Parallel lines, signs and conditions for parallelism of lines. Parallel lines

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Parallel lines concept

Definition 1

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

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

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

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

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

Definition 2

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

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

$ c \ parallel d $.

The concept of parallel lines is often considered.

Definition 3

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

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

$ AB \ parallel CD $.

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

$ MN ∦ AB $ and $ MN ∦ CD $.

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

Historical reference

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

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

Parallel lines in life

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

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

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

Parallel lines in architecture

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

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

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

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Parallel lines - basic information.

Definition.

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

Definition.

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

Note that the clause "if they lie in the same plane" in the definition of parallel lines in space is very important. Let us clarify this point: two straight lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel straight lines. The straight lines along which the plane of the house wall intersects the planes of the ceiling and floor are parallel. Railroad tracks on level ground can also be viewed as parallel straight lines.

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

Note: if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.

Let us voice a statement that plays an important role in the study of parallel straight lines on a plane: through a point that does not lie on a given straight line, there is a single straight line parallel to a given one. This statement is taken as a fact (it cannot be proved on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the following theorem is true: through any point in space that does not lie on a given straight line, there is a single straight line parallel to the given one. This theorem can be easily proved using the above axiom of parallel lines (you can find its proof in the geometry textbook for grades 10-11, which is indicated at the end of the article in the bibliography).

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

Parallelism of lines - signs and conditions of parallelism.

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

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

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

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

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

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

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

Theorem.

If two straight lines on a plane are intersected by a secant, then for their parallelism it is necessary and sufficient that the cross-lying angles are equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphic illustration of this necessary and sufficient condition for the parallelism of straight lines on a plane.

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

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

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

Theorem.

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

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

Theorem.

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

Let us illustrate the stated theorems.

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

Theorem.

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

There is a similar theorem for lines in space.

Theorem.

If two straight lines in three-dimensional space are perpendicular to the same plane, then they are parallel.

Let us draw pictures corresponding to these theorems.

All the theorems, criteria and necessary and sufficient conditions formulated above are perfect for proving the parallelism of straight lines by the methods of geometry. That is, in order to prove the parallelism of two given lines, it is necessary to show that they are parallel to the third line, or to show the equality of the intersecting angles, etc. Many similar problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the method of coordinates to prove the parallelism of straight lines on a plane or in three-dimensional space. Let us formulate necessary and sufficient conditions for the parallelism of straight lines, which are given in a rectangular coordinate system.

Parallelism of straight lines in a rectangular coordinate system.

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

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

Theorem.

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

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

In particular, if the straight line a in the rectangular coordinate system Oxy on the plane is defined by the general equation of the straight line of the form , and line b - , then the normal vectors of these lines have coordinates and, respectively, and the condition for parallelism of lines a and b will be written as.

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

If straight line a and straight line b in a rectangular coordinate system are determined by the canonical equations of a straight line in the plane of the form and , or parametric equations of a straight line on the plane of the form and accordingly, the direction vectors of these lines have coordinates and, and the condition for parallelism of lines a and b is written as.

Let's look at the solutions of several examples.

Example.

Are the lines parallel and ?

Solution.

Let us rewrite the equation of a straight line in segments in the form of a general equation of a straight line: ... Now you can see that is the normal vector of the straight line , a is the normal vector of a straight line. These vectors are not collinear, since there is no real number t for which the equality ( ). Consequently, the necessary and sufficient condition for parallelism of lines on the plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are straight and parallel?

Solution.

Let us bring the canonical equation of the straight line to the equation of the straight line with the slope:. Obviously, the equations of the straight lines and are not the same (in this case, the given straight lines would be the same) and the slope coefficients of the straight lines are equal, therefore, the original straight lines are parallel.

Defining Parallel Lines. Parallel are two straight lines that lie in the same plane and do not intersect along their entire length.

Straight lines AB and CD (Fig. 57) will be parallel. The fact that they are parallel is sometimes expressed in writing: AB || CD.

Theorem 34. Two straight lines perpendicular to the same third are parallel.

Given straight lines CD and EF perpendicular to AB (Fig. 58)

CD ⊥ AB and EF ⊥ AB.

It is required to prove that CD || EF.

