All possible angles. The concept and types of angles

Students get acquainted with the concept of an angle in elementary school. But as a geometric figure with certain properties, they begin to study it from the 7th grade in geometry. Seems, pretty simple figure what can be said about her. But, acquiring new knowledge, schoolchildren more and more understand that they can learn quite interesting facts about her.

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When are studied

The school geometry course is divided into two sections: planimetry and solid geometry. Each of them has a lot of attention. paid to corners:

• In planimetry, their basic concept is given, an acquaintance with their species in size occurs. The properties of each type of triangles are studied in more detail. New definitions for students appear - these are geometric shapes formed by the intersection of two straight lines with each other and the intersection of several secant lines.
• In stereometry, spatial angles are studied - dihedral and triangular.

Attention! This article discusses all types and properties of angles in planimetry.

Definition and measurement

When starting to study, they initially determine what is angle in planimetry.

If we take a certain point on the plane and draw two arbitrary rays from it, we get a geometric figure - an angle, consisting of the following elements:

• vertex - the point from which the rays were drawn, denoted by a capital letter of the Latin alphabet;
• sides are half-lines drawn from the vertex.

All the elements that form the figure we are considering break the plane into two parts:

• internal - in planimetry does not exceed 180 degrees;
• external.

Angle measurement principle in planimetry explain on an intuitive basis. To begin with, they introduce students to the concept of an expanded angle.

Important! An angle is called unfolded if the half-lines going out from its vertex form a straight line. An undeveloped corner is all other cases.

If it is divided into 180 equal parts, then it is customary to consider the measure of one part equal to 10. In this case, they say that the measurement is made in degrees, and the degree measure of such a figure is 180 degrees.

Main types

The types of angles are classified according to such criteria as the degree measure, the nature of their formation and the categories presented below.

The largest

Given the magnitude, the angles are divided into:

• deployed;
• straight;
• blunt;
• spicy.

What angle is called unfolded was presented above. Let's define the concept of direct.

It can be obtained by dividing the expanded into two equal parts. In this case, it is easy to answer the question: a right angle, how many degrees is it?

We divide 180 degrees of unfolded by 2 and we get that right angle is 90 degrees... This is a wonderful figure, since many facts in geometry are associated with it.

It also has its own characteristics in the designation. To show a right angle in the figure, it is indicated not by an arc, but by a square.

The angles that are obtained by dividing an arbitrary straight ray are called acute. Logically, it follows that the acute angle is less than the right angle, but its measure is different from 0 degrees. That is, it has a value between 0 and 90 degrees.

An obtuse angle is greater than a right angle, but less than a deployed one. Its degree measure ranges from 90 to 180 degrees.

This element can be divided into different types of the considered figures, excluding the expanded one.

Regardless of how the non-deployed angle is broken, the basic axiom of planimetry is always used - "the main property of measurement".

At dividing an angle with one beam or several, the degree measure of this figure is equal to the sum of the measures of the angles into which it is divided.

At the 7th grade level, the types of angles in terms of their size end there. But to increase erudition, we can add that there are other varieties that have a degree of more than 180 degrees. They are called convex.

Shapes at the intersection of straight lines

The next types of angles that students become familiar with are the elements formed when two lines intersect. Shapes that are placed opposite each other are called vertical. Their distinguishing feature is that they are equal.

Elements that are adjacent to the same line are called adjacent. The theorem reflecting their property says that adjacent angles add up to 180 degrees.

Elements in a triangle

If we consider the figure as an element in a triangle, then the corners are divided into internal and external. The triangle is bounded by three line segments and consists of three vertices. The angles located inside the triangle at each vertex are called internal.

If we take any internal element at any vertex and extend any side, then the angle that is formed and is adjacent to the internal one is called external. This pair of elements has the following property: their sum is 180 degrees.

Intersection of two straight secants

Intersection of straight lines

When two straight secants intersect, angles are also formed, which are usually distributed in pairs. Each pair of elements has its own name. It looks like this:

• internal criss-crossing: ∟4 and ∟6, ∟3 and ∟5;
• internal one-sided: ∟4 and ∟5, ∟3 and ∟6;
• corresponding: ∟1 and ∟5, ∟2 and ∟6, ∟4 and ∟8, ∟3 and ∟7.

