On a plane, straight lines are called parallel if they have no common points, that is, they do not intersect. To indicate parallelism, use the special icon || (parallel lines a || b).
For straight lines lying in space, the requirement of the absence of common points is not enough - for them to be parallel in space, they must belong to the same plane (otherwise they will be crossing).
There is no need to go far for examples of parallel straight lines, they accompany us everywhere, in the room - these are the lines of intersection of the wall with the ceiling and floor, on a notebook sheet - opposite edges, etc.
It is quite obvious that, having parallelism of two lines and a third line parallel to one of the first two, it will be parallel to the second.
Parallel straight lines on a plane are connected by a statement that cannot be proved using the axioms of planimetry. It is taken as a fact, as an axiom: for any point on the plane that does not lie on a straight line, there is a single straight line that passes through it parallel to the given one. Every sixth grader knows this axiom.
Its spatial generalization, that is, the statement that for any point in space that does not lie on a straight line, there is a single straight line that passes through it parallel to a given one, is easily proved using the already known axiom of parallelism on the plane.
Parallel Line Properties
- If any of the parallel two straight lines is parallel to the third, then they are mutually parallel.
This property is possessed by parallel lines both on the plane and in space.
As an example, consider its justification in stereometry.
Let us admit parallelism of straight lines b and with straight line a.
The case when all straight lines lie in the same plane, we will leave the planimetry.
Suppose a and b belong to the beta plane, and gamma is the plane to which a and c belong (by the definition of parallelism in space, straight lines must belong to the same plane).
If we assume that the betta and gamma planes are different and mark a certain point B on the line b from the betta plane, then the plane drawn through the point B and the line c must intersect the betta plane in a straight line (denote it by b1).
If the resulting straight line b1 intersected the gamma plane, then, on the one hand, the intersection point would have to lie on a, since b1 belongs to the beta plane, and on the other hand, it should also belong to c, since b1 belongs to the third plane.
But parallel lines a and c should not intersect.
Thus, the straight line b1 must belong to the betta plane and, at the same time, have no points in common with a, therefore, according to the axiom of parallelism, it coincides with b.
We got the line b1 coinciding with the line b, which belongs to the same plane with the line c and does not intersect it, that is, b and c are parallel
- Through a point that does not lie on a given straight line, only one single straight line can pass parallel to the given one.
- Lying on a plane perpendicular to the third, two straight lines are parallel.
- Provided that the plane intersects one of the parallel two straight lines, the second straight line intersects the same plane.
- The corresponding and criss-crossing internal angles formed by the intersection of parallel two straight lines of the third are equal, the sum of the internal one-sided angles formed in this case is 180 °.
The converse statements are also true, which can be taken as signs of parallelism of two straight lines.
Parallelism condition for straight lines
The properties and features formulated above are the conditions for the parallelism of straight lines, and they can be fully proved by the methods of geometry. In other words, to prove the parallelism of two existing straight lines, it is sufficient to prove their parallelism to the third straight line or the equality of the angles, whether corresponding or crosswise, etc.
For the proof, the method “by contradiction” is mainly used, that is, with the assumption that the straight lines are not parallel. Proceeding from this assumption, it is easy to show that in this case the specified conditions are violated, for example, cross-lying internal angles turn out to be unequal, which proves the incorrectness of the assumption made.
Lesson objectives: In this lesson, you will get acquainted with the concept of "parallel lines", you will learn how to make sure that straight lines are parallel, and also what properties the angles formed by parallel lines and a secant have.
Parallel lines
You know that the concept of "straight line" is one of the so-called undefined concepts of geometry.
You already know that two straight lines can coincide, that is, have all common points, can intersect, that is, have one common point. Straight lines intersect at different angles, while the angle between the straight lines is considered the smallest of the angles that they form. A special case of intersection can be considered the case of perpendicularity, when the angle formed by the straight lines is 90 0.
But two straight lines may not have common points, that is, they do not intersect. Such straight lines are called parallel.
Work with electronic educational resource « ».
To get acquainted with the concept of "parallel lines", work in the video tutorial materials
Thus, you now know the definition of parallel lines.
From the materials of the video tutorial fragment, you learned about different types angles that are formed when two straight lines intersect the third.
Pairs of angles 1 and 4; 3 and 2 are called inner one-sided corners(they lie between the straight lines a and b).
Pairs of corners 5 and 8; 7 and 6 call external one-sided corners(they lie outside the straight lines a and b).
Pairs of angles 1 and 8; 3 and 6; 5 and 4; 7 and 2 are called one-sided corners for straight lines a and b and secant c... As you can see, out of a pair of corresponding angles, one lies between the right a and b and the other is outside of them.
