Parallel lines. Definition
Two lines in a plane are called parallel if they do not intersect.
The parallelism of lines a and b is denoted as follows: a||b. Figure 1 shows lines a and b perpendicular to line c. Such lines a and b do not intersect, i.e. they are parallel.
Along with parallel lines, they are often considered parallel lines. Two segments are called parallel if they lie on parallel lines. In the figure (Fig. 2, a) the segments AB and CD are parallel (AB||CO) and the segments MN and CD are not parallel. The parallelism of a segment and a straight line (Fig. 2, b), a ray and a straight line, a segment and a ray, and two rays (Fig. 2, c) is determined similarly.
Signs of parallelism of two lines
Line c is called a secant to lines a and b if it intersects them at two points (Fig. 3). When lines a and b intersect with transversal c, eight angles are formed, which are indicated by numbers in Figure 3.
Some pairs of these angles have special names:
crosswise angles: 3 and 5, 4 and 6;
one-sided angles: 4 and 5, 3 and 6;
corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7.
Let's consider three signs of parallelism of two straight lines associated with these pairs of angles.
Theorem. If, when two lines intersect crosswise, the angles involved are equal, then the lines are parallel.
Proof. Let the intersecting lines a and b crosswise the angles AB be equal: ∠1=∠2 (Fig. 4, a).
Let us show that a||b. If angles 1 and 2 are right (Fig. 4, b), then lines a and b are perpendicular to line AB and, therefore, parallel. Let's consider the case when angles 1 and 2 are not right. From the middle O of segment AB we draw a perpendicular OH to straight line a (Fig. 4, c). On straight line b from point B we plot the segment ВН1 equal to the segment AH, as shown in Figure 4, c, and draw the segment OH1. Triangles OHA and OH1B are equal on both sides and the angle between them (AO=VO. AN=BN1 ∠1=∠2), therefore ∠3=∠4 and ∠15=∠16. From the equality ∠3=∠4 it follows that point H1 lies on the continuation of the ray OH, i.e. points H, O and H1 lie on the same straight line, and from the equality ∠5=∠6 it follows that angle 6 is a straight line (so as angle 5 is a right angle). This means that lines a and b are perpendicular to line HH1, so they are parallel. The theorem has been proven.
Theorem. If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.
Proof. Suppose that when lines a and b intersect with transversal c, the corresponding angles are equal, for example ∠1=∠2 (Fig. 5). Since angles 2 and 3 are vertical, then ∠2=∠3. From these two equalities it follows that ∠1=∠3. But angles 1 and 3 are crosswise, so lines a and b are parallel. The theorem has been proven.
Theorem. If, when two lines intersect with a transversal, the sum of the one-sided angles is 180°, then the lines are parallel.
Proof. Let the intersection of straight lines a and b with transversal c sum the one-sided angles equal to 180°, for example ∠1+∠4=180° (see Fig. 5). Since angles 3 and 4 are adjacent, then ∠3+∠4=180°. From these two equalities it follows that the crosswise angles 1 and 3 are equal, therefore lines a and b are parallel. The theorem has been proven.
Practical ways to construct parallel lines
Signs of parallel lines underlie the methods of constructing parallel lines using various tools used in practice. Consider, for example, the method of constructing parallel lines using a drawing square and a ruler. To construct a straight line passing through the point M and parallel to the given straight line a, we will apply a drawing square to the straight line a, and a ruler to it as shown in Figure 103. Then, moving the square along the ruler, we will ensure that the point M is on the side square, and draw straight line b. Straight lines a and b are parallel, since the corresponding angles, indicated in Figure 103 by the letters alpha and beta, are equal.
There is also a way to construct parallel lines using a crossbar. This method is used in drawing practice.
A similar method is used when performing carpentry work, where a block (two wooden planks fastened with a hinge) is used to mark parallel lines.
Occupies a special place in the history of mathematics Euclid's fifth postulate (axiom of parallel lines). For a long time mathematicians tried unsuccessfully to deduce the fifth postulate from the remaining postulates of Euclid and only in the middle of the 19th century thanks to research N. I. Lobachevsky, B. Riman And Y. Bolyai it became clear that the fifth postulate cannot be deduced from the others, and the system of axioms proposed by Euclid is not the only possible one.
Axiom of parallel lines
Even the ancient Greeks came up with a simple way: how to draw a compass and ruler through a point A lying outside a given line l, another line m that does not intersect the line l. But is there the only solution to this problem? Or can you draw several different lines through point A that do not intersect the original line m?
Euclid, apparently, was the first among the Hellenes to understand that the answer to this question cannot be obtained based on other properties of lines and points - those that he formulated in the form of axioms and postulates. It is necessary to introduce an additional postulate about the uniqueness of the desired line m - and call this line parallel!
