Dot. Section

A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the framework of the task, only its location is important

The point is indicated by a number or a capital (large) Latin letter. Several dots - different numbers or different letters so that they can be distinguished

point A, point B, point C

A B C

point 1, point 2, point 3

1 2 3

You can draw three "A" points on a piece of paper and invite the child to draw a line through the two "A" points. But how to understand through which? A A A

A line is a set of points. She only measures length. It has no width or thickness.

Indicated by lowercase (small) with Latin letters

line a, line b, line c

a b c

The line could be

1. closed if its beginning and end are at the same point,
2. open if its beginning and end are not connected

closed lines

open lines

You left the apartment, bought bread in the store and returned back to the apartment. What line did you get? That's right, closed. You have returned to the starting point. You left the apartment, bought bread in the store, went into the entrance and talked to your neighbor. What line did you get? Open. You have not returned to the starting point. You left the apartment, bought bread in the store. What line did you get? Open. You have not returned to the starting point.

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken line
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that does not curve, has neither beginning nor end, it can be extended indefinitely in both directions

Even when seen small plot straight, it is assumed that it continues indefinitely in both directions

It is denoted by a lowercase (small) Latin letter. Or two capital (large) Latin letters - points lying on a straight line

straight line a

a

straight line AB

B A

straight lines can be

1. intersecting if they have a common point. Two lines can only intersect at one point.
   • perpendicular if they intersect at a right angle (90°).
2. parallel, if they do not intersect, they do not have a common point.

parallel lines

intersecting lines

perpendicular lines

A ray is a part of a straight line that has a beginning but no end, it can be extended indefinitely in only one direction

The starting point for the beam of light in the picture is the sun.

Sun

The point divides the line into two parts - two rays A A

The beam is indicated by a lowercase (small) Latin letter. Or two capital (large) Latin letters, where the first is the point from which the beam begins, and the second is the point lying on the beam

beam a

a

beam AB

B A

The beams match if

1. located on the same straight line
2. start at one point
3. directed to one side

rays AB and AC coincide

rays CB and CA coincide

C B A

A segment is a part of a straight line that is bounded by two points, that is, it has both a beginning and an end, which means that its length can be measured. The length of a segment is the distance between its start and end points.

Any number of lines can be drawn through one point, including straight lines.

Through two points - unlimited number of curves, but only one straight line

curved lines passing through two points

B A

straight line AB

B A

A piece was “cut off” from the straight line and a segment remained. From the example above, you can see that its length is the shortest distance between two points. ✂ B A ✂

A segment is denoted by two capital (large) Latin letters, where the first is the point from which the segment begins, and the second is the point from which the segment ends

segment AB

B A

Task: where is the line, ray, segment, curve?

A broken line is a line consisting of successively connected segments not at an angle of 180°

A long segment was “broken” into several short ones.

The links of a polyline (similar to the links of a chain) are the segments that make up the polyline. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The tops of the polyline (similar to the tops of mountains) are the point from which the polyline begins, the points at which the segments forming the polyline are connected, the point where the polyline ends.

A polyline is denoted by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

link of broken line AB, link of broken line BC, link of broken line CD, link of broken line DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

A B C D E 64 62 127 52

The length of a polyline is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

A task: which broken line is longer, but which one has more peaks? At the first line, all the links are of the same length, namely 13 cm. The second line has all the links of the same length, namely 49 cm. The third line has all the links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (they will help you remember the expressions: "go to all four sides", "run towards the house", "which side of the table will you sit on?") are the links of the broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of the polygon are the vertices of the polyline. Neighboring vertices are endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

side CD and side DE are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

A B C D E F 120 60 58 122 98 141

The perimeter of a polygon is the length of the polyline: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, and so on.

Point and line are basic geometric shapes on surface.

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.

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

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

In the figure above, we see that points A and C are located on a straight line. but. In such cases, we say that the points A and C belong to the line a. Or they say that the line passes through points A and C. When writing, the belonging of a point to a 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, the points B and D do not belong to the line a.

As noted above, in the figure, points A and C belong to the line a. The part of a line that consists of all points on that line that lie between two given points is called segment. In other words, a segment is a part of a straight 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 membership of points and lines is the following property: through any two points you can draw a line, and moreover, 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 line can be drawn through two points.

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

Point in math

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

In the figure, you can see the 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 a set of all points lying on a straight line between the ends of a 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.

Straight lines in mathematics

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

The figure shows two straight lines: CD and EF.

Ray 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 point of the beginning of the beam, so you cannot swap the letters.

The figure shows the beams: DC, KC, EF, MT, MS. Beams KC and KD - 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 a number line, as well as a ray OD and ED

The course uses geometric language, made up of notations and symbols adopted in the course of mathematics (in particular, in the new geometry course in high school).