Signs of parallelism of straight lines
Obviously, using the definition, it is impossible to draw a conclusion about the parallelism of two straight lines. Therefore, in order to conclude that two lines are parallel, use signs.
You can already formulate one of them by reading the materials of the first part of the video lesson:
Theorem 1... Two straight lines, perpendicular to the third, do not intersect, that is, they are parallel.
You will get acquainted with other signs of parallelism of straight lines based on the equality of certain pairs of angles by working with the materials of the second part of the video lesson"Signs of parallelism of straight lines".
Thus, you should know three more signs of parallelism of straight lines.
Theorem 2 (the first criterion for parallelism of lines)... If at the intersection of two intersecting straight lines, the lying angles are equal, then the straight lines are parallel.
Rice. 2. Illustration for the first sign parallelism of straight linesOnce again, repeat the first sign of parallelism of straight lines by working with an electronic educational resource « ».
Thus, in the proof of the first criterion for the parallelism of straight lines, the criterion for the equality of triangles (along two sides and the angle between them) is used, as well as the criterion for the parallelism of straight lines as perpendicular to one straight line.
Exercise 1.
Write down the formulation of the first criterion for parallelism of straight lines and its proof in your notebooks.
Theorem 3 (second criterion for parallelism of lines)... If at the intersection of two straight secant the corresponding angles are equal, then the straight lines are parallel.
Once again, repeat the second sign of parallelism of straight lines by working with an electronic educational resource « ».
When proving the second criterion for parallelism of straight lines, the property of vertical angles and the first criterion for parallelism of straight lines are used.
Task 2.
Write down the formulation of the second criterion for parallelism of straight lines and its proof in your notebooks.
Theorem 4 (third criterion for parallelism of lines)... If, at the intersection of two straight secant lines, the sum of one-sided angles is 180 0, then the straight lines are parallel.
Once again, repeat the third sign of parallelism of straight lines by working with an electronic educational resource « ».
Thus, in the proof of the first criterion for parallelism of straight lines, we use the property of adjacent angles and the first criterion for parallelism of straight lines.
Task 3.
Write down the formulation of the third criterion for parallelism of straight lines and its proof in your notebooks.
In order to practice solving the simplest tasks, work with the materials of the electronic educational resource « ».
The signs of parallelism of straight lines are used when solving problems.
Now consider examples of solving problems for signs of parallelism of straight lines, having worked with the materials of the video lesson“Solving problems on the topic“ Signs of parallelism of straight lines ”.
Now test yourself by completing the tasks of the control electronic educational resource « ».
Anyone who wants to work with the solution of more complex problems can work with the materials of the video lesson "Problems on the signs of parallelism of straight lines."
Parallel Line Properties
Parallel lines have a number of properties.
You will find out what these properties are by working with the materials of the video tutorial. "Properties of Parallel Lines".
Thus, important fact which you should know is the axiom of parallelism.
Parallelism axiom... Through a point that does not lie on a given straight line, you can draw a straight line parallel to the given one, and, moreover, only one.
As you learned from the materials of the video lesson, 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 the third, then they are parallel to each other.
Task 4.
Write down the wording of the formulated consequences and their proofs in your notebooks.
The properties of the angles formed by parallel lines and a secant are theorems opposite to the corresponding criteria.
So, from the materials of the video tutorial, you learned the property of criss-crossing corners.
Theorem 5 (theorem converse to the first criterion for parallelism of lines)... When two parallel intersecting straight lines intersect, the lying angles are equal.
Task 5.
Once again repeat the first property of parallel straight lines after working with the electronic educational resource « ».
Theorem 6 (theorem converse to the second criterion for parallelism of lines)... When two parallel straight lines intersect, the corresponding angles are equal.
Task 6.
Write down the statement of this theorem and its proof in your notebooks.
Once again, repeat the second property of parallel straight lines by working with an electronic educational resource « ».
Theorem 7 (theorem converse to the third criterion for parallelism of lines)... When two parallel straight lines intersect, the sum of the 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 straight lines once again by working with an electronic educational resource « ».
All properties of parallel lines are also used in solving problems.
Consider typical examples solving problems by working with the materials of the video lesson "Parallel lines and problems on the angles between them and the secant."
This chapter is devoted to the study of parallel lines. This is the name of two straight lines on a plane that do not intersect. We see the segments of parallel straight lines in environment- these are two edges of a rectangular table, two edges of a book cover, two trolley bars, etc. Parallel lines play a very important role... In this chapter, you will learn about what the axioms of geometry are and what the axiom of parallel lines is - one of the most famous axioms of geometry.
