Lesson objectives: In this lesson, you will become familiar with the concept of “parallel lines”, learn how you can verify the parallelism of lines, as well as what properties the angles formed by parallel lines and a transversal have.

Parallel lines

You know that the concept of “straight line” is one of the so-called indefinable concepts of geometry.

You already know that two lines can coincide, that is, have all common points, or intersect, that is, have one common point. Straight lines intersect at different angles, and the angle between the straight lines is considered to be the smallest of the angles formed by them. A special case of intersection can be considered the case of perpendicularity, when the angle formed by straight lines is equal to 90 0.

But two straight lines may not have common points, that is, do not intersect. Such lines are called parallel.

Work with electronic educational resource « ».

To get acquainted with the concept of “parallel lines”, work with the video lesson materials

Thus, now you know the definition of parallel lines.

From the materials of the video lesson fragment you learned about various types angles that are formed when two straight lines intersect with a third.

Pairs of corners 1 and 4; 3 and 2 are called internal one-sided corners(they lie between straight lines a And b).

Pairs of angles 5 and 8; 7 and 6 are called external one-sided corners(they lie outside the lines a And b).

Pairs of angles 1 and 8; 3 and 6; 5 and 4; 7 and 2 are called one-sided angles at right angles a And b and secant c. As you can see, out of a pair of corresponding angles, one lies between the right angle a And b, and the other is outside them.

Signs of parallel lines

It is obvious that using the definition it is impossible to conclude that two lines are parallel. Therefore, in order to conclude that two lines are parallel, use signs.

You can already formulate one of them after getting acquainted with the materials of the first part of the video lesson:

Theorem 1. Two lines perpendicular to the third do not intersect, that is, they are parallel.

You will become familiar with other signs of parallelism of lines based on the equality of certain pairs of angles by working with the materials in the second part of the video lesson"Signs of parallel lines."

Thus, you should know three more signs of parallel lines.

Theorem 2 (the first sign of parallel lines). If, when two lines intersect crosswise, the angles involved are equal, then the lines are parallel.

Rice. 2. Illustration for the first sign parallelism of lines

Repeat the first sign of parallel lines once again by working with the electronic educational resource « ».

Thus, when proving the first sign of parallelism of lines, the sign of equality of triangles is used (on two sides and the angle between them), as well as the sign of parallelism of lines as perpendicular to one straight line.

Task 1.

Write down the formulation of the first sign of parallel lines and its proof in your notebooks.

Theorem 3 (second sign of parallel lines). If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.

Repeat the second sign of parallel lines once again by working with the electronic educational resource « ».

When proving the second sign of parallelism of lines, the property of vertical angles and the first sign of parallelism of lines are used.

Task 2.

Write down the formulation of the second criterion for the parallelism of lines and its proof in your notebooks.

Theorem 4 (third sign of parallel lines). If, when two lines intersect with a transversal, the sum of one-sided angles is equal to 180 0, then the lines are parallel.

Repeat the third sign of parallel lines once again by working with the electronic educational resource « ».

Thus, when proving the first sign of parallelism of lines, the property of adjacent angles and the first sign of parallelism of lines are used.

Task 3.

Write down the formulation of the third criterion for parallel lines and its proof in your notebooks.

In order to practice solving simple problems, work with the materials of the electronic educational resource « ».

Signs of parallelism of lines are used in solving problems.

Now look at examples of solving problems on the signs of parallel lines, working with the materials of the video lesson“Solving problems on the topic “Signs of parallel lines.”

Now test yourself by completing the tasks of the control electronic educational resource « ».

Anyone who wants to work with the solution more complex tasks, can work with video lesson materials "Tasks on signs of parallelism of lines."

Properties of parallel lines

Parallel lines have a set of properties.

You will learn what these properties are by working with the video tutorial materials "Properties of parallel lines."

So, an important fact that you should know is the concurrency axiom.

Axiom of parallelism. Through a point not lying on a given line, it is possible to draw a line parallel to the given one, and, moreover, only one.

As you learned from the video tutorial, based on this axiom, two consequences can be formulated.

Corollary 1. If a line intersects one of the parallel lines, then it also intersects the other parallel line.

Corollary 2. If two lines are parallel to a third, then they are parallel to each other.

Task 4.

Write down the formulation of the stated corollaries and their proofs in your notebooks.

The properties of angles formed by parallel lines and a transversal are theorems that are inverse to the corresponding properties.

So, from the video lesson materials you learned the property of cross-lying angles.

Theorem 5 (theorem inverse to the first criterion for parallel lines). When two parallel lines intersect crosswise, the angles involved are equal.

Task 5.

Repeat the first property of parallel lines once again by working with the electronic educational resource « ».

