Beam and angle measurement of angles. Point, line, straight line, ray, segment, broken line

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

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

point A, point B, point C

point 1, point 2, point 3

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

A line is a set of points. Only the length is measured. It has no width or thickness

Indicated by lowercase (small) in Latin letters

line a, line b, line c

The line may 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

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. direct
2. broken
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

straight line a

straight line AB

Direct may be

1. intersecting if they have common point. Two lines can intersect only at one point.
   • perpendicular if they intersect at right angles (90°).
2. Parallel, if they do not intersect, 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 continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

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

ray a

beam AB

The rays coincide if

1. located on the same straight line
2. start at one point
3. directed in one direction

rays AB and AC coincide

rays CB and CA coincide

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

Through one point you can draw any number of lines, including straight lines

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

curved lines passing through two points

straight line AB

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 (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

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

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

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. 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 vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

A broken line is designated 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

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

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

Task: which broken line is longer, A which has more vertices? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all links of the same length, namely 41 cm.

A polygon is a closed polyline

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

The vertices of a polygon are the vertices of a broken line. Neighboring Peaks- these are the points of the ends 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

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

The perimeter of a polygon is the length of the broken line: 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, etc.

Lesson 14

Beam. Number beam. Corner. Types of angles. Construction right angle using a compass and ruler

Goals : Recognition and image of geometric shapes: points, straight lines, right angles. Measuring the length of a segment and constructing a segment of a given length Constructing a right angle on checkered paper

Planned results :

Know concepts of “ray”, “numerical ray”.Be able to recognize geometric shapes and draw them on lined paper, draw a ray and a number rayKnow concept of “angle”, types of angles.Be able to recognize geometric shapes and draw them on lined paper, construct a right angle.

Lesson progress

1. Organizational moment

2. Updating knowledge

Checking homework

3. Work on the topic of the lesson:

In this lesson we will look at ray and number ray. First, we will recall the concepts of “straight line”, “segment” and “ray”, and consider their differences. Let's introduce the concept of a number ray, get acquainted with the history of its origin and solve a number of examples.

Look at the first drawing (Fig. 1) and say what is the difference between a ray and a straight line and a segment.

Rice. 1. Segment, ray and straight line

Solution : 1. Straight can be continued as much as desired in both directions - an endless line that has no ends or boundaries.

2. Segment - part of a straight line that is limited on both sides. So, in Figure 1, segment is.

3. Part of a straight line bounded by a point on one side –beam . The drawing (Fig. 1) shows a ray with a beginning at the point. The beam can be extended in a straight line only in one direction.

Consider a ray with origin at point(Fig. 2). Let's put it on it equal segments – single segments . Unit segments can be equal to any value: one cell, one centimeter, three centimeters. The main thing is that every next unit segment was equal to the previous one. If we number these segments with numbers, we getnumber ray .

Rice. 2. Number beam

You can use the number line to represent any number because it is infinite. It is also very easy to compare numbers: the further to the right a point is from the beginning of the ray, the larger the number we are faced with.

Corner. Types of angles. Constructing a right angle using a compass and ruler

Beam - this is a part of a straight line, limited on one side by a point. In the figure you can see a beam with a beginning at a point and a beam with a beginning at a point (Fig. 1).

Rice. 1. Rays

A figure formed by two rays with the same origin is called angle. The rays that form an angle are called sides of the angle, and their general beginning – vertex of the angle(Fig. 2).

Rice. 2. Angles

An angle can be named by one capital Latin letter based on its vertex. In Fig. 2 you can see the angle and angle . But angles can be designated in another way.

The angle of a polygon is denoted by three in capital letters. Naming an angle starts with the letter on one side, then names the letter at the apex, and ends with the letter on the other side. For example, in a triangle, the angle with the vertex is the angle (Fig. 3) or in reverse order – .

In a triangle, the angle with a vertex is the angle or.

Rice. 3. Angles in a triangle

It must be remembered that in the middle of the name of the angle there should be the letter that indicates the vertex of the angle.

Sometimes an angle is indicated by a small letter or number, placing them inside the angle (Fig. 4). For clarity, a bow is drawn between the sides of the angle.

Rice. 4. Designating an angle with a letter or number

Rice. 5. Types of angles

There are various types corners

1. If the sides of an angle lie on the same straight line, then such an angle is called expanded. In Fig. 6 corner M – unfolded (comparison with an unfolded fan is appropriate).

Rice. 6. Full angle

2. Direct An angle is the angle that is half of the unfolded angle (Fig. 7). For example, a right angle can be obtained by folding paper (if the sheet is folded twice).