Proof... If lines CD and EF were not parallel, they would intersect at some point M. In this case, two perpendiculars would be dropped from point M onto line AB, which is impossible (Theorem 11), hence line CD || EF (CHTD).

Theorem 35. Two straight lines, one of which is perpendicular and the other inclined to the third, always intersect.

Two straight lines EF and CG are given, of which EF ⊥ AB, and CG is inclined to AB (Fig. 59).

You want to prove that CG will meet line EF or that CG is not parallel to EF.

Proof... From point C, we raise the perpendicular CD to line AB, then at point C an angle DCG is formed, which we will repeat so many times so that line CK falls below line AB. Suppose that for this we repeat the angle DCG n times, as that

In the same way, we postpone on line AB line CE also n times so that CN = nCE.

Reconstruct the perpendiculars LL ", MM", NN "from points C, E, L, M, N. The space contained between the two parallel segments CD, NN” and the segment CN will be n times larger than the space between the two perpendiculars CD, EF and segment CE, so DCNN "= nDCEF.

The space contained in the corner DCK contains the space DCNN ", therefore,

DCK> CDNN "or
nDCG> nDCEF, whence
DCG> DCEF.

The last inequality can take place only when the line CG leaves the limits of the space DCEF during its continuation, that is, when the line CG meets the line EF, hence the line CG is not parallel to CF (CGT).

Theorem 36. A straight line perpendicular to one of the parallel, perpendicular to the other.

Given two parallel straight lines AB and CD and a straight line EF perpendicular to CD (Fig. 60).

AB || CD, EF ⊥ CD

It is required to prove that EF ⊥ AB.

Proof... If line AB were inclined to EF, then two lines CD and AB would intersect, because CD ⊥ EF and AB is inclined to EF (Theorem 35), and lines AB and CD would not be parallel, which would contradict this condition, therefore, straight line EF is perpendicular to CD (CTD).

Angles formed by the intersection of two lines of the third line... At the intersection of two straight lines AB and CD of the third straight line EF (Fig. 61), eight angles α, β, γ, δ, λ, μ, ν, ρ are formed. These corners are given special names.

The four angles α, β, ν and ρ are called external.

The four angles γ, δ, λ, μ are called internal.

The four angles β, γ, μ, ν and the four angles α, δ, λ, ρ are called unilateral, because they lie on one side of the straight line EF.

In addition, the angles, when taken in pairs, are named as follows:

The angles β and μ are called the respective ... In addition to this pair, the same corresponding angles will be pairs of angles:γ and ν, α and λ, δ and ρ.

The pairs of angles δ and μ, as well as γ and λ, are called internal cross-lying .

The pairs of angles β and ρ, as well as α and ν, are called external cross-lying .

The pairs of angles γ and μ, as well as δ and λ, are called domestic unilateral .

The pairs of angles β and ν, as well as α and ρ, are called external unilateral .

Conditions for parallelism of two lines

Theorem 37. Two straight lines are parallel if at the intersection of their third one they have equal: 1) the corresponding angles, 2) the inner cross-lying, 3) the outer cross-lying, and, finally, if 4) the sum of the inner one-sided is equal to two straight lines, 5) the sum of the outer one-sided is equal to two straight lines.

Let us prove each of these parts of the theorem separately.

1st case. The corresponding angles are equal(Fig. 62).

Given. The angles β and μ are equal.

Proof... If lines AB and CD intersected at point Q, then we would get a triangle GQH, whose external angle β would be equal to the internal angle μ, which would contradict Theorem 22, therefore, lines AB and CD do not intersect or AB || CD (CHTD).

2nd case. The inner cross-lying corners are equal, that is, δ = μ.

Proof... δ = β as vertical, δ = μ by hypothesis, hence β = μ. That is, the corresponding angles are equal, and in this case the lines are parallel (1st case).

3rd case. Outside cross-lying corners are equal, that is, β = ρ.

Proof... β = ρ by hypothesis, μ = ρ as vertical, therefore, β = μ, since the corresponding angles are equal. Hence it follows that AB || CD (1st case).

4th case. The sum of inner one-sided is equal to two straight lines or γ + μ = 2d.

Proof... β + γ = 2d as the sum of adjacent ones, γ + μ = 2d by hypothesis. Therefore, β + γ = γ + μ, whence β = μ. The corresponding angles are equal, therefore, AB || CD.