In the case when the secant intersects two

In this article, we will comprehensively analyze one of the main geometric shapes - the angle. Let's start with the auxiliary concepts and definitions that will lead us to the definition of the angle. After that, we present the accepted ways of denoting angles. Next, let's take a closer look at the process of measuring angles. In conclusion, we will show how you can mark the corners in the drawing. We have provided all the theory with the necessary drawings and graphic illustrations for better memorization of the material.

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Determination of the angle.

The angle is one of the most important figures in geometry. The definition of the angle is given through the definition of the beam. In turn, the idea of ​​a ray cannot be obtained without knowledge of such geometric shapes as a point, a straight line and a plane. Therefore, before getting acquainted with the definition of the angle, we recommend that you brush up on the theory from sections and.

So, we will start from the concepts of a point, a straight line on a plane and a plane.

Let us first give the definition of a ray.

Let us be given some straight line on the plane. Let us denote it by the letter a. Let O be some point of the straight line a. Point O divides line a into two parts. Each of these parts, together with point O, is called ray, and point O is called the beginning of the ray... You can still hear that the beam is called semidirect.

For brevity and convenience, the following designations for rays have been introduced: a ray is denoted either by a small Latin letter (for example, ray p or ray k), or by two large Latin letters, the first of which corresponds to the beginning of the ray, and the second designates some point of this ray (for example, ray OA or beam CD). Let's show the image and the designation of the rays in the drawing.

Now we can give the first definition of an angle.

Definition.

Injection Is a flat geometric figure (that is, lying entirely in a certain plane), which is made up of two non-coincident rays with a common origin. Each of the rays is called side corner, the common origin of the sides of the corner is called top of the corner.

It is possible that the sides of the corner form a straight line. This corner has its own name.

Definition.

If both sides of an angle lie on one straight line, then such an angle is called deployed.

We bring to your attention a graphic illustration of the unfolded corner.

To indicate the angle, use the angle icon "". If the sides of the corner are indicated by small Latin letters (for example, one side of the corner is k and the other is h), then to designate this angle, after the angle sign, letters corresponding to the sides are written in succession, and the order of writing does not matter (that is, or). If the sides of the corner are indicated by two large Latin letters (for example, one side of the corner is OA, and the other side of the corner is OB), then the angle is denoted as follows: after the angle sign, three letters are written that take part in the designation of the sides of the angle, and the letter corresponding to the apex of the angle, is located in the middle (in our case, the angle will be designated as or). If the vertex of the corner is not the vertex of some other angle, then such an angle can be denoted by the letter corresponding to the vertex of the angle (for example,). Sometimes you can see that the corners in the drawings are marked with numbers (1, 2, etc.), these angles are designated as and so on. For clarity, we will give a figure showing the angles.

Any angle divides the plane into two parts. Moreover, if the angle is not developed, then one part of the plane is called inner corner and the other is outside corner area... The following image explains which part of the plane is inside the corner and which is outside.

Any of the two parts into which the flattened corner divides the plane can be considered the inner region of the flattened corner.

The definition of the inner region of the corner leads us to the second definition of the angle.

Definition.

Injection- this is a geometric figure, which is made up of two non-coincident rays with a common origin and the corresponding inner region of the corner.

It should be noted that the second definition of the angle is stricter than the first, since it contains more conditions. However, the first definition of angle should not be dismissed, and the first and second definitions of angle should not be considered separately. Let us clarify this point. When we talk about an angle as a geometric figure, then an angle means a figure composed of two rays with a common origin. If it becomes necessary to carry out any actions with this angle (for example, measuring the angle), then two rays with a common origin and an inner region should already be understood at an angle (otherwise a double situation would arise due to the presence of both the inner and outer regions of the angle ).

Let us give more definitions of adjacent and vertical angles.

Definition.

Adjacent corners- these are two corners in which one side is common, and the other two form a developed angle.

It follows from the definition that adjacent angles complement each other up to a deployed angle.

Definition.

Vertical corners- these are two corners, in which the sides of one corner are the continuation of the sides of the other.

The illustration shows the vertical corners.

Obviously, two intersecting straight lines form four pairs of adjacent angles and two pairs of vertical angles.

Comparison of angles.

In this paragraph of the article, we will deal with the definitions of equal and unequal angles, and also in the case of unequal angles, we will explain which angle is considered large and which is smaller.

Recall that two geometric shapes are called equal if they can be superimposed.

Let us be given two angles. Let's give some reasoning that will help us get an answer to the question: "Are these two angles equal or not?"