Are other formulations of the postulate about parallel lines possible - incompatible with Euclid’s postulate? For example, we can assume the existence of several different lines that do not intersect a given line l and pass through common point A. Will such an assumption lead to a logical contradiction or not? If not, then geometries other than Euclidean are possible!
The first non-Euclidean geometry was invented in the 1820s by three talented mathematics: German Carl Gauss, Russian Nikolai Lobachevsky and Hungarian Janos Bolyai. The Russian mathematician turned out to be the most courageous and persistent of the three discoverers. He was the first to publish his book with a prediction remarkable properties non-Euclidean figures. For example, on the Lobachevsky plane the sum internal corners a triangle is always less than 180 degrees. She accepts different meanings for different triangles; with two similar to a triangle necessarily equal!
At the end of the 19th century, geometers Klein and Poincaré invented quite simple models surfaces on which Lobachevsky's geometry is embodied. Even earlier, Riemann noticed that the ordinary sphere embodies the third possible geometry (projective): there are no “parallel” lines in it at all, and the sum of the internal angles of a triangle is always greater than 180 degrees.
Until the early 20th century, it was believed that non-Euclidean geometries could only be useful internally mathematical science. But in the 1910s, Einstein created General Theory Relativity: it turned out to be a four-dimensional embodiment of Lobachevsky’s non-Euclidean geometry. Since then, physicists have believed that every consistent mathematical construct is embodied somewhere in Nature. This may be true.
Historical reference
In ancient times, literally 2500 years ago, in famous school Pythagoras Greek word“parallelos” began to be used as a geometric term, although the definition of parallel lines was not yet known at that time. But historical facts they say that the ancient Greek scientist Euclid in the third century BC, in his books, nevertheless revealed the meaning of such a concept as parallel lines.
As you already know, from the material covered in previous classes, the term “parallelos” translated from Greek language means walking next to or held near each other.
In mathematics, there is a special sign to indicate parallel lines. True, the parallelism sign did not always have its current form. For example, the ancient Greek mathematician Pappus in the third century AD used the equal sign “=” to indicate parallelism. And only in the eighteenth century, thanks to William Oughtred, they began to use the sign “//” to denote parallel lines. If there are, for example, parallel a and b, then they should be written in writing as a//b
But the “=” sign was introduced into general circulation by Record and began to be used as an equals sign.
Parallel lines in everyday life
We often encounter parallel lines in the life around us, although, as a rule, we rarely focus our attention on it. During music lessons, when we open a music book, we immediately see the lines of the staff with the naked eye. But you can see parallel lines not only in music books and songbooks, but also if you look closely at musical instruments. After all, the strings of a guitar, harp or organ are also parallel.
Looking up on the street, you see electrical wires running parallel. Finding yourself on the subway or railway, it is also not difficult to notice that the rails are located parallel to each other.
Parallel lines can be found everywhere. We constantly encounter them in everyday life and painting. Architecture cannot do without them, since the concept of parallelism is strictly taken into account in the construction of buildings.
If you look closely at the image, you will immediately notice the presence of parallel lines in these architectural structures. Perhaps they last so long and remain beautiful because the architects and engineers used parallel lines when creating these iconic buildings.
Have you ever wondered why the wires in power lines are arranged in parallel? And imagine what would happen if they were not parallel and intersected or touched each other. And this would lead to bad consequences, in which a short circuit could occur, interruptions and lack of electricity. What could happen to the train if the rails were not parallel? It's scary to even think about it.
You all know well that parallel lines never intersect. But if you look into the distance for a long time, into infinity, you may eventually see how parallel lines intersect. In this case, we are faced with an illusion of vision. Perhaps it was only thanks to such illusions and visual distortions that painting appeared.
Homework
1. Give your examples of where you are Everyday life, in everyday life or in nature, you come across moments or facts of parallelism.
2. What methods do you know by which you can draw parallel lines? Name these methods.
3. Draw parallel lines in your notebook using methods that you know.
4. Under what conditions can straight lines be called parallel?
Questions:
1. Which lines are called parallel?
2. What practical ways constructions of parallel lines exist.?
In any design training course, they teach you to use thin auxiliary lines when creating drawings. Previously, they were applied on a drawing board and then erased from the finished document. Currently in use electronic programs for a drawing, but the need for auxiliary lines is not even discussed. Although in Compass 3D it is even easier to work with them than on a classic drawing board. Auxiliary lines are used to form necessary connections, marking the drawing, creating certain boundaries.