5th case. The sum of the outer one-sided is equal to two straight lines, that is, β + ν = 2d.

Proof... μ + ν = 2d as the sum of adjacent ones, β + ν = 2d by hypothesis. Therefore, μ + ν = β + ν, whence μ = β. The corresponding angles are equal, therefore, AB || CD.

Thus, in all cases AB || CD (CHTD).

Theorem 38(reverse 37). If two straight lines are parallel, then when their third straight line intersects, they will be equal: 1) internal cross-lying angles, 2) external cross-lying, 3) corresponding angles and are equal to two straight lines 4) the sum of internal one-sided and 5) the sum of external one-sided angles.

Two parallel lines AB and CD are given, that is AB || CD (Fig. 63).

It is required to prove that all of the above conditions are met.

1st case... We intersect two parallel lines AB and CD of the third inclined line EF. Let G and H denote the intersection points of lines AB and CD of line EF. From the point O of the midpoint of the line GH we drop the perpendicular to the line CD and extend it to the intersection with the line AB at the point P. The line OQ perpendicular to CD is also perpendicular to AB (Theorem 36). Right-angled triangles OPG and OHQ are equal, because OG = OH by construction, ∠ HOQ = ∠ POG as vertical angles, hence OP = OQ.

Hence it follows that δ = μ, i.e. the inner cross-lying angles are.

2nd case... If AB || CD, then δ = μ, and since δ = β, and μ = ρ, then β = ρ, i.e. the outer cross-lying corners are equal.

3rd case... If AB || CD, then δ = μ, and since δ = β, then β = μ, therefore, the corresponding angles are.

4th case... If AB || CD, then δ = μ, and since δ + γ = 2d, then μ + γ = 2d, that is, the sum of inner one-sided is equal to two straight lines.

5th case... If AB || CD, then δ = μ.

Since μ + ν = 2d, μ = δ = β, therefore, ν + β = 2d, i.e. the sum of the outer one-sided is equal to two straight lines.

These theorems imply consequence. Only one straight line can be drawn through a point, parallel to another straight line.

Theorem 39. Two straight lines, parallel to the third, are parallel to each other.

Three straight lines (Fig. 64) AB, CD and EF are given, of which AB || EF, CD || EF.

It is required to prove that AB || CD.

Proof... We intersect these lines with the fourth line GH.

If AB || EF then ∠ α = ∠ γ as appropriate. If CD || EF then ∠ β = ∠ γ as well as the corresponding ones. Hence, ∠ α = ∠ β .

If the corresponding angles are equal, then the straight lines are parallel, therefore, AB || CD (CHTD).

Theorem 40. The angles of the same name with parallel sides are equal.

Given are angles of the same name (both acute or both obtuse) angles ABC and DEF, their sides are parallel, that is, AB || DE, BC || EF (Fig. 65).

It is required to prove that ∠ B = ∠ E.

Proof... Extend the side DE to its intersection with the line BC at the point G, then

∠ E = ∠ G as corresponding from the intersection of the sides parallel to BC and EF of the third straight line DG.

∠ B = ∠ G as corresponding from the intersection of parallel sides AB and DG of line BC, therefore,

∠ E = ∠ B (CHTD).

Theorem 41. Opposite angles with parallel sides complement each other up to two straight lines.

Given two opposite angles ABC and DEF (Fig. 66) with parallel sides, therefore, AB || DE and BC || EF.

It is required to prove that ABC + DEF = 2d.

Proof... Extend line DE to the intersection with line BC at point G.

∠ B + ∠ DGB = 2d as the sum of the inner one-sided angles formed by the intersection of the third line BC parallel to AB and DG.

∠ DGB = ∠ DEF as corresponding, therefore

∠ B + ∠ DEF = 2d (CHTD).

Theorem 42. Angles of the same name with perpendicular sides are equal and opposite angles complement each other up to two straight lines.

Consider two cases: when A) the angles are of the same name and when B) they are opposite.

1st case... The sides of two corners of the same name DEF and ABC (Fig. 67) are perpendicular, ie DE ⊥ AB, EF ⊥ BC.

It is required to prove that ∠ DEF = ∠ ABC.

Proof... Draw from point B lines BM and BN parallel to lines DE and EF so that

BM || DE, BN || EF.

These lines are also perpendicular to the sides of a given angle ABC, i.e.