Obviously, we can always match the vertices of two corners, as well as one side of the first corner to either side of the second corner. Align the side of the first corner with that side of the second corner so that the remaining sides of the corners are on one side of the straight line on which the aligned sides of the corners lie. Then, if the other two sides of the corners coincide, then the corners are called equal.

If the other two sides of the corners do not coincide, then the corners are called unequal, and smaller the angle that is part of the other ( big is the corner that completely contains the other corner).

Obviously, the two unfolded corners are equal. It is also obvious that a flattened angle is greater than any non-flattened corner.

Measurement of angles.

Angle measurement is based on comparing the measured angle with the angle taken as the unit of measure. The process of measuring angles looks like this: starting from one of the sides of the measured angle, its inner area is sequentially filled with single angles, tightly stacking them one to the other. At the same time, the number of laid corners is remembered, which gives the measure of the measured angle.

In fact, any angle can be taken as the unit of measurement for angles. However, there are many generally accepted units for measuring angles related to various fields of science and technology, they have received special names.

One of the units of measure for angles is degree.

Definition.

One degree Is an angle equal to one hundred and eightieth of the expanded angle.

A degree is denoted by "", therefore one degree is denoted as.

Thus, in an unfolded corner, we can fit 180 angles in one degree. It will look like a half of a round pie cut into 180 equal pieces. It is very important: "pieces of pie" are tightly stacked one to another (that is, the sides of the corners are aligned), and the side of the first corner is aligned with one side of the unfolded corner, and the side of the last unit corner coincides with the other side of the unfolded corner.

When measuring angles, it is found out how many times a degree (or other unit of measurement of angles) fits in the measured angle until the inner region of the measured angle is completely covered. As we have already seen, in the unfolded angle, the degree fits exactly 180 times. Below are examples of angles in which an angle of one degree fits exactly 30 times (this angle is one sixth of the expanded angle) and exactly 90 times (half of the expanded angle).

To measure angles less than one degree (or another unit of measurement of angles) and in cases where the angle cannot be measured with an integer number of degrees (taken units of measurement), you have to use parts of a degree (parts of taken units of measurement). Certain parts of the degree have received special names. The most widespread are the so-called minutes and seconds.

Definition.

Minute Is one sixtieth of a degree.

Definition.

Second Is one sixtieth of a minute.

In other words, a minute contains sixty seconds, and a degree contains sixty minutes (3600 seconds). The symbol “” is used to denote the minutes, and the symbol “” is used to denote the seconds (do not confuse with the signs of the derivative and the second derivative). Then, with the introduced definitions and notations, we have, and the angle at which 17 degrees 3 minutes and 59 seconds fit can be designated as.

Definition.

Degree measure of angle a positive number is called, which shows how many times a degree and its parts fit in a given angle.

For example, the degree measure of the unfolded angle is one hundred and eighty, and the degree measure of the angle is .

There are special measuring devices for measuring angles, the most famous of which is the protractor.

If you know both the designation of the angle (for example,) and its degree measure (let 110), then use a short notation of the form and they say: "The angle of the AOB is one hundred and ten degrees."

From the definitions of the angle and the degree measure of the angle it follows that in geometry the measure of the angle in degrees is expressed by a real number from the interval (0, 180] (in trigonometry, angles with an arbitrary degree measure are considered, they are called). The angle of ninety degrees has a special name, it is called right angle... An angle less than 90 degrees is called acute angle... An angle greater than ninety degrees is called obtuse angle... So, the measure of an acute angle in degrees is expressed by a number from the interval (0, 90), the measure of an obtuse angle is a number from the interval (90, 180), a right angle is ninety degrees. Here are illustrations of an acute angle, an obtuse angle, and a right angle.

From the principle of measuring angles, it follows that the degree measures of equal angles are the same, the degree measure of the larger angle is greater than the degree measure of the smaller one, and the degree measure of the angle, which is made up of several angles, is equal to the sum of the degree measures of the constituent angles. The figure below shows the angle AOB, which is made by the angles AOC, COD and DOB, while.

Thus, the sum of adjacent angles is one hundred and eighty degrees since they constitute a flat angle.

This statement implies that. Indeed, if the angles AOB and COD are vertical, then the angles AOB and BOC are adjacent and the angles COD and BOC are also adjacent, therefore, equalities and are true, whence equality follows.

