 are the same, since ∠ 2 = ∠ 4, and ∠ 4 = ∠ 6. We also make sure that the other corresponding angles are equal.

4. Sum internal one-sided corners 3 and 6 will be 2d because the sum adjacent corners 3 and 4 is equal to 2d = 180 0 , and ∠ 4 can be replaced by the identical ∠ 6. Also make sure that sum of angles 4 and 5 is equal to 2d.

5. Sum external one-sided corners will be 2d because these angles are equal respectively internal one-sided corners like corners vertical.

From the justification proved above, we obtain inverse theorems.

When, at the intersection of two lines of an arbitrary third line, we obtain that:

1. Internal cross lying angles are the same;

or 2. External cross lying angles are the same;

or 3. The corresponding angles are the same;

or 4. The sum of internal one-sided angles is equal to 2d = 180 0 ;

or 5. The sum of the outer one-sided is 2d = 180 0 ,

then the first two lines are parallel.

They do not intersect, no matter how long they continue. The parallelism of lines in writing is indicated as follows: AB|| WITHE

The possibility of the existence of such lines is proved by a theorem.

Theorem.

Through any point taken outside a given line, one can draw a parallel to this line..

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

For the proof, we assume the opposite, i.e., that CE 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, CE cannot intersect 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 a straight line WITHD, drawn through the point WITH parallel to a straight line AB, then any other line WITHE through the same point WITH, cannot be parallel AB, i.e. she continues intersect 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(SW), then it intersects with the other ( AB), because otherwise through the same point WITH 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 are parallel between themselves.

Indeed, if we assume that A And B intersect at some point M, then two different straight lines, parallel to each other, would pass through this point. WITH, which is impossible.

Theorem.

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

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

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

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

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

Definition 1

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

Definition 2

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

It should be noted that in order to determine parallel lines in space, the clarification “lying in the same plane” is extremely important: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

To denote parallel lines, it is common to use the symbol ∥ . That is, if the given lines a and b are parallel, this condition should be briefly written as follows: a ‖ b . Verbally, the parallelism of lines is indicated as follows: lines a and b are parallel, or line a is parallel to line b, or line b is parallel to line a.

Let us formulate a statement that plays an important role in the topic under study.

Axiom

Through a point that does not belong to a given line, there is only one line parallel to the given line. This statement cannot be proved on the basis of the known axioms of planimetry.

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

Theorem 1

Through any point in space that does not belong to a given line, there will be only one line parallel to the given one.

This theorem is easy to prove on the basis of the above axiom (geometry program for grades 10-11).

The sign of parallelism is a sufficient condition under which parallel lines are guaranteed. In other words, the fulfillment of this condition is sufficient to confirm the fact of parallelism.

In particular, there are necessary and sufficient conditions for the parallelism of lines in the plane and in space. Let us explain: necessary means the condition, the fulfillment of which is necessary for parallel lines; if it is not satisfied, the lines are not parallel.

Summarizing, a necessary and sufficient condition for the parallelism of lines is such a condition, the observance of which is necessary and sufficient for the lines to be parallel to each other. On the one hand, this is a sign of parallelism, on the other hand, a property inherent in parallel lines.

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

Definition 3

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

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

Theorem 2

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

Let us graphically illustrate the necessary and sufficient condition for parallel lines on the plane:

The proof of these conditions is present in the geometry program for grades 7-9.

In general, these conditions are also applicable for three-dimensional space, provided that the two lines and the secant belong to the same plane.

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

Theorem 3

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

Theorem 4

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

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

We give an illustration of these theorems:

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

Theorem 5

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

Let us formulate a similar one for a three-dimensional space.

Theorem 6

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

Let's illustrate:

All the above theorems, signs and conditions make it possible to conveniently prove the parallelism of lines by the methods of geometry. That is, to prove the parallelism of lines, one can show that the corresponding angles are equal, or demonstrate the fact that two given lines are perpendicular to the third, and so on. But we note that it is often more convenient to use the coordinate method to prove the parallelism of lines in a plane or in three-dimensional space.