Rice. 7. Right angle

To make it easier to determine whether a right angle is right or not, there is a special tool - right triangle, in which one of the angles is straight (Fig. 8).

Rice. 8. Right triangle and its application

3. Oblique angles are divided into stupid And spicy.

An angle that is less than a right angle is spicy angle (Fig. 9).

Rice. 9. Acute angle
An angle that is greater than a right angle but less than a straight angle is blunt angle (Fig. 10).

Rice. 10. Obtuse angle

Find straight, obtuse and acute angles in the drawing (Fig. 11).

Rice. 11. Illustration for the task

A tool will help us find a solution - a right triangle, which will be applied to each of the vertices of the triangle by combining one of the sides. If it coincides with an angle, then this angle is right. If the angle is less than the right angle of the tool, then this angle is acute. And if the angle is greater than the right angle of the tool, then this obtuse angle.

Right angles:

Obtuse angles:

Sharp corners: , , ,

Explanation of new material

So we have reached the land of Geometry. And the queen of this country, Dot, meets us. Without it, not a single figure can be built.

Once upon a time there was a Point. She was very curious and wanted to know everything. Dot will see an unfamiliar line and will certainly ask:

What is this line called, is it long or short?

One day Dot thought: “How will I know everything if I sit in one place all the time. I’ll go on a trip.” No sooner said than done. The Point came out on a straight line and walked along this line.

She walked, walked, walked for a long time. Tired. And the Dot says: “How long will I continue to walk along this line?”

Guys! Is the straight line coming to an end soon?

Are you saying that a straight line has no end? Then I'll turn back, I probably went in the wrong direction.

Guys! Will the Point be able to find the end of a straight line?

Of course he can’t, a straight line has no end.

Without end and edge

The line is straight!

Walk along it for at least a hundred years

You can't find the end of the road.

But the Point did not know about this. She walked, tired, sad. A point stood on a straight line and decided to call the scissors for help. Then, out of nowhere, scissors appeared and snapped right in front of Dot’s nose. And they cut straight.

Hooray! - Dot shouted. - This is the end! But now there are two, I don’t know what to call them...

The news spreads about a new figure:
Let there be no end to it,
But there is a beginning.
And the sun, quietly rising from behind the clouds,
He said: “Friends, let’s call it ray!”

I like them! - Dot shouted. They look like rays of sunshine.

Geometric figure - the ray can have different directions. The main thing to remember is that the beginning of the beam is a point. Let's call this point letter A.

The beam is limited on one side and can be extended in a straight line only in one direction as far as desired.

Let's build a beam together. What tools will we need?

Of course, a ruler and a pencil will help us build the beam.
Where do we start building the beam?

That's right, let's put an end to it.
All constructions and measurements start from scratch. Align the point with the “0” mark on the ruler. Let's draw a straight line. Choose the length and direction yourself.
We also built a beam. Do you agree with me? (There is a number beam on the screen.)
Yes, this is also a beam, but it is called numerical. Why?
What are the numbers on the beam for? Now we will learn to use the number beam, we will count, calculate.
Divide your number line into equal sections and place dots.
Label the points with numbers in order. What number will we use to denote the very first point – the origin of the count?

That's right, let's start counting from scratch. Which of school supplies reminds us of a number ray?

Well done guys. It looks like a ruler.

Any number can be depicted on a number line by denoting it with a dot, since the line is infinite.

With the help of a number beam, numbers are easy to compare: the further to the right the point is from the beginning of the beam, the more it corresponds to the less to the left.

Tell me guys which way number ray need to move to find all numbers that are less than ten?

Right, left. How about finding all numbers greater than ten?

Yes, you need to move to the right of the number ten.

Now place point A and draw two rays AB and AC from this point.

We got a new one geometric figure. It's called an angle. Point A is the vertex of the angle. Each corner has a name. It can consist of one letter - the vertex of the angle, or of three letters that indicate the rays, with the letter of the vertex of the angle in the middle. Read like this: angle A or angle ABC

From the top along the beam

It's like I'm going down a hill.

Only the beam now is her.

And it's called "side".

We see that the rays are now the sides of the angle. These are sides AB and AC. Remember that the ray starts from a point.

There are several types of angles: straight, acute and obtuse. An angle such as that on a square is called a right angle. In the figure, this is angle K. An angle that is less than a right angle is called an acute angle. In the figure, this is angle B.

An angle that is larger than a right angle is called an obtuse angle. This is angle C.

In order to correctly determine the type of angle, we will use a square.

Take rulers and pencils.

Draw a right angle using a square, call it M.

Now try to draw acute angle, which is less than a right angle. Call him T.

Now draw an obtuse angle that is larger than a right angle. Call it N.