The program allows you to create auxiliary lines in several ways, again, this is very convenient, since sometimes one is used, and in another situation a different method of drawing auxiliary lines is used.
1. Create a straight line using two points.
One of the most popular methods. To activate, you must open the main menu Tools – Geometry - Auxiliary lines - Auxiliary line.
Or you can click in the panel Geometry-Auxiliary line.
Let's set our line by left-clicking on the sheet, so defining the first point, then specify end point lines. At the same time, the program itself will generate the required angle of inclination for the created straight line. However, you can change the angle by entering your values in the box below, then just click Enter.
The auxiliary line has been formed, now you need to click on the familiar icon Abort command, located in the properties panel. However, you can activate this command after finishing working with the line by simply right-clicking the mouse and then selecting the appropriate item in the drop-down menu.
Using a base point you can create infinite number straight lines going at any angles. By the way, if you have the coordinates or with coordinate grid work more conveniently, then you can always ask required values in the menu below. You will place a straight line, without any adjustments, on the sheet. Worth paying attention to the group Modes, it has two important switches. The first one is active during standard startup - Don't put intersection points, and you can choose the second one yourself - Set intersection points. Using this setting, you can automatically place points at any intersections, without additional options or manual placement.
However, here you need to specify the style Auxiliary. By the way, to remove all auxiliary elements, with finished drawing just activate the item in the main menu Editor-Delete-Auxiliary curves and points. We discussed working with points on curves in detail in lesson #3.
2.Draw a horizontal line
You can build auxiliary lines using horizontal lines. Let's open the already familiar menu Tools-Geometry-Auxiliary Lines-Horizontal Line.
A faster option, using a compact panel, select Geometry - Horizontal straight line. However, the basic panel will not be visible on the screen; to correct the situation, press the auxiliary lines button and hold it for a while.
All that remains is to use a left-click to indicate the desired point through which we will pass our straight line. You can create any number horizontal lines. To finish the job, just click Abort command in the properties panel or in the drop-down menu, right-click.
You also need to remember that a horizontal straight line is always parallel to the current x-axis. However, when setting horizontal lines using a rotated coordinate system, they will not be horizontal on the sheet.
3. Draw a vertical straight line.
The general mechanism for calling the line drawing mechanism is absolutely identical to that described above, with the exception of the choice Vertical straight.
However, there are a few important things to remember here. The created vertical straight line is always parallel only to the actual coordinate axis; here the case is identical to the horizontal straight line. Therefore, if you have a modified coordinate system, vertical straight lines will not be parallel to the sheet.
4. Create a parallel straight line.
You can build a parallel straight line only if there is any object on the sheet. It is to these lines that we will create a parallel. Moreover, absolutely any object can act as objects for snapping, from straight and auxiliary lines to the faces of polygonal objects. So, as part of the lesson, let’s take as the main one the horizontal line that goes from the origin of coordinates on our sheet.
Calling a parallel straight line is identical, open Tools – Geometry - Auxiliary lines - Parallel line.
Or use a compact panel, here you need to call Geometry-Parallel Line.
Now let’s indicate the base object to which we will conduct parallel line. As agreed, the object is a horizontal straight line, select it with the mouse. Then, we need to set the distance at which our parallel line will be located. Below you can specify numeric value, for example 30 mm, or pull it straight with the mouse to the desired distance.
When specifying the distance in numbers, the system will offer two phantom lines at the same distance. This can be disabled if in the properties Number of lines - Two lines remove the activation, transforming it into the creation of one straight line. To fix the created line, just select the active phantom using the mouse and click on the create object button. When you need to create both lines, click Create Object again and then abort the command.
When you need to build a new parallel line, but near another object, just press the button Specify again. Now you can specify new object and build a line in the manner described in this chapter of the lesson.
That's all, in this lesson we covered the basics of creating auxiliary straight lines.
The methods for constructing parallel lines using various tools are based on the signs of parallel lines.
Constructing parallel lines using a compass and ruler
Let's consider the principle of constructing a parallel line passing through a given point, using a compass and ruler.
Let a line be given and some point A that does not belong to the given line.
It is necessary to construct a line passing through a given point $A$ parallel to the given line.
In practice, it is often necessary to construct two or more parallel lines without a given line and point. In this case, it is necessary to draw a straight line arbitrarily and mark any point that will not lie on this straight line.
Let's consider stages of constructing a parallel line:
In practice, they also use the method of constructing parallel lines using a drawing square and a ruler.
Constructing parallel lines using a square and ruler
For constructing a line that will pass through point M parallel to a given line a, necessary:
- Apply the square to the straight line $a$ diagonally (see figure), and attach a ruler to its larger leg.
- Move the square along the ruler until given point$M$ will not be on the diagonal of the square.