Parallelism of lines in a rectangular coordinate system

In a given rectangular coordinate system, a straight line is determined by the equation of a straight line on a plane of one of the possible types. Similarly, a straight line given in a rectangular coordinate system in three-dimensional space corresponds to some equations of a straight line in space.

Let us write the necessary and sufficient conditions for the parallelism of lines in a rectangular coordinate system, depending on the type of equation describing the given lines.

Let's start with the condition of parallel lines in the plane. It is based on the definitions of the direction vector of the line and the normal vector of the line in the plane.

Theorem 7

For two non-coincident lines to be parallel on a plane, it is necessary and sufficient that the direction vectors of the given lines be collinear, or the normal vectors of the given lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the other line.

It becomes obvious that the condition of parallel lines on the plane is based on the condition of collinear vectors or the condition of perpendicularity of two vectors. That is, if a → = (a x , a y) and b → = (b x , b y) are the direction vectors of lines a and b ;

and n b → = (n b x , n b y) are normal vectors of lines a and b , then we write the above necessary and sufficient condition as follows: a → = t b → ⇔ a x = t b x a y = t b y or n a → = t n b → ⇔ n a x = t n b x n a y = t n b y or a → , n b → = 0 ⇔ a x n b x + a y n b y = 0 , where t is some real number. The coordinates of the directing or direct vectors are determined by the given equations of the lines. Let's consider the main examples.

1. The line a in a rectangular coordinate system is determined by the general equation of the line: A 1 x + B 1 y + C 1 = 0 ; line b - A 2 x + B 2 y + C 2 = 0 . Then the normal vectors of the given lines will have coordinates (A 1 , B 1) and (A 2 , B 2) respectively. We write the condition of parallelism as follows:

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

1. The straight line a is described by the equation of a straight line with a slope of the form y = k 1 x + b 1 . Straight line b - y \u003d k 2 x + b 2. Then the normal vectors of the given lines will have coordinates (k 1 , - 1) and (k 2 , - 1), respectively, and we write the parallelism condition as follows:

k 1 = t k 2 - 1 = t (- 1) ⇔ k 1 = t k 2 t = 1 ⇔ k 1 = k 2

Thus, if parallel lines on a plane in a rectangular coordinate system are given by equations with slope coefficients, then the slope coefficients of the given lines will be equal. And the converse statement is true: if non-coinciding lines on a plane in a rectangular coordinate system are determined by the equations of a line with the same slope coefficients, then these given lines are parallel.

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

Then the direction vectors of the given lines will be: a x , a y and b x , b y respectively, and we write the parallelism condition as follows:

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

Let's look at examples.

Example 1

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

Solution

We write the equation of a straight line in segments in the form of a general equation:

x 1 2 + y 5 = 1 ⇔ 2 x + 1 5 y - 1 = 0

We see that n a → = (2 , - 3) is the normal vector of the line 2 x - 3 y + 1 = 0 , and n b → = 2 , 1 5 is the normal vector of the line x 1 2 + y 5 = 1 .

The resulting vectors are not collinear, because there is no such value of t for which the equality will be true:

2 = t 2 - 3 = t 1 5 ⇔ t = 1 - 3 = t 1 5 ⇔ t = 1 - 3 = 1 5

Thus, the necessary and sufficient condition of parallelism of lines on the plane is not satisfied, which means that the given lines are not parallel.

Answer: given lines are not parallel.

Example 2

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

Solution

Let's transform the canonical equation of the straight line x 1 \u003d y - 4 2 to the equation of a straight line with a slope:

x 1 = y - 4 2 ⇔ 1 (y - 4) = 2 x ⇔ y = 2 x + 4

We see that the equations of the lines y = 2 x + 1 and y = 2 x + 4 are not the same (if it were otherwise, the lines would be the same) and the slopes of the lines are equal, which means that the given lines are parallel.

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

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

The normal vector of the line y = 2 x + 1 is the vector n a → = (2 , - 1) , and the direction vector of the second given line is b → = (1 , 2) . The scalar product of these vectors is zero:

n a → , b → = 2 1 + (- 1) 2 = 0

Thus, the vectors are perpendicular: this demonstrates to us the fulfillment of the necessary and sufficient condition for the original lines to be parallel. Those. given lines are parallel.