What to do if you don’t have a square, but you need to draw a right angle on unlined paper? This can be done using a ruler and compass. Let's try to do this together.

To use sharp tools correctly, you need to remember

safety rules:

You cannot put the compass close to your face; there is a needle at the end, you can prick yourself.

You cannot pass the compass forward with the needle, you can prick your friend.

There should be order on the desktop.

And now that you know the safety rules, let's draw a straight line

put two points A and B on it
draw two circles to make points
A and B became the centers of circles
points of intersection of circles
designate with letters C and D
through the obtained points C and D
draw a straight line
point of intersection of two lines
mark the lines with the letter O

Name the angles you get.

Let's read them together, corner OWL, corner

BOD, angle AOC and angle AOD

The definition of the concept of a ray is based on two basic concepts of geometry: a point and a straight line. Let's take an arbitrary straight line and choose an arbitrary point on it. Such a point will divide this straight line into two parts (Fig. 1).

Definition 1

A ray will be called a part of a line that is limited by some point on this line, but only on one side.

Definition 2

The point to which the ray is limited within the framework of Definition 1 is called the beginning of this ray.

Note 1

Note that the angle that was obtained in Figure 1 is called unfolded.

We will denote the ray by two points: its beginning and any other arbitrary point on it. Note that here, in the notation, the order in which these points are designated is important. We always put the beginning of the ray in first place (Fig. 2)

The concept of a ray is associated with the following axiom of geometry:

Axiom 1: Any arbitrary point on a line will divide it into two rays, and any arbitrary points one and the same of them will lie on one side of this point, and two points from different rays will lie on different sides from this point.

The following axiom is also associated with the concept of a ray and a segment.

Axiom 2: From the beginning of any ray a segment can be plotted, which is obviously equal to this segment, and such a segment will be unique.

Corner

Let us be given two arbitrary rays. Let's put them on top of each other. Then

Definition 3

We will call an angle two rays that have the same origin.

Definition 4

The point that is the beginning of the rays within the framework of Definition 3 is called the vertex of this angle.

We will denote the angle by its following three points: the vertex, a point on one of the rays and a point on the other ray, and the vertex of the angle is written in the middle of its designation (Fig. 3).

The following axiom is also associated with the concept of ray and angle.

Axiom 3: From any arbitrary ray an angle can be plotted into a certain half-plane, which is obviously equal to this angle, and such an angle will be unique.

Angle comparison

Let's consider two arbitrary angle. Obviously, they can be either equal or unequal.

So, to compare the angles we have chosen (let’s denote them as angle 1 and angle 2), we will superimpose the vertex of angle 1 on the vertex of angle 2, so that one of the rays of these angles overlaps each other, and the other two are on the same side of these rays . After such an overlay, the following two cases are possible:

Angle size

In addition to comparing one angle to another, measuring angles is also often necessary. To measure an angle means to find its magnitude. To do this, we need to select some kind of “reference” angle, which we will take as a unit. Most often, this angle is the angle that is equal to the $\frac(1)(180)$ part of the unfolded angle. This quantity is called a degree. After choosing such an angle, we compare the angles with it, the value of which needs to be found.

The most in a simple way Measuring the magnitude of angles is a measurement using a protractor.

Example 1

Find the value of the following angle:

We use a protractor:

Answer: $30^0$.

After determining the magnitude of the angles, we have a second way to compare angles. If, with the same choice of unit of measurement, angle 1 and angle 2 will have the same size, then such angles will be called equal. If, without loss of generality, angle 1 has a value of numerical value is less than angle 2, then angle 1 will be less than angle 2.

Beam and angle- basic information.

Beam goes from one point to infinity (and is called, for example, “outgoing and point A”).

A ray in geometry is an analogy to a light ray in real life.

Many rays can emanate from one point.

Each ray is named either in small Latin letters: a, b, c, d,..., or by starting point and any other point on this ray, for example: AK

These are two rays ( sides of the corner), which come out from one point ( corner vertices). In the corner, as a rule, an arc is placed, which indicates the angle.

The angle can be:

Denote by dots: ∠AOB

Denote by straight lines: ∠ab

Actually straight, only B is the vertex, DC and DA are rays.

Any corner divides the plane into 2 parts: internal And external. At a rotated angle, any plane can be considered internal or external.

The inner part of the angle can be divided into 2 new angles by drawing a new ray in the inner part.

If a ray divides an angle into two equal angles, then this ray is called bisector. For memorization, a rhyme is used: “a bisector is a rat that runs around the corners and divides the corner in half.”

It's logical that each point of a bisector is equidistant from right angles.

Please note how the angles are indicated in the figure below - they are drawn with identical arcs, which means in the drawings that these angles are equal.