The whole variety of designations and symbols, as well as the connections between them, can be divided into two groups:

group I - designations of geometric figures and relations between them;

group II designations of logical operations, constituting the syntactic basis of the geometric language.

The following is full list mathematical symbols used in this course. Special attention is given to symbols that are used to designate projections of geometric shapes.

Group I

SYMBOLS DESIGNATED GEOMETRIC FIGURES AND RELATIONSHIPS BETWEEN THEM

A. Designation of geometric shapes

1. The geometric figure is denoted - F.

2. Points are indicated capital letters Latin alphabet or Arabic numerals:

A, B, C, D, ... , L, M, N, ...

1,2,3,4,...,12,13,14,...

3. Lines arbitrarily located in relation to the projection planes are indicated by lowercase letters of the Latin alphabet:

a, b, c, d, ... , l, m, n, ...

Level lines are indicated: h - horizontal; f- frontal.

The following notation is also used for straight lines:

(AB) - a straight line passing through the points A and B;

[AB) - a ray with the beginning at point A;

[AB] - a straight line segment bounded by points A and B.

4. Surfaces are denoted by lowercase letters of the Greek alphabet:

α, β, γ, δ,...,ζ,η,ν,...

To emphasize the way the surface is defined, you should specify the geometric elements by which it is defined, for example:

α(a || b) - plane α is determined by parallel lines a and b;

β(d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2 , the generatrix g and the plane of parallelism α.

5. Angles are indicated:

∠ABC - angle with apex at point B, as well as ∠α°, ∠β°, ... , ∠φ°, ...

6. Angular: the value (degree measure) is indicated by the sign, which is placed above the angle:

The value of the angle ABC;

The value of the angle φ.

A right angle is marked with a square with a dot inside

7. Distances between geometric figures are indicated by two vertical segments - ||.

For example:

|AB| - distance between points A and B (length of segment AB);

|Aa| - distance from point A to line a;

|Aα| - distances from point A to surface α;

|ab| - distance between lines a and b;

|αβ| distance between surfaces α and β.

8. For projection planes, the following designations are accepted: π 1 and π 2, where π 1 is the horizontal projection plane;

π 2 -fryuntal plane of projections.

When replacing projection planes or introducing new planes, the latter denote π 3, π 4, etc.

9. Projection axes are denoted: x, y, z, where x is the x-axis; y is the y-axis; z - applicate axis.

The constant line of the Monge diagram is denoted by k.

10. Projections of points, lines, surfaces, any geometric figure are indicated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they were obtained:

A", B", C", D", ... , L", M", N", horizontal projections of points; A", B", C", D", ... , L", M" , N", ... frontal projections of points; a" , b" , c" , d" , ... , l", m" , n" , - horizontal projections of lines; a" ,b" , c" , d" , ... , l" , m " , n" , ... frontal projections of lines; α", β", γ", δ",...,ζ",η",ν",... horizontal projections of surfaces; α", β", γ", δ",...,ζ" ,η",ν",... frontal projections of surfaces.

11. Traces of planes (surfaces) are indicated by the same letters as the horizontal or frontal, with the addition of a subscript 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α.

So: h 0α - horizontal trace of the plane (surface) α;

f 0α - frontal trace of the plane (surface) α.

12. Traces of straight lines (lines) are indicated by capital letters that begin words that define the name (in Latin transcription) of the projection plane that the line crosses, with a subscript indicating belonging to the line.

For example: H a - horizontal trace of a straight line (line) a;

F a - frontal trace of a straight line (line) a.

13. The sequence of points, lines (of any figure) is marked with subscripts 1,2,3,..., n:

A 1, A 2, A 3,..., A n;

a 1 , a 2 , a 3 ,...,a n ;

α 1 , α 2 , α 3 ,...,α n ;

F 1 , F 2 , F 3 ,..., F n etc.

The auxiliary projection of the point, obtained as a result of the transformation to obtain the actual value of the geometric figure, is denoted by the same letter with the subscript 0:

A 0 , B 0 , C 0 , D 0 , ...

Axonometric projections

14. Axonometric projections of points, lines, surfaces are indicated by the same letters as nature with the addition of the superscript 0:

A 0, B 0, C 0, D 0, ...

1 0 , 2 0 , 3 0 , 4 0 , ...

a 0 , b 0 , c 0 , d 0 , ...

α 0 , β 0 , γ 0 , δ 0 , ...

15. Secondary projections are indicated by adding a superscript 1:

A 1 0 , B 1 0 , C 1 0 , D 1 0 , ...

1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ...

a 1 0 , b 1 0 , c 1 0 , d 1 0 , ...

α 1 0 , β 1 0 , γ 1 0 , δ 1 0 , ...