Theorem 6 (theorem converse to the second criterion for the parallelism of lines). When two parallel lines intersect, the corresponding angles are equal.

Task 6.

Write down the statement of this theorem and its proof in your notebooks.

Repeat the second property of parallel lines once again by working with the electronic educational resource « ».

Theorem 7 (theorem converse to the third criterion for the parallelism of lines). When two parallel lines intersect, the sum of one-sided angles is 180 0.

Task 7.

Write down the statement of this theorem and its proof in your notebooks.

Repeat the third property of parallel lines once again by working with the electronic educational resource « ».

All properties of parallel lines are also used in solving problems.

Consider typical examples of problem solving by working with the video lesson materials “Parallel lines and problems on the angles between them and the transversal.”

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 the 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 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 converse 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 a converse, i.e. if this theorem is true, then converse theorem may be incorrect.

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 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. To make the theoretical material clearer, we will use illustrations and solutions to typical examples.

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. Verbally, parallelism of lines is denoted 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 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 the case when we are talking 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).

The parallelism criterion is a sufficient condition, the fulfillment of which guarantees parallelism of lines. 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 the lines to be parallel; 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 coordinate system, a straight line is determined by the equation of a straight line on a plane of one of the 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 for 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.

  1. Straight a in a rectangular coordinate system is defined general equation straight 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

  1. 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 slope coefficients 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.

  1. 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 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 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 the canonical equation of the straight line x 1 = y - 4 2 to the equation of the straight line with the 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 identical) and the slopes 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) . Dot 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 canonical equations 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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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 opposite signs parallelism of 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 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 with 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.

CHAPTER III.
PARALLEL DIRECT

§ 35. SIGNS OF PARALLEL TWO LINES.

The theorem that two perpendiculars to one line are parallel (§ 33) gives a sign that two lines are parallel. You can withdraw more general signs parallelism of two lines.

1. The first sign of parallelism.

If, when two straight lines intersect a third, the internal angles lying crosswise are equal, then these lines are parallel.

Let straight lines AB and CD be intersected by straight line EF and / 1 = / 2. Take point O - the middle of the segment KL of the secant EF (Fig. 189).

Let us lower the perpendicular OM from point O onto the straight line AB and continue it until it intersects with the straight line CD, AB_|_MN. Let us prove that CD_|_MN.
To do this, consider two triangles: MOE and NOK. These triangles are equal to each other. In fact: / 1 = / 2 according to the conditions of the theorem; ОK = ОL - by construction;
/ MOL = / NOK, like vertical angles. Thus, the side and two adjacent angles of one triangle are respectively equal to the side and two adjacent angles of another triangle; hence, /\ MOL = /\ NOK, and hence
/ LMO = / KNO, but / LMO is direct, which means / KNO is also straight. Thus, the lines AB and CD are perpendicular to the same line MN, therefore, they are parallel (§ 33), which was what needed to be proven.

Note. The intersection of straight lines MO and CD can be established by rotating the triangle MOL around point O by 180°.

2. The second sign of parallelism.

Let's see whether straight lines AB and CD are parallel if, when they intersect the third straight line EF, the corresponding angles are equal.

Let some corresponding angles be equal, for example / 3 = / 2 (drawing 190);
/ 3 = / 1, as the angles are vertical; Means, / 2 will be equal / 1. But angles 2 and 1 are intersecting interior angles, and we already know that if when two straight lines intersect the third, the intersecting interior angles are equal, then these lines are parallel. Therefore, AB || CD.

If, when two lines intersect a third, the corresponding angles are equal, then these two lines are parallel.

The construction of parallel lines using a ruler and a drawing triangle is based on this property. This is done as follows.

Let's attach the triangle to the ruler as shown in drawing 191. We will move the triangle so that one of its sides slides along the ruler, and draw several straight lines along some other side of the triangle. These lines will be parallel.

3. The third sign of parallelism.

Let us know that when two straight lines AB and CD intersect with a third straight line, the sum of any internal one-sided angles is equal to 2 d(or 180°). Will the straight lines AB and CD be parallel in this case (Fig. 192).

Let / 1 and / 2 are interior one-sided angles and add up to 2 d.
But / 3 + / 2 = 2d as adjacent angles. Hence, / 1 + / 2 = / 3+ / 2.

From here / 1 = / 3, and these internal angles lie crosswise. Therefore, AB || CD.

If, when two straight lines intersect a third, the sum of the internal one-sided angles is equal to 2 d, then these two lines are parallel.

Exercise.

Prove that the lines are parallel:
a) if external crosswise angles are equal (Fig. 193);
b) if the sum of external one-sided angles equals 2 d(drawing 194).