The rule is what is a ray and a segment. Gaps in geometry (line, angle, ray, segment, straight line, curve, closed line)

Point and line are the main geometric figures on the plane.

The ancient Greek scientist Euclid said: “a point” is that which has no parts.” The word "point" in translation from Latin means the result of an instant touch, a prick. The point is the basis for constructing any geometric figure.

A straight line or just a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire line and measure it.

The dots are capitalized. with Latin letters A, B, C, D, E, etc., and straight lines with the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on it. For example, the line a can be denoted by AB.

We can say that the points AB lie on the line a or belong to the line a. And we can say that the line a passes through the points A and B.

The simplest geometric figures on a plane are a line segment, a ray, broken line.

A segment is a part of a line, which consists of all points of this line, bounded by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.

A ray or half-line is a part of a line, which consists of all points of this line, lying on one side of its given point. This point is called the starting point of the half-line or the beginning of the ray. A ray has a start point but no end point.

Half-lines or rays are denoted by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.

It turns out that the line is infinite: it has neither beginning nor end; a ray has only a beginning but no end, while a segment has a beginning and an end. Therefore, we can only measure a segment.

Several segments that are connected in series with each other so that the segments (adjacent) having one common point are not located on the same straight line represent a broken line.

The polyline can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if not, an open one.

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Point and line are the main geometric figures on the plane.

The ancient Greek scientist Euclid said: “a point” is that which has no parts.” The word "point" in Latin means the result of an instant touch, a prick. The point is the basis for constructing any geometric figure.

A straight line or just a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire line and measure it.

Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, the line a can be denoted by AB.

We can say that the points AB lie on the line a or belong to the line a. And we can say that the line a passes through the points A and B.

The simplest geometric figures on a plane are a segment, a ray, a broken line.

A segment is a part of a line, which consists of all points of this line, bounded by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.

A ray or half-line is a part of a line, which consists of all points of this line, lying on one side of its given point. This point is called the starting point of the half-line or the beginning of the ray. A ray has a start point but no end point.

Half-lines or rays are denoted by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.

It turns out that the line is infinite: it has neither beginning nor end; a ray has only a beginning but no end, while a segment has a beginning and an end. Therefore, we can only measure a segment.

Several segments that are connected in series with each other so that the segments (adjacent) having one common point are not located on the same straight line represent a broken line.

The polyline can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if not, an open one.

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visiting extra classes we realized that we can’t operate with the concepts of a point, line, angle, ray, segment, straight line, curve, closed line and draw them, we can draw more precisely, but we can’t identify them.

Children must distinguish between lines, curves, circles. This develops their graphics and a sense of correctness when drawing, appliqué. It is important to know what basic geometric shapes exist, what they are. Lay out the cards in front of the child, ask them to draw exactly the same as in the picture. Repeat several times.

During the course, we were given the following materials:

A small fairy tale.

In the country of Geometry there lived a point. She was small. It was left by a pencil when it stepped on a sheet of notebook, and no one noticed it. So she lived until she came to visit the lines. (Drawing on the board.)

Look at the lines. (Straight and curved.)

Straight lines are like strings stretched, and strings that are not pulled are crooked lines.

How many straight lines? (2.)

How many curves? (3.)

The straight line began to show off, “I am the longest! I have no beginning and no end! I am infinite!

It became very interesting to look at her point. The point itself is tiny. She went out and was so carried away that she did not notice how she stepped on a straight line. And suddenly the straight line disappeared. A beam appeared in its place.

It was also very long, but still not like a straight line. He got a start.

The point was frightened: “What have I done!” She wanted to run away, but as luck would have it, she stepped on the beam again.

And a segment appeared in place of the beam. He didn't brag about how big he was, he already had a beginning and an end.

This is how a small dot could change the life of large lines.

So who guessed who came to visit us with the cat? (straight line, ray, segment and point)

Correctly, along with the cat, a straight line, a ray, a segment and a point came to our lesson.

Who guessed what we will do in this lesson? (Learn to recognize and draw a straight line, ray, segment.)

What lines did you hear about? (About a straight line, a ray, a segment.)

What did you learn about the straight line? (It has neither beginning nor end. It is endless.)

(We take two spools of thread, pull them, depicting a straight line, and unwinding one or the other, demonstrates that the straight line can be continued in both directions to infinity.)