BM ⊥ AB and BN ⊥ BC.

Because ∠ NBC = d, ∠ MBA = d, then

∠ NBC = ∠ MBA (a)

Subtracting from both sides of the equality (a) in the NBA corner, we find

∠ MBN = ∠ ABC

Since the angles MBN and DEF are of the same name and with parallel sides, they are equal (Theorem 40).

∠ MBN = ∠ DEF (b)

Equalities (a) and (b) imply the equality

∠ ABC = ∠ DEF (CHTD).

2nd case... Angles GED and ABC with perpendicular sides are of different dimensions.

It is required to prove that ∠ GED + ∠ ABC = 2d (Fig. 67).

Proof... The sum of the angles GED and DEF is equal to two straight lines.

GED + DEF = 2d
DEF = ABC, therefore
GED + ABC = 2d (CHTD).

Theorem 43. Parts of parallel lines between other parallel lines are equal.

There are four straight lines AB, BD, CD, AC (Fig. 68), of which AB || CD and BD || AC.

It is required to prove that AB = CD and BD = AC.

Proof... Connecting point C with point B by segment BC, we get two equal triangles ABC and BCD, because

BC - common side,

∠ α = ∠ β (as internal cross-lying from the intersection of parallel lines AB and CD of the third line BC),

∠ γ = ∠ δ (as internal criss-crossing lines from the intersection of parallel lines BD and AC of line BC).

Thus, triangles have an equal side and two equal angles lying on it.

Equal sides AC and BD lie opposite equal angles α and β, and equal sides AB and CD lie against equal angles γ and δ, therefore,

AC = BD, AB = CD (CHTD).

Theorem 44. Parallel lines along their entire length are at an equal distance from each other.

The distance of a point from a straight line is determined by the length of the perpendicular dropped from a point to a straight line. To determine the distance of any two points A and B parallel to AB from CD, we drop the perpendiculars AC and BD from points A and B.

Given line AB parallel to CD, segments AC and BD are perpendicular to line CD, ie AB || CD, AC ⊥ DC, BD ⊥ CD (Fig. 69).

It is required to prove that AC = BD.

Proof... Lines AC and BD, being both perpendicular to CD, are parallel, and therefore, AC and BD as parts of parallel between parallel, are equal, that is, AC = BD (BD).

Theorem 45(reverse 43). If the opposite parts of four intersecting lines are equal, then these parts are parallel.

Given four intersecting straight lines, the opposite parts of which are equal: AB = CD and BD = AC (Fig. 68).

It is required to prove that AB || CD and BD || AC.

Proof... Let us connect points B and C with line BC. Triangles ABC and BDC are equal because

BC - common side,
AB = CD and BD = AC by condition.

From here

∠ α = ∠ β , ∠ γ = ∠ δ

Hence,

AC || BD, AB || CD (CHTD).

Theorem 46. The sum of the angles of a triangle is equal to two right angles.

Given a triangle ABC (Fig. 70).

It is required to prove that A + B + C = 2d.

Proof... Draw from point C a line CF parallel to side AB. At point C, there are three angles BCA, α and β. Their sum is equal to two straight lines:

BCA + α + β = 2d

α = B (as internal cross-lying angles at the intersection of parallel lines AB and CF of line BC);

β = A (as the corresponding angles at the intersection of lines AB and CF of line AD).

Replacing the angles α and β by their values, we get:

BCA + A + B = 2d (CHTD).

The following corollaries follow from this theorem:

Corollary 1. The outer corner of a triangle is equal to the sum of the inner ones not adjacent to it.

Proof... Indeed, from drawing 70,

∠ BCD = ∠ α + ∠ β

Since ∠ α = ∠ B, ∠ β = ∠ A, then

∠ BCD = ∠ A + ∠ B.

Corollary 2. In a right-angled triangle, the sum of the acute angles is equal to the right angle.

Indeed, in a right-angled triangle (Fig. 40)

A + B + C = 2d, A = d, therefore
B + C = d.

Corollary 3. A triangle cannot have more than one right or one obtuse angle.

Corollary 4. In an equilateral triangle, each angle is 2/3 d .

Indeed, in an equilateral triangle

A + B + C = 2d.

Since A = B = C, then

3A = 2d, A = 2/3 d.