- Draw the required straight line $b$ through the point $M$.
We have obtained a line passing through a given point $M$, parallel to a given line $a$:
$a \parallel b$, i.e. $M \in b$.
The parallelism of straight lines $a$ and $b$ is evident from the equality of the corresponding angles, which are marked in the figure with the letters $\alpha$ and $\beta$.
Construction of a parallel line spaced at a specified distance from a given line
If it is necessary to construct a straight line parallel to a given straight line and spaced from it at a given distance, you can use a ruler and a square.
Let a straight line $MN$ and a distance $a$ be given.
- On the given straight line $MN$ we mark arbitrary point and let's call it $B$.
- Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
- On the straight line $AB$ from the point $B$ we plot the segment $BC=a$.
- Using a square and a ruler, we draw a straight line $CD$ through the point $C$, which will be parallel to the given straight line $AB$.
If we plot the segment $BC=a$ on the straight line $AB$ from point $B$ in the other direction, we get another parallel line to the given one, spaced from it by specified distance$a$.
Other ways to construct parallel lines
Another way to construct parallel lines is to construct using a crossbar. More often this method used in drawing practice.
When performing carpentry work for marking and constructing parallel lines, a special drawing tool is used - a clapper - two wooden planks that are fastened with a hinge.
The construction of a straight line parallel to a given plane is based on
the following position, known from geometry: a straight line is parallel to the plane,
if this line is parallel to any line in the plane.
Through a given point in space one can draw countless
a set of straight lines parallel to a given plane: To obtain
The only solution requires some additional condition.
For example, through a point (Fig. 180) you need to draw a straight line,
parallel to the plane given triangle ABC, and projection planes!
(additional condition).
Obviously, the desired straight line must be parallel to the intersection line
both planes, i.e. must be parallel to the horizontal track
planes, given by a triangle ABC. To determine the direction of this
trace, you can use the horizontal plane defined by the triangle
ABC. In Fig. 180 draw horizontal line DC and then draw through point M
line parallel to this horizontal line.
Let us pose the inverse problem: draw a plane through a given point,
parallel to a given straight line. Planes passing through some
point A parallel to some straight line BC, form a bundle of planes, the axis
which is a straight line passing through point A parallel to straight line BC.
To obtain a unique solution, some additional
For example, you need to draw a plane parallel to straight line CD, not through
point, and through straight line AB (Fig. 181). Direct AB and CD are crossing. If
it is required to draw a plane through one of two intersecting lines,
parallel-
Rice. 180 Fig. 181
different, then the problem has a unique solution. Through point B
a straight line parallel to straight CD is drawn; straight lines AB and BE determine
plane parallel to straight line CD.
How to determine whether a given line is parallel to a given plane?
You can try to draw a parallel line in this plane
this line. If such a straight line in the plane cannot be constructed, then
the given straight line and plane are not parallel to each other.
You can also try to find the point of intersection of a given line with a given one.
flat. If such a point cannot be found, then the given straight line and
plane are mutually parallel.
§ 28. Construction of mutually parallel planes
Let a point K be given through which a plane must be drawn,
parallel to some plane defined by the intersecting lines AF and BF
Obviously, if through point K we draw straight lines SK and DK, respectively
parallel to the lines AF and BF, then the plane defined by the lines CK and DK,
will be parallel given plane.
Another example of construction is given in Fig. 183 on the right. Through point A
carried out pl. parallel to the square A. First, a straight line is drawn through point A,
obviously parallel square. . This is a horizontal line with projections "" and "",
and A"N"\\h "o. So
Rice. 182 Fig. 183
since point N is the frontal trace of the horizontal line AN, then through this
the trace f"o% f"o will pass the point, and the trace h"o || h"o will pass through X. Planes
and are mutually parallel, since their intersecting traces of the same name are mutually
parallel.
In Fig. 184 shows two planes parallel to each other - one
one of them is given by the triangle LAN, the other by parallel lines DE and FG.
How is the parallelism of these planes established? Those that are in the plane,
given by the lines DE and FG, it turned out to be possible to draw two intersecting
straight lines KN and KM, respectively parallel to the intersecting straight lines AC and
Sun of a different plane.
Of course, one could try to find the intersection point at least
line DE with the plane of triangle ABC. Failure would confirm
parallelism of planes.
QUESTIONS FOR §§ 27-28
1. What is the basis for constructing a straight line, which should be
parallel to some plane?
2. How to draw a plane through a line parallel to a given line?
3. What determines the mutual parallelism of two planes?
4. How to draw a plane parallel to a given plane through a point?
5. How to check in a drawing whether the given values are parallel to one another