Along with the degree, a convenient unit for measuring angles is called radian... The radian measure is widely used in trigonometry. Let's define a radian.

Definition.

One radian angle- This center corner, which corresponds to the length of the arc, equal to the length of the radius of the corresponding circle.

Let's give a graphic illustration of a one radian angle. In the drawing, the length of the radius OA (as well as the radius OB) is equal to the length of the arc AB, therefore, by definition, the angle AOB is equal to one radian.

Radians are abbreviated as "rad". For example, writing 5 rad means 5 radians. However, in writing, the designation "glad" is often omitted. For example, when it is written that the angle is pi, it means pi rad.

It should be noted separately that the value of the angle, expressed in radians, does not depend on the length of the radius of the circle. This is due to the fact that the figures bounded by a given angle and an arc of a circle centered at the apex of a given angle are similar to each other.

Measuring angles in radians can be performed in the same way as measuring angles in degrees: find out how many times an angle of one radian (and its parts) fits into a given angle. Or you can calculate the length of the arc of the corresponding central angle, and then divide it by the length of the radius.

For the needs of practice, it is useful to know how the degree and radian measures relate to each other, since quite a part has to be carried out. In this article, a connection is established between the degree and radian measure of an angle, and examples of converting degrees to radians and vice versa are given.

Designation of angles in the drawing.

In the drawings, for convenience and clarity, the corners can be marked with arcs, which are usually drawn in the inner region of the corner from one side of the corner to the other. Equal angles are marked with the same number of arcs, unequal angles - with a different number of arcs. Right angles in the drawing are denoted by a symbol of the form "", which is depicted in the inner region of a right angle from one side of the corner to the other.

If you have to mark many different angles in the drawing (usually more than three), then when marking angles, in addition to ordinary arcs, it is permissible to use arcs of some special type. For example, you can draw jagged arcs, or something similar.

It should be noted that you should not get carried away with the designation of angles in the drawings and do not clutter the drawings. We recommend that you mark only those angles that are necessary in the process of solving or proving.

Bibliography.

• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. 7 - 9 grades: a textbook for educational institutions.
• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. A textbook for grades 10-11 of secondary school.
• Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.

This article will look at one of the main geometric shapes - the angle. After a general introduction to this concept, we will focus on a separate type of such a figure. Flat angle is an important concept of geometry, which will be the main topic of this article.

Introduction to the concept of geometric angle

In geometry, there are a number of objects that form the basis of all science. The angle just refers to them and is determined using the concept of a ray, so let's start with it.

Also, before proceeding with the definition of the angle itself, you need to remember several equally important objects in geometry - this is a point, a straight line on a plane and the plane itself. A straight line is the simplest geometric figure that has no beginning or end. A plane is a surface that has two dimensions. Well, a ray (or half-line) in geometry is a part of a straight line that has a beginning but no end.

Using these concepts, we can make a statement that an angle is a geometric figure that lies completely in a certain plane and consists of two non-coincident rays with a common origin. Such rays are called the sides of the corner, and the common origin of the sides is its vertex.

Types of angles and geometries

We know that angles can be very different. Therefore, a little below will be a small classification that will help you better understand the types of angles and their main features. So, there are several types of angles in geometry:

1. Right angle. It is characterized by a size of 90 degrees, which means that its sides are always perpendicular to each other.
2. Sharp corner. These angles include all their representatives with a size less than 90 degrees.
3. Obtuse angle. All angles with a value from 90 to 180 degrees can also be here.
4. Expanded corner. It has a size of strictly 180 degrees and outwardly its sides make up one straight line.

Flat angle concept

Now let's take a closer look at the unfolded corner. This is the case when both sides lie on the same straight line, which can be clearly seen in the picture below. This means that we can say with confidence that at the unfolded corner, one of its sides is essentially a continuation of the other.

It is worth remembering the fact that such an angle can always be divided using a ray that exits from its top. As a result, we get two corners, which are called adjacent in geometry.

Also, the unfolded corner has several features. In order to talk about the first of them, you need to remember the concept of "angle bisector". Recall that this is a ray that divides any angle strictly in half. As for the unfolded angle, its bisector divides it in such a way that two 90-degree right angles are formed. It is very easy to calculate mathematically: 180˚ (degree of unfolded angle): 2 = 90˚.