In Section 1, we noted that two lines either have one common point, that is, they intersect, or do not have a single point in common, that is, they do not intersect.
Definition
The parallelism of straight lines a and b is denoted as follows: a || b.
Figure 98 shows straight lines a and b, perpendicular to line c. In Section 12 we established that such lines a and b do not intersect, that is, they are parallel.
Rice. 98
Along with parallel lines, parallel lines are often considered. The two segments are called parallel if they lie on parallel lines. In Figure 99, and segments AB and CD are parallel (AB || CD), and segments MN and CD are not parallel. Similarly, the parallelism of a segment and a straight line (Fig. 99, b), a ray and a straight line, a segment and a ray, two rays (Fig. 99, c) is determined.
Rice. 99 Signs of parallelism of two straight lines
Straight with is called secant in relation to straight lines a and b, if it intersects them at two points (Fig. 100). At the intersection of straight lines a and b secant c, eight corners are formed, which are indicated by numbers in Figure 100. Some pairs of these angles have special names:
criss-cross corners: 3 and 5, 4 and 6;
one-sided corners: 4 and 5, 3 and 6;
corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7.
Rice. 100
Consider three signs of parallelism of two straight lines associated with these pairs of angles.
Theorem
Proof
Suppose that at the intersection of lines a and b secant AB, the intersecting angles are equal: ∠1 = ∠2 (Fig. 101, a).
Let us prove that a || b. If angles 1 and 2 are straight (Fig. 101, b), then straight lines a and b are perpendicular to line AB and, therefore, are parallel.
Rice. 101
Consider the case where angles 1 and 2 are not straight.
From the middle O of the segment AB we draw the perpendicular OH to the straight line a (Fig. 101, c). On the straight line b from point B, we postpone the segment BH 1, equal to the segment AH, as shown in Figure 101, c, and draw the segment OH 1. The triangles ОНА and ОН 1 В are equal on two sides and the angle between them (AO = BO, AH = BH 1, ∠1 = ∠2), therefore ∠3 = ∠4 and ∠5 = ∠6. From the equality ∠3 = ∠4 it follows that the point H 1 lies on the extension of the ray OH, that is, the points H, O and H 1 lie on one straight line, and from the equality ∠5 = ∠6 it follows that the angle 6 is a straight line (since angle 5 is straight). So, straight lines a and b are perpendicular to the straight line HH 1 so they are parallel. The theorem is proved.
Theorem
Proof
Let at the intersection of straight lines a and b secant with the corresponding angles are equal, for example ∠1 = ∠2 (Fig. 102).
Rice. 102
Since corners 2 and 3 are vertical, then ∠2 = ∠3. It follows from these two equalities that ∠1 = ∠3. But angles 1 and 3 are crosswise, so lines a and b are parallel. The theorem is proved.
Theorem
Proof
Let at the intersection of straight lines a and b secant with the sum of one-sided angles equal to 180 °, for example ∠1 + ∠4 = 180 ° (see Fig. 102).
Since angles 3 and 4 are adjacent, ∠3 + ∠4 = 180 °. From these two equalities it follows that the cross-lying angles 1 and 3 are equal, therefore the straight lines a and b are parallel. The theorem is proved.
Practical ways to build parallel lines
The signs of parallelism of straight lines underlie the methods of constructing parallel straight lines using various tools used in practice. Consider, for example, a method for constructing parallel lines using a drawing square and a ruler. To build a straight line passing through point M and parallel to a given straight line a, we apply a drawing square to line a, and a ruler to it, as shown in Figure 103. Then, moving the square along the ruler, we will achieve that point M is on the side of the square , and draw a line b. Lines a and b are parallel, since the corresponding angles, designated in Figure 103 by the letters α and β, are equal.
Rice. 103 Figure 104 shows a method for constructing parallel lines using a flight bus. This method is used in drawing practice.
Rice. 104 A similar method is used when performing carpentry work, where a malka is used to mark parallel straight lines (two wooden planks, fastened with a hinge, Fig. 105).
Rice. 105
Tasks
186. In Figure 106, straight lines a and b are crossed by straight line c. Prove that a || b if:
a) ∠1 = 37 °, ∠7 = 143 °;
b) ∠1 = ∠6;
c) ∠l = 45 °, and angle 7 is three times larger than angle 3.
Rice. 106
187. According to Figure 107, prove that AB || DE.
Rice. 107
188. Segments AB and CD intersect in their common middle. Prove that lines AC and BD are parallel.
189. Using the data in Figure 108, prove that ВС || AD.
Rice. 108
190. In Figure 109 AB = BC, AD = DE, ∠C = 70 °, ∠EAC = 35 °. Prove that DE || AC.
Rice. 109
191. Segment BK - bisector of triangle ABC. A straight line is drawn through point K, intersecting the side BC at point M so that BM = MK. Prove that lines KM and AB are parallel.