To facilitate the reading of the drawings in the textbook, several colors were used in the design of the illustrative material, each of which has a certain meaning: black lines (dots) indicate the initial data; green color used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements to which special attention should be paid.

B. Symbols Denoting Relations Between Geometric Figures

no.	Designation	Content	Symbolic notation example
1	≡	Match	(AB) ≡ (CD) - a straight line passing through points A and B, coincides with the line passing through points C and D
2	≅	Congruent	∠ABC≅∠MNK - angle ABC is congruent to angle MNK
3	∼	Similar	ΔABS∼ΔMNK - triangles ABC and MNK are similar
4	||	Parallel	α||β - plane α is parallel to plane β
5	⊥	Perpendicular	a⊥b - lines a and b are perpendicular
6		interbreed	with d - lines c and d intersect
7		Tangents	t l - line t is tangent to line l. βα - plane β tangent to surface α
8	→	Are displayed	F 1 → F 2 - the figure F 1 is mapped onto the figure F 2
9	S	projection center. If the projection center is not a proper point, its position is indicated by an arrow, indicating the direction of projection	-
10	s	Projection direction	-
11	P	Parallel projection	p s α Parallel projection - parallel projection to the plane α in the direction s

B. Set-theoretic notation

no.	Designation	Content	Symbolic notation example	An example of symbolic notation in geometry
1	M,N	Sets	-	-
2	A,B,C,...	Set elements	-	-
3	{ ... }	Consists of...	F(A, B, C,... )	Ф(A, B, C,...) - figure Ф consists of points A, B, C, ...
4	∅	Empty set	L - ∅ - the set L is empty (contains no elements)	-
5	∈	Belongs to, is an element	2∈N (where N is the set natural numbers) - the number 2 belongs to the set N	A ∈ a - point A belongs to the line a (point A lies on line a)
6	⊂	Includes, contains	N⊂M - the set N is a part (subset) of the set M of all rational numbers	a⊂α - line a belongs to the plane α (understood in the sense: the set of points of the line a is a subset of the points of the plane α)
7	∪	Union	C \u003d A U B - set C is a union of sets A and B; (1, 2. 3, 4.5) = (1.2.3)∪(4.5)	ABCD = ∪ [BC] ∪ - broken line, ABCD is union of segments [AB], [BC],
8	∩	Intersection of many	М=К∩L - the set М is the intersection of the sets К and L (contains elements belonging to both the set K and the set L). M ∩ N = ∅- intersection of sets M and N is the empty set (sets M and N do not have common elements)	a = α ∩ β - line a is the intersection planes α and β and ∩ b = ∅ - lines a and b do not intersect (have no common points)

Group II SYMBOLS DESIGNATED LOGICAL OPERATIONS

no.	Designation	Content	Symbolic notation example
1	∧	conjunction of sentences; corresponds to the union "and". Sentence (p∧q) is true if and only if p and q are both true	α∩β = ( K:K∈α∧K∈β) The intersection of surfaces α and β is a set of points (line), consisting of all those and only those points K that belong to both the surface α and the surface β
2	∨	Disjunction of sentences; corresponds to the union "or". Sentence (p∨q) true when at least one of the sentences p or q is true (i.e. either p or q or both).	-
3	⇒	Implication is a logical consequence. The sentence p⇒q means: "if p, then q"	(a||c∧b||c)⇒a||b. If two lines are parallel to a third, then they are parallel to each other.
4	⇔	The sentence (p⇔q) is understood in the sense: "if p, then q; if q, then p"	А∈α⇔А∈l⊂α. A point belongs to a plane if it belongs to some line belonging to that plane. The converse is also true: if a point belongs to some line, belonging to the plane, then it also belongs to the plane itself.
5	∀	The general quantifier reads: for everyone, for everyone, for anyone. The expression ∀(x)P(x) means: "for any x: property P(x)"	∀(ΔABC)( = 180°) For any (for any) triangle, the sum of the values ​​of its angles at the vertices is 180°
6	∃	The existential quantifier reads: exists. The expression ∃(x)P(x) means: "there is x that has the property P(x)"	(∀α)(∃a). For any plane α, there exists a line a not belonging to the plane α and parallel to the plane α
7	∃1	The uniqueness of existence quantifier, reads: there is a unique (-th, -th)... The expression ∃1(x)(Px) means: "there is a unique (only one) x, having the property Rx"	(∀ A, B)(A≠B)(∃1a)(a∋A, B) For any two various points A and B there is a single line a, passing through these points.
8	(px)	Negation of the statement P(x)	ab(∃α )(α⊃а, b). If lines a and b intersect, then there is no plane a that contains them
9	\	Negative sign	≠ - the segment [AB] is not equal to the segment .a? b - the line a is not parallel to the line b