What did you learn about the beam? (It has a beginning, but no end.) (The teacher takes the scissors, cuts the thread. Shows that now the line can only be continued in one end.)

What did you learn about the segment? (It has both a beginning and an end.) (The teacher cuts off the other end of the thread and shows that the thread does not stretch. It has both a beginning and an end.)

How to draw a straight line? (Draw a line along the ruler.)

How to draw a line? (Put two dots and connect them.)

And of course the prescription:

In the lesson, you will get acquainted with the concept of a plane, with various minimal figures that are in geometry, and study their properties. Learn what a line, line segment, ray, angle, etc.

We depict all geometric shapes on a sheet of paper with a pencil, on a school board with chalk or a marker. Often in the summer we draw figures on the pavement with chalk or a white stone. And always, before we start drawing our plans, we evaluate whether there is enough space for us. And since we rarely know exact dimensions our future drawing, then you always need to take places with a margin, and better with a large margin. Usually we are not afraid that the drawing space will run out if the drawing field is many times larger than the drawing itself. So the asphalt in the yard is quite enough to draw a field for jumping. A notebook sheet is enough to draw two intersecting segments in the middle.

In mathematics, such a field on which we depict everything is a plane (Fig. 1).

Rice. 1. Plane

It has two qualities:

1. On it you can depict any figure that we have already talked about, or we will still talk about.

2. We will not reach the edge. Its dimensions can be considered much larger than the dimensions of the figure.

The fact that we never reach the edge of the plane can be understood as the absence of edges at all. We do not need its edges, so we agreed to consider that they do not exist (Fig. 2).

Rice. 2. The plane is infinite

In this sense, the plane is infinite in any direction.

We can represent it as big leaf paper, a large flat asphalt pad or a huge drawing board.

There are an infinite number of geometric shapes, and it is absolutely impossible to study them all. But geometry is arranged much like a constructor. There are several types of basic parts from which you can build everything else, any most complex building.

This principle can be compared to words and letters: we know all the letters, but we do not know all the words. Having met an unfamiliar word, we will be able to read it, because we know how letters are written and how the corresponding sounds are pronounced.

So in mathematics - there are very few basic geometric shapes that you and I need to know well.

Consider a segment (Fig. 3). The cut is shortest line connecting two points.

Rice. 3. Cut

We continue the segment in both directions to infinity. We will continue straight ahead.

What does "straight" mean? Consider the segments and (Fig. 4).

Rice. 4. Segments and

Let's continue on both sides. The top line is straight, but the bottom line is not (Fig. 5).

Let's add one more point to the upper and lower lines and (Fig. 6). The part of the upper line between the points and is also a segment, but the part of the lower line between the points and the segment is not, since it does not connect these points along the shortest path.

Rice. 6. Continuation of lines and

A straight line is a line that continues indefinitely in both directions, any part of which, bounded by two points, is a segment.

A straight line is a type of line, and like any line, a straight line is a shape. And, as for any line, a given point either belongs to a given line or does not (Fig. 7).

Rice. 7. Points and belonging to the line, and points and not belonging to the line

1. The straight line divides the plane into two parts, into two half-planes. In figure 8, the points and lie in the same half-plane, and and - in different half-planes.

Rice. 8. Two half-planes

2. It is always possible to draw a straight line through two points, and only one (Fig. 9).

A straight line, like any line, can be marked with one lower case Latin alphabet or a sequence of points that lie on it. To designate a line through the points lying on it, two points are sufficient.

Extending the segment in both directions to infinity, we got a straight line. If we also extend the segment, but only in one direction to infinity, we get a figure called a ray (Fig. 10). This geometric ray is very similar to a light ray, hence its name. If you take it in hand laser pointer, then the beam of light will start at the pointer and go to infinity in a straight line.

Rice. 10. Beam

The point is called the beginning of the beam. Ray is denoted.

If you mark a point on a straight line, then it divides this straight line into two rays (Fig. 11). Both rays originate at point , but are directed in different directions. These two rays make up a straight line, are its halves. Therefore, the beam is often also called "half-line".

Rice. 11. A point divides a line into two rays

Consider Figure 12.