If we divide the unfolded angle with a completely arbitrary ray, then as a result we always get two angles, one of which will be acute and the other obtuse.

Flattened Corner Properties

It will be convenient to consider this angle, bringing together all its main properties, which we did in this list:

1. The sides of the unfolded corner are antiparallel and form a straight line.
2. The unfolded angle is always 180˚.
3. Together, two adjacent corners always make up a flat angle.
4. The full angle, which is 360˚, consists of two unfolded ones and is equal to their sum.
5. Half of the unfolded corner is a right angle.

So, knowing all these characteristics of a given type of angles, we can use them to solve a number of geometric problems.

Problems with expanded corners

In order to understand whether you have learned the concept of a wide angle, try to answer the following few questions.

1. What is the unfolded angle if its sides form a vertical line?
2. Will two corners be adjacent if the first is 72˚ and the other is 118˚?
3. If a full angle consists of two unfolded, how many right angles are there?
4. The unfolded angle was divided by the beam into two angles such that their degree measures are 1: 4. Calculate the resulting angles.

Solutions and Answers:

1. No matter how the unfolded angle is located, it is always 180˚ by definition.
2. Adjacent corners have one side in common. Therefore, in order to calculate the size of the angle that they make up together, you just need to add the value of their degree measures. This means that 72 + 118 = 190. But by definition, the unfolded angle is 180˚, which means that these two angles cannot be adjacent.
3. A flattened corner accommodates two right angles. And since there are two unfolded in the complete, it means that there will be 4 straight lines in it.
4. If we call the sought angles a and b, then let x be the coefficient of proportionality for them, which means that a = x, and accordingly b = 4x. The unfolded angle in degrees is 180˚. And according to its properties, that the degree measure of an angle is always equal to the sum of the degree measures of those angles into which it is split by any arbitrary ray that passes between its sides, we can conclude that x + 4x = 180˚, which means 5x = 180˚ ... From here we find: x = a = 36˚ and b = 4x = 144˚. Answer: 36˚ and 144˚.

If you managed to answer all these questions without prompting and without peeping at the answers, then you are ready to move on to the next lesson in geometry.

"Little son came to his father and asked Little Tiny:" What angles are there? " But father, I forgot the answer. This is very bad!".

In our article, we propose to recall the lessons of mathematics and find answers to Crumbs' questions.

What is angle

What an angle is, of course, is easier to show than to explain. From elementary school, we know that a flat angle:

1. This is a geometric figure.
2. It is formed by two sides - rays.
3. The rays come out from one vertex - a point.
4. Measured in degrees.

That is, if you put a point on any plane, and then derive two rays from this point (a ray is a straight line with a beginning but no end), then we get an angle, and not one, but two. This is because the rays have divided the plane in two. We have formed two corners - an internal and an external one.

Angle designation

The angle in mathematics is denoted by this symbol - "" and Greek letters: β, δ, φ. You can also designate the corners in small or large Latin letters. Lowercase (d, c, b) denote rays forming an angle, therefore, the name will consist of two letters and an icon - ˪ab. Large latin letters represent three corner points: two on the sides and one vertex (˪ DEF). Moreover, the letter of the vertex will always be in the middle of the name, but it makes no difference how to read DEF or FED.

Types of angles

Depending on the degrees (measured value), the angles are divided into:

• Sharp (> 90 degrees);
• Straight lines (exactly 90);
• Blunt (180);
• Expanded (equal to 180);
• Non-convex (more than 180, but less than 360);
• Full (360);

All angles that are not straight or unfolded are called oblique.

Also, what are the corners?

• Adjacent - they have one side in common, while the others lie, not coinciding, on the same plane. The sum of these angles will always be 180.
• Vertical - angles formed by two intersecting straight lines and they do not have common sides, but their rays come out from one point. That is, the side of one corner is a continuation of the other. These angles are equal.
• Central is the corner whose vertex is the center of the circle.
• Inscribed corner. Its top is on a circle, and the rays that form it intersect this circle.

Now you know which angle is right, and you can also tell which angle is acute. It is not difficult to remember this, and other types of corners also have characteristic names.

The corner is the main geometric figure, which we will analyze throughout the entire topic. Definitions, methods of assignment, designations and measurements of the angle. Let's look at the principles of selecting corners in drawings. The whole theory is illustrated and has a large number of visual drawings.