Answer: these lines are parallel.

To prove the parallelism of lines in a rectangular coordinate system of three-dimensional space, the following necessary and sufficient condition is used.

Theorem 8

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

Those. for given equations of lines in three-dimensional space, the answer to the question: are they parallel or not, is found by determining the coordinates of the direction vectors of the given lines, as well as checking the condition of their collinearity. In other words, if a → = (a x, a y, a z) and b → = (b x, b y, b z) are the direction vectors of the lines a and b, respectively, then in order for them to be parallel, the existence of such a real number t is necessary, so that equality holds:

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

Example 3

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

Solution

The conditions of the problem are the canonical equations of one straight line in space and the parametric equations of another straight line in space. Direction vectors a → and b → given lines have coordinates: (1 , 0 , - 3) and (2 , 0 , - 6) .

1 = t 2 0 = t 0 - 3 = t - 6 ⇔ t = 1 2 , then a → = 1 2 b → .

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

Answer: the parallelism of the given lines is proved.

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First, let's look at the difference between the concepts of attribute, property, and axiom.

Definition 1

sign called a certain fact by which it is possible to determine the truth of a judgment about an object of interest.

Example 1

Lines are parallel if their secant forms equal cross-lying angles.

Definition 2

Property is formulated in the case when there is confidence in the validity of the judgment.

Example 2

With parallel lines, their secant forms equal cross-lying angles.

Definition 3

axiom call such a statement that does not require proof and is accepted as true without it.

Each science has axioms on which subsequent judgments and their proofs are built.

Axiom of parallel lines

Sometimes the axiom of parallel lines is taken as one of the properties of parallel lines, but at the same time other geometric proofs are built on its validity.

Theorem 1

Through a point that does not lie on a given line, only one line can be drawn on the plane, which will be parallel to the given one.

The axiom does not require proof.

Properties of parallel lines

Theorem 2

Property1. Property of transitivity of parallel lines:

When one of two parallel lines is parallel to the third, then the second line will also be parallel to it.

Properties require proof.

Proof:

Let there be two parallel lines $a$ and $b$. Line $c$ is parallel to line $a$. Let us check whether in this case the line $с$ is also parallel to the line $b$.

For the proof, we will use the opposite proposition:

Imagine that there is such a variant in which line $c$ is parallel to one of the lines, for example, line $a$, and the other - line $b$ - intersects at some point $K$.

We obtain a contradiction according to the axiom of parallel lines. It turns out a situation in which two lines intersect at one point, moreover, they are parallel to the same line $a$. Such a situation is impossible, hence the lines $b$ and $c$ cannot intersect.

Thus, it is proved that if one of the two parallel lines is parallel to the third line, then the second line is also parallel to the third line.

Theorem 3

Property 2.

If one of two parallel lines intersects with a third, then the second line will also intersect with it.

Proof:

Let there be two parallel lines $a$ and $b$. Also, let there be some line $c$ that intersects one of the parallel lines, for example, the line $a$. It is necessary to show that the line $c$ also intersects the second line, the line $b$.

Let us construct a proof by contradiction.

Imagine that the line $c$ does not intersect the line $b$. Then two lines $a$ and $c$ pass through the point $K$ and do not intersect the line $b$, i.e., they are parallel to it. But this situation contradicts the axiom of parallel lines. Hence, the assumption was wrong and the line $c$ will intersect the line $b$.

The theorem has been proven.

Corner Properties, which form two parallel lines and a secant: crosswise angles are equal, the corresponding angles are equal, * the sum of one-sided angles is equal to $180^(\circ)$.

Example 3

Given two parallel lines and a third line perpendicular to one of them. Prove that this line is perpendicular to another of the parallel lines.

Proof.

Let we have lines $a \parallel b$ and $c \perp a$.

Since the line $c$ intersects the line $a$, then, according to the property of parallel lines, it will also intersect the line $b$.

The secant $c$, intersecting the parallel lines $a$ and $b$, forms equal interior cross-lying angles with them.

Because $c \perp a$, then the angles will be $90^(\circ)$.

Hence $c \perp b$.

The proof is complete.