Rice. 12. Segment, line and beam

Let's figure out how a segment, a straight line and a ray are similar and not similar to each other:

The segment and the beam are easily completed to a straight line, for this the segment must be continued in both directions, and the beam in one;

On a straight line, you can always select a segment or a ray;

A point divides a line into two rays, into two half-lines;

Points and limit on a straight line segment ;

All these figures: a segment, a ray, a straight line - are "straight lines". They differ in the presence of ends. A segment has two, a ray has one, and a straight line has none. Otherwise, we can also say this: both the ray and the segment are part of a straight line;

We know that the length of a segment can be measured. Two segments can be compared, find out which one is longer;

The straight line continues indefinitely in both directions, the ray - in one direction. For this reason, it is impossible to measure the length of a straight line or a beam, and it is also impossible to compare two straight lines or two beams in length. They are all equally endless.

Two rays, having their origins at one point, form another geometric figure from the main set - angle. The point at the beginning of both rays is called the apex of the angle. The rays themselves are called the sides of the angle.

So, an angle is a figure consisting of two rays coming out of one point (Fig. 13).

Rice. 13. Angle

Designate the corner with one letter corresponding to the designation of the vertex. In this case, the angle can be called an angle (Fig. 14). To make it clear that we are talking about an angle, and not about a point, you must write the word “angle” before its name or put a special angle symbol (“”).

Rice. 14. Angle

If at the top it is difficult to understand which particular angle in question, as in Figure 15, then use two more points on both sides of the corner.

If we simply name the angle in this figure, then it is not clear which one we are talking about, because with the vertex at the point we see several angles. Therefore, we add a point to the sides of the angle we need and denote the angle as (Fig. 15).

Rice. 15. Angle

It is possible to go in the opposite direction when designating, but so that again the vertex is in the middle of the record.

Another common designation is one Greek letter: alpha, beta, gamma and so on (Fig. 16). In this case, the letter is usually entered inside the corner (Fig. 17).

Rice. 16. Greek alphabet

Rice. 17. The name of the corner written inside the corner

So, in Figure 18, the designations , , are equivalent, they denote the same angle.

Rice. 18. , , - the same angle

Let two lines and intersect at a point (Fig. 19). The point divides each line into two rays, that is, a total of 4 rays. Each pair of rays defines an angle.

Rice. 19. Straight and form 4 beams

For example, , , .

Through two points and you can always draw a line. Is it the same with three dots?

In Figure 20, a straight line can be drawn through three points, but not in Figure 21.

Rice. 20. A line can be drawn through three points

Rice. 21. You can't draw a straight line through three points

Three points in the figure are said to lie on the same straight line. So they say, even if the line itself is not drawn, simply implying that it can be drawn. In the second case, the points are said not to lie on the same line, implying that it is impossible to draw a line through all three points.

If we connect in series first the 1st and 2nd points, then the 2nd and 3rd, then the resulting line is called a broken line (Fig. 22). The name follows from its appearance.

Rice. 22. broken line

Similarly, a broken line can connect any number of points. Points , , , , are called polyline vertices, segments , , , are called polyline links.

The broken line is denoted by its vertices .

Rice. 23. broken line

If the last point is connected to the first, then the resulting polyline is called closed (Fig. 24).

Rice. 24. Closed polyline

What polyline can be constructed with minimum set vertices and links? If there are two points, then they can be connected by a segment. This will be the most simple example polyline: two vertices and one link connecting them. We can say that a segment is a minimal polyline.

If it is required that the polyline be closed, then the simplest such polyline is a triangle. If you take two points, then connect the last point to the first one only with the same segment that already exists. That is, the broken line will remain, as before, open. And if you add one more point that does not lie on the same line with the points and , connect all the points with three segments, you get a triangle (Fig. 25).

Rice. 25. Triangle

A triangle is a closed polyline with three vertices. Or even like this: triangle is the smallest closed polyline.

The points , and are the vertices of the triangle. The segments connecting them, the links of the broken line, are called the sides of the triangle.

A triangle is denoted by its vertices. For example, . Before the designation, you must put the word "triangle" or a special triangle symbol ("").

A triangle has three angles. Two sides come from each of the vertices, that is, the sides of the triangle are the sides of the corners (Fig. 26).

Rice. 26. Angles of a triangle

Thus, the triangle has three vertices (three points , and ), three sides (three segments , and ).