Yandex.RTB R-A-339285-1 Definition 1

Injection Is a simple important figure in geometry. The angle directly depends on the definition of the ray, which in turn consists of the basic concepts of point, line and plane. For a thorough study, you need to delve into the topics straight line on the plane - necessary information and plane - necessary information.

The concept of an angle begins with the concept of a point, a plane, and a line drawn on that plane.

Definition 2

You are given a straight line a on the plane. We denote some point O on it. The straight line is divided by a point into two parts, each of which has a name Ray, and point O - ray start.

In other words, a ray or semi-straight - it is a part of a straight line, consisting of points of a given straight line, located on the same side relative to the starting point, that is, point O.

The designation of the beam is permissible in two variations: one lowercase or two uppercase letters of the Latin alphabet. When designated by two letters, the beam has a two-letter name. Let's take a closer look at the drawing.

Let's move on to the concept of determining the angle.

Definition 3

Injection Is a figure located in a given plane, formed by two mismatched rays having a common origin. Corner side is a ray, vertex- the common origin of the parties.

There is a case when the sides of the corner can act as a straight line.

Definition 4

When both sides of an angle are located on the same straight line or its sides serve as additional half-lines of one straight line, then such an angle is called deployed.

The figure below shows a flattened corner.

A point on a straight line is the apex of the angle. Most often, it is designated by the point O.

An angle in mathematics is denoted by the "∠" sign. When the sides of the corner are denoted by small Latin, then for the correct definition of the angle, letters are written in succession corresponding to the sides. If the two sides are designated k and h, then the angle is designated as ∠ k h or ∠ h k.

When the designation is in capital letters, the sides of the corner are respectively named O A and O B. In this case, the corner has the name of three letters of the Latin alphabet, written in a row, in the center with the vertex - ∠ A O B and ∠ B O A. There is a designation in the form of numbers when the corners do not have names or letters. Below is a picture where angles are indicated in different ways.

The angle divides the plane into two parts. If the angle is not unfolded, then one part of the plane has the name inner corner, the other is outer corner area... Below is an image explaining which parts of the plane are external and which are internal.

When divided by a flattened corner on a plane, any of its parts is considered to be the inner region of the flattened corner.

Inner area of ​​the corner - the element used for the second definition of the angle.

Definition 5

Corner is called a geometric figure consisting of two mismatched rays having a common origin and the corresponding inner region of the corner.

This definition is more stringent than the previous one, since it has more conditions. It is not advisable to consider both definitions separately, because an angle is a geometric figure transformed with the help of two rays emanating from one point. When it is necessary to perform actions with an angle, then the definition means the presence of two rays with a common origin and an inner region.

The two corners are called related, if there is a common side, and the other two are additional half-lines or form a developed angle.

The figure shows that adjacent corners complement each other, as they are a continuation of each other.

Definition 7

The two corners are called vertical if the sides of one are complementary half-lines of the other or are extensions of the sides of the other. The figure below shows an image of the vertical corners.

When straight lines intersect, 4 pairs of adjacent and 2 pairs of vertical angles are obtained. Below is shown in the figure.

The article shows the definitions of equal and unequal angles. Let us analyze which angle is considered large, which is smaller and other properties of the angle. Two shapes are considered equal if, when overlapping, they completely coincide. The same property applies to comparing angles.

Two angles are given. It is necessary to come to the conclusion whether these angles are equal or not.

It is known that the vertices of two corners and the side of the first corner overlap with any other side of the second. That is, in case of complete coincidence, when the angles are superimposed, the sides of the given angles will be completely combined, the angles equal.

It may be that when overlapping the sides may not match, then the corners unequal, less of which it consists of another, and more incorporates a complete different angle. Shown below are unequal angles that are not aligned when overlaid.

The flattened corners are equal.

Measurement of angles begins with measuring the side of the measured angle and its inner region, filling which with unit angles, are applied to each other. It is necessary to count the number of laid corners, and they predetermine the measure of the measured angle.

Angle units can be expressed as any measurable angle. There are generally accepted units of measurement that are used in science and technology. They specialize in other names.

The most commonly used concept is degree.

Definition 8

One degree called an angle that has one hundred and eightieth of a flat angle.

The standard notation for a degree is "°", then one degree is 1 °. Therefore, the unfolded angle consists of 180 such angles, consisting of one degree. All available corners are tightly packed to each other and the sides of the previous one are aligned with the next.

It is known that the number of degrees in an angle is the same measure of the angle. The expanded corner has 180 stowed corners in its composition. The figure below shows examples where the angle is laid 30 times, that is, one-sixth of the expanded, and 90 times, that is, half.

Minutes and seconds are used to accurately determine angles. They are used when the angle value is not an integer degree designation. Such parts of a degree allow for more accurate calculations.

Definition 9

A minute call one sixtieth of a degree.

Definition 10

In a second call one sixtieth of a minute.

The degree contains 3600 seconds. Minutes stand for "" "and seconds for" "" ".

1 ° = 60 "= 3600" ", 1" = (1 60) °, 1 "= 60" ", 1" "= (1 60)" = (1 3600) °,

and the designation of the angle 17 degrees 3 minutes and 59 seconds is 17 ° 3 "59" ".

Definition 11

Here is an example of the designation of the degree measure of the angle equal to 17 ° 3 "59" ". The record has another form 17 + 3 60 + 59 3600 = 17 239 3600.

A measuring device such as a protractor is used to accurately measure angles. When designating the angle ∠ A O B and its degree measure of 110 degrees, a more convenient notation ∠ A O B = 110 ° is used, which reads "Angle A O B is equal to 110 degrees."

In geometry, an angle measure from the interval (0, 180] is used, and in trigonometry, an arbitrary degree measure is called angles of rotation. Angles are always expressed as real numbers. Right angle Is an angle that has 90 degrees. Sharp corner- an angle that is less than 90 degrees, and blunt- more.

An acute angle is measured in the interval (0, 90), and an obtuse one - (90, 180). Three types of angles are clearly shown below.

Any degree measure of any angle has the same meaning. A larger angle accordingly has a larger degree measure than a smaller one. The degree unit of one angle is the sum of all available internal angle degrees. Below is a figure that shows the angle AOB, consisting of the angles AOC, COD and DOB. In detail it looks like this: ∠ A O B = ∠ A O C + ∠ D O B = 45 ° + 30 ° + 60 ° = 135 °.

Based on this, we can conclude that sum of all adjacent angles is 180 degrees, because they all make up the unfolded corner.

Hence it follows that any vertical angles are equal... If we consider this by an example, we get that the angle A O B and C O D are vertical (in the drawing), then the pairs of angles A O B and B O C, C O D and B O C are considered adjacent. In this case, the equality ∠ A O B + ∠ B O C = 180 ° together with ∠ C O D + ∠ B O C = 180 ° are considered unambiguously true. Hence we have that ∠ A O B = ∠ C O D. Below is an example of the image and notation of vertical catch.

In addition to degrees, minutes and seconds, another unit of measurement is used. It is called radian... Most often it can be found in trigonometry when denoting the corners of polygons. What is called a radian.

Definition 12

Angle of one radian called the central angle, which has the length of the radius of the circle equal to the length of the arc.

In the figure, the radian is depicted in the form of a circle, where there is a center indicated by a point, with two points on the circle, connected and converted into radii O A and O B. By definition, this triangle AOB is equilateral, which means that the length of the arc AB is equal to the lengths of the radii O B and About A.

Angle designation is taken as "rad". That is, an entry of 5 radians is abbreviated as 5 rad. Sometimes you can find a designation called pi. Radians do not depend on the length of a given circle, since the figures have a certain limitation with the help of an angle and its arc with a center located at the vertex of a given angle. They are considered similar.

Radians have the same meaning as degrees, only the difference is in their magnitude. To determine this, it is necessary to divide the calculated length of the arc of the central angle by the length of its radius.

In practice, use converting degrees to radians and radians to degrees for more convenient problem solving. The specified article has information on the relationship of the degree measure with the radian, where you can study in detail the translations from the degree to the radian and vice versa.

For a clear and convenient image of arcs, angles, drawings are used. It is not always possible to correctly depict and mark this or that angle, arc or name. Equal angles are designated in the form of the same number of arcs, and unequal ones in the form of different ones. The drawing shows the correct designation of acute, equal and unequal angles.

When more than 3 corners need to be marked, special arc symbols are used, such as wavy or jagged. This is not so important. Below is a figure showing their designation.

The notation of the angles should be simple so as not to interfere with other values. When solving the problem, it is recommended to select only the corners necessary for the solution, so as not to clutter up the entire drawing. This will not interfere with the solution and proof, and will also give an aesthetic appearance to the drawing.

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