If 2 are parallel. Parallel lines, signs and conditions for parallel lines

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 limit 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.

CHAPTER III.
PARALLEL DIRECT

§ 38. DEPENDENCE BETWEEN ANGLES,
FORMED BY TWO PARALLEL LINES AND A SECONDARY.

We know that two lines are parallel if, when they intersect a third line, the corresponding angles are equal, or internal or external angles lying crosswise are equal, or the sum of internal, or the sum of external one-sided angles is equal to 2 d. Let us prove that the converse theorems are also true, namely:

If two parallel lines are crossed by a third, then:

1) corresponding angles are equal;
2) internal crosswise angles are equal;
3) external crosswise angles are equal;
4) the sum of internal one-sided angles is equal to 2 d ;
5) the sum of external one-sided angles is equal to 2 d .

Let us prove, for example, that if two parallel lines are intersected by a third line, then the corresponding angles are equal.

Let straight lines AB and CD be parallel, and MN be their secant (Fig. 202). Let us prove that the corresponding angles 1 and 2 are equal to each other.

Let's assume that / 1 and / 2 are not equal. Then at point O we can construct / IOC, corresponding and equal / 2 (drawing 203).

But if / MOQ = / 2, then straight line OK will be parallel to CD (§ 35).

We found that two straight lines AB and OK were drawn through point O, parallel to straight line CD. But this cannot be (§ 37).

We arrived at a contradiction because we assumed that / 1 and / 2 are not equal. Therefore, our assumption is incorrect and / 1 must be equal / 2, i.e. the corresponding angles are equal.

Let us establish the relationships between the remaining angles. Let straight lines AB and CD be parallel, and MN be their secant (Fig. 204).

We have just proved that in this case the corresponding angles are equal. Let us assume that any two of them have 119° each. Let's calculate the size of each of the other six angles. Based on the properties of adjacent and vertical angles, we find that four of the eight angles will have 119° each, and the rest will have 61° each.

It turned out that both internal and external crosswise angles are equal in pairs, and the sum of internal or external one-sided angles is equal to 180° (or 2 d).

The same will take place for any other value of equal corresponding angles.

Corollary 1. If each of two lines AB and CD is parallel to the same third line MN, then the first two lines are parallel to each other (drawing 205).

In fact, by drawing the secant EF (Fig. 206), we obtain:
A) / 1 = / 3, since AB || MN; b) / 2 = / 3, since CO || MN.

Means, / 1 = / 2, and these are the angles corresponding to the lines AB and CD and the secant EF, therefore, the lines AB and CD are parallel.

Corollary 2. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other (drawing 207).

Indeed, if EF _|_ AB, then / 1 = d; if AB || CD, then / 1 = / 2.

Hence, / 2 = d i.e. EF _|_ CD .

1) If, when two lines intersect with a transversal, the lying angles are equal, then the lines are parallel.

2) If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.

3) If, when two straight lines intersect with a transversal, the sum of one-sided angles is equal to 180°, then the straight lines are parallel.

3. Through a point not lying on a given line there passes only one line parallel to the given one.

4 If a line intersects one of two parallel lines, then it also intersects the other.

5. If two lines are parallel to a third line, then they are parallel.

Properties of parallel lines

1) If two parallel lines are intersected by a transversal, then the intersecting angles are equal.

2) If two parallel lines are intersected by a transversal, then the corresponding angles are equal.

3) If two parallel lines are intersected by a transversal, then the sum of the one-sided angles is 180°.

7. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

8. Solving a system of two equations with two Such a pair of numbers is called unknown X And at , which, when substituted into this system, turn each of its equations into a correct numerical equality.

9.Solve the system of equations- means to find all its solutions or establish that there are none.

1. Methods for solving a system of equations:

a) substitution

b) addition;

c) graphic.

10. The sum of the angles of a triangle is 180°.

11.External corner of a triangle is an angle adjacent to some angle of this triangle.

An exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.

12. In any triangle, either all the angles are acute, or two angles are acute, and the third is obtuse or straight.

13If all three angles of a triangle are acute, then the triangle is called acute-angled.

14.If one of the angles of a triangle is obtuse, then the triangle is called obtuse-angled.

15. If one of the angles of a triangle is right, then the triangle is called rectangular.

16. The side of a right triangle lying opposite the right angle is called hypotenuse, and the other two sides are legs.

17. In a triangle: 1) the larger angle lies opposite the larger side; 2) back, the larger side lies opposite the larger angle.

18. In a right triangle, the hypotenuse is longer than the leg.

19. If two angles of a triangle are equal, then the triangle is isosceles (sign of an isosceles triangle).

20. Each side of a triangle is less than the sum of the other two sides.

21 The sum of two acute angles of a right triangle is 90°.

22. A leg of a right triangle lying opposite an angle of 30° is equal to half the hypotenuse.

Signs of equality of right triangles: 1) on two sides; 2) along the hypotenuse and acute angle; 3) along the hypotenuse and leg; 4) along the leg and acute angle

The length of a perpendicular drawn from a point to a line is called the distance from this point to the line.

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.

Yandex.RTB R-A-339285-1 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 an 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 parallel lines; 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 of 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. 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; 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 the angular coefficients of the 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 coincident) and the angular coefficients 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 met.

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 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. given the 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 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 by the canonical equations of one line in space and the parametric equations of 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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corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7;

internal crosswise angles: 3 and 5, 4 and 6;

external crosswise angles: 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 according to what has been proven, ∠ 4 = ∠ 6.

Therefore, ∠ 2 =∠ 8.

3. Corresponding angles 2 and 6 are the same, since ∠ 2 = ∠ 4, and ∠ 4 = ∠ 6. Let’s 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. We 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 above proven justification we obtain converse theorems.

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

1. Internal crosswise angles are the same;

or 2. External crosswise angles are identical;

or 3. Corresponding angles are equal;

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

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

then the first two lines are parallel.