192. In triangle ABC, angle A is equal to 40 °, and angle BCE adjacent to angle ACB is equal to 80 °. Prove that the bisector of angle ALL is parallel to line AB.
193. In triangle ABC ∠A = 40 °, ∠B = 70 °. Line BD is drawn through vertex B so that ray BC is the bisector of angle ABD. Prove that lines AC and BD are parallel.
194. Draw a triangle. Through each vertex of this triangle, using a drawing square and a ruler, draw a straight line parallel to the opposite side.
195. Draw triangle ABC and mark point D on the AC side. Through point D, using a drawing square and a ruler, draw straight lines parallel to the other two sides of the triangle.
§ 1. Signs of parallelism of two straight lines - Geometry grade 7 (Atanasyan L.S.)
Short description:
You will learn about what parallel lines are in this paragraph. You will get a simple definition, but at the same time somewhat unusual - two straight lines on a plane are called parallel if they do not intersect. In other words, if two lines do not intersect, then they will be parallel. Or, if the lines do not have intersection points, then they are parallel.
The unusualness of this definition lies in the fact that if there are two straight lines in front of you and you do not see their intersection point, this does not mean that there is none. This means that you may simply not see it.
Therefore, this definition cannot be used directly to prove that two lines are parallel. After all, you cannot endlessly follow the continuation of straight lines in order to make sure that they do not intersect.
But this is not necessary. There are signs by which one can judge the parallelism of straight lines. There are three of them. In accordance with each of them, special angles or their combinations are considered, which are formed at the intersection of these two investigated straight lines of the third straight line - the secant. These angles are used to judge the parallelism of straight lines.
The proofs of these signs - the theorem on the parallelism of lines - are based on the theorem that you already considered in Chapter 1 of the textbook - two lines perpendicular to the third do not intersect. Only now this theorem looks different - two lines perpendicular to the third are parallel.
Signs of parallelism of two straight lines
Theorem 1. If at the intersection of two secant lines:
criss-crossing angles are equal, or
the corresponding angles are equal, or
the sum of the one-sided angles is 180 °, then
straight lines are parallel(fig. 1).
Proof. We restrict ourselves to the proof of case 1.
Suppose that at the intersection of lines a and b secant AB, the intersecting angles are 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 outer corner of the triangle ABM. Let, for definiteness, ∠ 4 be the outer corner of the triangle ABM, and ∠ 6 - the inner one. From the theorem on the outer angle of a triangle it follows that ∠ 4 is greater than ∠ 6, and this contradicts the condition, which means that the lines a and 6 cannot intersect, so they are parallel.
Corollary 1. Two different straight lines in a plane perpendicular to the same straight line are parallel(fig. 2).
Comment. The way we have just proved case 1 of Theorem 1 is called contradiction or reduction to absurdity. This method got its first name because at the beginning of the reasoning, an assumption is made that is opposite (opposite) to what is required to be proved. It is called a reduction to absurdity due to the fact that, arguing on the basis of the assumption made, we come to an absurd conclusion (to an absurdity). The receipt of such a conclusion forces us to reject the assumption made at the beginning and accept the one that was required to be proved.
Objective 1. Construct a straight line passing through a given point M and parallel to a given straight line a, not passing through a point M.
Solution. Draw through the point M a straight line p perpendicular to a straight line a (Fig. 3).
Then we draw a straight line b through point M perpendicular to a straight line p. Line b is parallel to line a according to the corollary to Theorem 1.
An important conclusion follows from the considered problem:
through a point that does not lie on a given straight line, you can always draw a straight line parallel to a given.
The main property of parallel lines is as follows.
Axiom of parallel lines. Through a given point, which does not lie on a given straight line, only one straight line, parallel to the given one, passes.
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 the third line, then they are parallel (Fig. 5).
The following theorem is also true.
Theorem 2. If two parallel lines are intersected by a secant, then:
criss-crossing angles are equal;
the corresponding angles are equal;
the sum of the one-sided angles is 180 °.
Corollary 2. If a line is perpendicular to one of two parallel lines, then it is 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 the converse, that is, if this theorem is true, then the converse of the theorem may not be true.
Let us explain this by the example of the theorem on vertical corners... This theorem can be formulated as follows: if two angles are vertical, then they are equal. The theorem converse to it would be as follows: if two angles are equal, then they are vertical. And this, of course, is not true. Two equal angle do not 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 corners.
Solution. Let Figure 6 meet the condition.
