How to denote a straight line. The relative position of lines on a plane. Introduction to new material

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§1. Control questions
Question 1. Give examples of geometric shapes.
Answer. Examples of geometric shapes: triangle, square, circle.

Question 2. Name the basic geometric shapes on a plane.
Answer. The main geometric figures on a plane are a point and a straight line.

Question 3. How are points and lines designated?
Answer. Points are designated in capital Latin letters: A, B, C, D, …. Direct lines are designated by lowercase Latin letters: a, b, c, d, ….
A straight line can be denoted by two points lying on it. For example, line a in Figure 4 can be labeled AC, and line b can be labeled BC.

Question 4. Formulate the basic properties of membership of points and lines.
Answer. Whatever the line, there are points that belong to this line and points that do not belong to it.
Through any two points you can draw a straight line, and only one.
Question 5. Explain what a line segment with ends at these points is.
Answer. A segment is a part of a line that consists of all points of this line lying between two given points. These points are called the ends of the segment. A segment is indicated by indicating its ends. When they say or write: “segment AB,” they mean a segment with ends at points A and B.

Question 6. Formulate the basic property of the location of points on a straight line.
Answer. Of the three points on a line, one and only one lies between the other two.
Question 7. Formulate the basic properties of measuring segments.
Answer. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
Question 8. What is the distance between two given points?
Answer. The length of segment AB is called the distance between points A and B.
Question 9. What properties does the division of a plane into two half-planes have?
Answer. Partitioning a plane into two half-planes has the following property. If the ends of a segment belong to the same half-plane, then the segment does not intersect the line. If the ends of a segment belong to different half-planes, then the segment intersects a line.

We will look at each of the topics, and at the end there will be tests on the topics.

Point in mathematics

What is a point in mathematics? A mathematical point has no dimensions and is designated by capital letters: A, B, C, D, F, etc.

In the figure you can see an image of points A, B, C, D, F, E, M, T, S.

Segment in mathematics

What is a segment in mathematics? In mathematics lessons you can hear the following explanation: a mathematical segment has a length and ends. A segment in mathematics is the set of all points lying on a straight line between the ends of the segment. The ends of the segment are two boundary points.

In the figure we see the following: segments ,,,, and , as well as two points B and S.

Direct in mathematics

What is a straight line in mathematics? The definition of a straight line in mathematics is that a straight line has no ends and can continue in both directions indefinitely. A line in mathematics is denoted by any two points on a line. To explain the concept of a straight line to a student, you can say that a straight line is a segment that does not have two ends.

The figure shows two straight lines: CD and EF.

Beam in mathematics

What is a ray? Definition of a ray in mathematics: a ray is a part of a line that has a beginning and no end. The name of the beam contains two letters, for example, DC. Moreover, the first letter always indicates the starting point of the beam, so letters cannot be swapped.

The figure shows the rays: DC, KC, EF, MT, MS. Beams KC and KD are one beam, because they have a common origin.

Number line in mathematics

Definition of a number line in mathematics: a line whose points mark numbers is called a number line.

The figure shows the number line, as well as the ray OD and ED

In geometry, the main geometric figures are the point and the line. To designate points, it is customary to use capital Latin letters: A, B, C, D, E, F.... To denote straight lines, lowercase Latin letters are used: a, b, c, d, e, f .... The figure below shows straight line a, and several points A, B, C, D.

To depict a straight line in the drawing, we use a ruler, but we do not depict the entire straight line, but only a piece of it. Since the straight line in our representation extends to infinity in both directions, the straight line is infinite.

In the figure presented above we see that points A and C are located on a straight line A. In such cases, they say that points A and C belong to line a. Or they say that a straight line passes through points A and C. When writing, the belonging of a point to a straight line is indicated by a special icon. And the fact that the point does not belong to the line is marked with the same icon, only crossed out.

In our case, points B and D do not belong to straight line a.

As noted above, in the figure points A and C belong to straight line a. The part of a line that consists of all the points of this line lying between two given points is called segment. In other words, a segment is a part of a line bounded by two points.

In our case we have a segment AB. Points A and B are called the ends of the segment. In order to designate a segment, its ends are indicated, in our case AB. One of the main properties of belonging of points and lines is the following property: through any two points you can draw a straight line, and only one.

If two lines have a common point, then the two lines are said to intersect. In the figure, lines a and b intersect at point A. Lines a and c do not intersect.

Any two lines have only one common point or no common points. If we assume the opposite, that two lines have two points in common, then two lines would pass through them. But this is impossible, since only one straight line can be drawn through two points.

Main geometric shapes on a plane there are a point and a straight line. Points are usually denoted in capital Latin letters:
A, B, C, D, ... .

Direct lines are indicated in lowercase Latin letters:
a, b, c, d
In Figure 3 you see point A and straight line a.
infinite. In the figure we depict only part of the line, but imagine it extended indefinitely in both directions.

Look at Figure 4. You see straight lines a, b and points A, B, C. Points A to C lie on straight line a. We can also say that points A and C belong to straight a or that line a passes through points A and C.

Point B lies on line b. It does not lie on line a. Point C lies on both line a and line b. Lines a and b intersect at point C. Point C is the point of intersection of lines a and b.
In Figure 5 you see how a straight line is constructed using a ruler passing through two given points A and B.

We will call the following properties the main properties of belonging of points and lines on a plane:

I. Whatever the line, there are points that belong to this line and points that do not belong to it.

Through any two points you can draw a straight line, and only one.

A straight line can be denoted by two points lying on it. For example, straight line o in Figure 4 can be designated AC, and straight line b can be designated BC.

Problem (3)". Can two lines have two points of intersection? Explain the answer.

Solution. If two lines had two points of intersection, then two lines would pass through these points. But this is impossible, since only one straight line can be drawn through two points. This means that two straight lines cannot have two points of intersection.

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions