Where angles apply. Expanded, obtuse, vertical and non-expanded: types of corners of the geometry. Intersection of two straight lines
The angle is the main geometric figure, which we will analyze throughout the topic. Definitions, methods of setting, notation and measurement of the angle. Let's analyze the principles of selecting corners in the drawings. The whole theory is illustrated and has a large number of visual drawings.
Yandex.RTB R-A-339285-1 Definition 1
Corner- a simple important figure in geometry. The angle directly depends on the definition of a ray, which in turn consists of the basic concepts of a point, a line and a plane. For a thorough study, you need to delve into the topics straight line on a plane - necessary information And plane - necessary information.
The concept of an angle begins with the concepts of a point, a plane, and a straight line depicted on this plane.
Definition 2
Given a line a on a plane. Denote some point O on it. The line is divided by a point into two parts, each of which has a name Ray, and the point O is beam start.
In other words, a beam or half-line - it is a part of a line, consisting of points of a given line, located on the same side relative to the starting point, that is, the point O.
The designation of the beam is allowed in two variations: one lowercase or two uppercase letters of the Latin alphabet. When denoted by two letters, the beam has a name consisting of two letters. Let's take a closer look at the drawing.
Let's move on to the concept of defining an angle.
Definition 3
Corner- this is a figure located in a given plane, formed by two mismatched rays that have a common origin. side corner is a beam vertex- the common beginning of the parties.
There is a case when the sides of an angle 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 vertex of the angle. Most often, it is denoted by the dot O.
An angle in mathematics is denoted by the sign "∠". When the sides of an angle are denoted by small Latin, then for the correct definition of the angle, letters are written in a row, respectively, according to the sides. If two sides are denoted k and h, then the angle is denoted as ∠ k h or ∠ h k .
When there is a designation in capital letters, then, respectively, the sides of the corner have the names O A and O B. In this case, the angle has a name of three letters of the Latin alphabet, written in a row, in the center with a 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 figure where angles are indicated in different ways.
An angle divides the plane into two parts. If the angle is not developed, then one part of the plane has the name inner corner area, the other - outer corner area. Below is an image explaining which parts of the plane are external and which are internal.
When divided by a straight angle on a plane, any of its parts is considered to be the interior of the straight angle.
The inner area of the corner is an element that serves for the second definition of the corner.
Definition 5
corner a geometric figure is called, consisting of two non-coinciding rays, having a common origin and a corresponding internal area of \u200b\u200bthe angle.
This definition is more rigorous than the previous one, as it has more conditions. It is not advisable to consider both definitions separately, because an angle is a geometric figure transformed using two rays coming out of 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 internal region.
Definition 6The two corners are called related, if there is a common side, and the other two are complementary half-lines or form a straight angle.
The figure shows that adjacent corners complement each other, as they are a continuation of one another.
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 crossing lines, 4 pairs of adjacent and 2 pairs of vertical angles are obtained. Below is shown in the picture.
The article shows the definitions of equal and unequal angles. We will analyze which angle is considered large, which is smaller, and other properties of the angle. Two figures are considered equal if, when superimposed, they completely coincide. The same property applies to comparing angles.
Given two angles. 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 coincide completely, the angles equal.
It may be that when superimposing the sides may not be combined, then the corners unequal, smaller of which consists of another, and more incorporates a complete other angle. Below are unequal angles not aligned when superimposed.
The developed angles are equal.
The measurement of angles begins with the measurement of the side of the measured angle and its inner region, filling which with unit angles, they are applied to each other. It is necessary to count the number of stacked corners, they predetermine the measure of the measured angle.
An angle unit can be expressed in any measurable angle. There are generally accepted units of measurement that are used in science and technology. They specialize in other titles.
The most commonly used concept degree.
Definition 8
one degree is called an angle that has one hundred and eightieth of a straightened angle.
The standard notation for a degree is "°", then one degree is 1°. Therefore, a straight angle consists of 180 such angles, consisting of one degree. All available corners are tightly stacked 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 developed corner has 180 stacked 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 angle measurements. They are used when the angle value is not an integer degree designation. Such parts of a degree allow you to perform more accurate calculations.
Definition 9
minute called one sixtieth of a degree.
Definition 10
second called one sixtieth of a minute.
A degree contains 3600 seconds. Minutes denote """, and seconds """". The designation takes place:
1°=60"=3600"", 1"=(160)°, 1"=60"", 1""=(160)"=(13600)°,
and the notation for the angle 17 degrees 3 minutes and 59 seconds is 17° 3 "59"".
Definition 11
Let's give an example of the notation of the degree measure of an angle equal to 17 ° 3 "59" ". The entry has another form 17 + 3 60 + 59 3600 \u003d 17 239 3600.
To accurately measure angles, a measuring device such as a protractor is used. When designating the angle ∠ A O B and its degree measure of 110 degrees, a more convenient notation is used ∠ A O B \u003d 110 °, 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 turning angles. The value of the angles is always expressed as a real number. Right angle is an angle that has 90 degrees. Sharp corner is an angle that is less than 90 degrees, and blunt- more.
An acute angle is measured in the interval (0, 90) , and an obtuse angle - (90, 180) . Three types of angles are clearly shown below.
Any degree measure of any angle has the same value. A larger angle, respectively, has a larger degree measure than a smaller one. The degree measure of one angle is the sum of all available degree measures of interior angles. The figure below 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, it can be concluded that sum all adjacent angles is 180 degrees because they all make up an expanded angle.
It follows from this that any vertical angles are equal. If we consider this with 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 such a case, the equality ∠ A O B + ∠ B O C = 180 ° together with ∠ C O D + ∠ B O C = 180 ° are considered uniquely true. Hence we have that ∠ A O B = ∠ C O D . Below is an example of the image and designation of vertical catches.
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 designating the angles of polygons. What is called a radian.
Definition 12
One radian angle called the central angle, which has a radius of a circle equal to the length of the arc.
In the figure, the radian is depicted as 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 A O B is equilateral, which means that the length of the arc A B is equal to the lengths of the radii O B and Oh A.
The designation of the angle is taken as "rad". That is, an entry in 5 radians is abbreviated as 5 rad. Sometimes you can find a designation that has the name pi. Radians do not depend on the length of a given circle, since the figures have some kind of restriction 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, they use convert degrees to radians and radians to degrees for easier problem solving. The specified article has information about the connection between the degree measure and the radian, where you can study in detail the translations from degree to radian and vice versa.
For a visual and convenient depiction of arcs, angles, drawings are used. It is not always possible to correctly depict and mark a particular angle, arc or name. Equal angles have the designation in the form of the same number of arcs, and unequal in the form of different ones. The drawing shows the correct designation of sharp, equal and unequal angles.
When more than 3 corners need to be marked, special arch symbols are used, such as wavy or jagged. It doesn't matter that much. The figure below shows their designation.
The designation of the angles should be simple so as not to interfere with other values. When solving a problem, it is recommended to select only the corners necessary for solving 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.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Students are introduced to the concept of 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 shape what can be said about her. But, acquiring new knowledge, schoolchildren understand more and more that you can learn quite interesting facts about her.
In contact with
When are studied
The school geometry course is divided into two sections: planimetry and solid geometry. Each of them has a lot of attention. given to the corners:
- In planimetry, their basic concept is given, acquaintance with their types in size takes place. The properties of each type of triangles are studied in more detail. New definitions for students appear - these are geometric shapes formed at the intersection of two lines with each other and the intersection of several lines of a secant.
- In stereometry, spatial angles are studied - dihedral and trihedral.
Attention! This article discusses all types and properties of angles in planimetry.
Definition and measurement
Starting to study, first determine, what is an 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:
- the vertex - the point from which the rays were drawn, is indicated by a capital letter of the Latin alphabet;
- the sides are half-line drawn from the top.
All the elements that form the figure we are considering divide the plane into two parts:
- internal - in planimetry does not exceed 180 degrees;
- external.
The principle of measuring angles in planimetry explained intuitively. To begin with, students are introduced to the concept of a developed angle.
Important! An angle is said to be developed if the half-lines emanating from its vertex form a straight line. An unfolded angle 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
Types of angles are subdivided according to such criteria as degree measure, the nature of their formation and the categories below.
By size
Given the magnitude, the angles are divided into:
- deployed;
- straight;
- blunt;
- spicy.
What angle is called deployed was presented above. Let's define the concept of a straight line.
It can be obtained by dividing the deployed into two equal parts. In this case, it is easy to answer the question: a right angle, how many degrees is it?
Divide 180 degrees by 2 to get 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 ray of a straight line are called acute. According to the logic of things, it follows that an acute angle is less than a right angle, but its measure is different from 0 degrees. That is, it has a value from 0 to 90 degrees.
An obtuse angle is greater than a right angle, but less than a straight angle. Its degree measure varies from 90 to 180 degrees.
This element can be divided into different types of figures under consideration, excluding the expanded one.
Regardless of how the non-rotated angle is broken, the basic axiom of planimetry is always used - “the main property of measurement”.
At dividing the angle with one beam or several, the degree measure of a given figure is equal to the sum of the measures of the angles into which it is divided.
At the level of the 7th grade, the types of angles in their magnitude end there. But to increase erudition, it can be added that there are other varieties that have a degree measure of more than 180 degrees. They are called convex.
Figures at the intersection of lines
The next types of angles that students are introduced to are the elements formed when two lines intersect. Figures 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 mapping 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 angles are divided into internal and external. The triangle is bounded by three segments and consists of three vertices. The angles located inside the triangle at each vertex, 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 lines
Line intersection
When two straight lines intersect, angles are also formed, which are usually distributed in pairs. Each pair of elements has its own name. It looks like this:
- internal cross-lying: ∟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 lines, all these pairs of angles have certain properties:
- Internal crosswise lying and corresponding figures are equal to each other.
- Internal one-sided elements add up to 180 degrees.
We study angles in geometry, their properties
Types of angles in mathematics
Conclusion
This article presents all the main types of angles that are found in planimetry and are studied in the seventh grade. In all subsequent courses, the properties relating to all the elements considered are the basis for further study of geometry. For example, studying, it will be necessary to recall all the properties of the angles formed at the intersection of two parallel lines of a secant. When studying the features of triangles, it is necessary to remember what adjacent angles are. Having switched to stereometry, all three-dimensional figures will be studied and built based on planimetric figures.
In this article, we will comprehensively analyze one of the main geometric shapes - the angle. Let's start with auxiliary concepts and definitions that will lead us to the definition of an angle. After that, we give the accepted methods for designating angles. Next, we will deal in detail with the process of measuring angles. In conclusion, we will show how you can mark the corners in the drawing. We provided all the theory with the necessary drawings and graphic illustrations for better memorization of the material.
Page navigation.
Angle definition.
Angle is one of the most important figures in geometry. The definition of an angle is given through the definition of a ray. In turn, the idea of a ray cannot be obtained without knowledge of such geometric figures as a point, a straight line and a plane. Therefore, before getting acquainted with the definition of the angle, we recommend refreshing 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's denote it with the letter a. Let O be some point of the line a . The point O divides the line a into two parts. Each of these parts together with the point O is called beam, and the point O is called the beginning of the beam. You can also hear that the beam is called semidirect.
For brevity and convenience, the following notation for rays has 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 denotes some point of this ray (for example, ray OA or beam CD). Let's show the image and designation of the rays in the drawing.
Now we can give the first definition of an angle.
Definition.
Corner- this is a flat geometric figure (that is, lying entirely in a certain plane), which is made up of two mismatched rays with a common origin. Each of the rays is called corner side, the common beginning of the sides of the angle is called top corner.
It is possible that the sides of an angle form a straight line. This angle has its own name.
Definition.
If both sides of an angle lie on the same line, then the angle is called deployed.
We bring to your attention a graphic illustration of a developed angle.
An angle symbol is used to denote an angle. If the sides of the angle are indicated in small Latin letters (for example, one side of the angle is k, and the other is h), then to designate this angle, after the angle sign, letters corresponding to the sides are written in a row, and the order of recording does not matter (that is, or). If the sides of the angle are indicated by two large Latin letters (for example, one side of the angle OA, and the second side of the angle OB), then the angle is denoted as follows: after the angle sign, three letters are written that participate in the designation of the sides of the angle, and the letter corresponding to the vertex of the angle, located in the middle (in our case, the angle will be indicated as or ). If the vertex of an angle 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 corners are denoted as and so on. For clarity, we present a figure in which the corners are shown and indicated.
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 area, and the other outside corner area. The following image explains which part of the plane corresponds to the inside of the corner and which part to the outside.
Any of the two parts into which a flattened angle divides a plane can be considered an interior region of the flattened angle.
The definition of the interior of an angle leads us to the second definition of an angle.
Definition.
Corner- this is a geometric figure, which is made up of two mismatched rays with a common origin and the corresponding inner region of the angle.
It should be noted that the second definition of the angle is stricter than the first, since it contains more conditions. However, one should not dismiss the first definition of the angle, nor should one consider the first and second definitions of the angle separately. Let's explain this point. When it comes to an angle as a geometric figure, then an angle is understood as 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 an angle), then an angle should already be understood as two rays with a common origin and an internal region (otherwise a twofold situation would arise due to the presence of both an internal and an external region of the angle ).
Let us give more definitions of adjacent and vertical angles.
Definition.
Adjacent corners- these are two angles in which one side is common, and the other two form a straight angle.
It follows from the definition that adjacent angles complement each other up to a straight angle.
Definition.
Vertical angles are two angles in which the sides of one angle are extensions of the sides of the other.
The figure shows vertical angles.
Obviously, two intersecting lines form four pairs of adjacent angles and two pairs of vertical angles.
Angle comparison.
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 figures are called equal if they can be superimposed.
Let us be given two angles. Let us give 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 with any of the sides of the second corner. Let's combine the side of the first corner with that side of the second corner so that the remaining sides of the corners are on the same side of the straight line on which the combined sides of the corners lie. Then, if the other two sides of the corners are aligned, then the corners are called equal.
If the other two sides of the angles do not match, then the angles are called unequal, and smaller the angle is considered to be part of another ( big is the angle that completely contains another angle).
Obviously, the two straight angles are equal. It is also obvious that a developed angle is greater than any non-developed angle.
Angle measurement.
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 stacked corners is remembered, which gives a measure of the measured angle.
In fact, any angle can be taken as the unit of measure 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 for measuring angles is degree.
Definition.
one degree is an angle equal to one hundred and eightieth of a straightened angle.
A degree is denoted by the symbol "", therefore, one degree is denoted as.
Thus, in a developed angle, we can fit 180 angles into one degree. It will look like half a round pie cut into 180 equal pieces. Very important: the "pieces of the pie" fit tightly together (that is, the sides of the corners are aligned), with the side of the first corner aligned with one side of the flattened corner, and the side of the last unit corner coincided with the other side of the flattened 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 area of the measured angle is completely covered. As we have already seen, in a developed angle, the degree fits exactly 180 times. Below are examples of angles in which a one-degree angle fits exactly 30 times (such an angle is a sixth of a straightened angle) and exactly 90 times (half a straightened angle).
To measure angles less than one degree (or another unit of measurement of angles) and in cases where the angle cannot be measured in an integer number of degrees (taken units), you have to use parts of a degree (parts of taken units of measurement). Certain parts of the degree received special names. The most common 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, there are sixty seconds in a minute, and sixty minutes (3600 seconds) in a degree. The symbol "" is used to denote minutes, and the symbol "" is used to denote seconds (do not confuse with the signs of the derivative and the second derivative). Then, with the introduced definitions and notation, we have , and the angle in which 17 degrees 3 minutes and 59 seconds fit can be denoted as .
Definition.
Degree measure of an angle a positive number is called, which shows how many times a degree and its parts fit into a given angle.
For example, the degree measure of a straightened angle is one hundred and eighty, and the degree measure of an angle is 
.
To measure angles, there are special measuring instruments, the most famous of which is a protractor.
If both the designation of the angle (for example,) and its degree measure (let 110) are known, then use a short notation of the form 
and say: "The angle 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). An 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 - by a number from the interval (90, 180), a right angle is equal to ninety degrees. Here are illustrations of an acute angle, an obtuse angle, and a right angle.
It follows from the principle of measuring angles that the degree measures of equal angles are the same, the degree measure of a larger angle is greater than the degree measure of a smaller one, and the degree measure of an angle that consists of several angles is equal to the sum of the degree measures of the component angles. The figure below shows the angle AOB, which is made up of the angles AOC, COD and DOB, while .
Thus, sum of adjacent angles is one hundred and eighty degrees, since they form a straight angle.
It follows from this assertion 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, the equalities and are valid, from which the 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 central corner, which corresponds to the length of the arc, equal to the length of the radius of the corresponding circle.
Let's give a graphical illustration of an angle of one radian. 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.
The abbreviation "rad" is used to denote radians. For example, writing 5 rad means 5 radians. However, in writing, the designation "rad" is often omitted. For example, when it is written that the angle is equal to 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 vertex of a given angle are similar to each other.
Measuring angles in radians can be done in the same way as measuring angles in degrees: find out how many times an angle of one radian (and its parts) fit into a given angle. And 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 relationship 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 corners in the drawing.
In the drawings, for convenience and clarity, 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 the right angle from one side of the corner to the other.
If the drawing has to mark many different angles (usually more than three), then when designating the angles, in addition to ordinary arcs, it is permissible to use arcs of some special type. For example, you can depict 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 up the drawings. We recommend marking 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. Grades 7 - 9: a textbook for educational institutions.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
An angle is a figure formed by two rays coming out of the same point.
The rays forming an angle are called the sides of the angle, and the point from which they come out is called the vertex of the angle.
In my drawing, the rays OB and OS are the sides of the angle, the vertex is point O, and the angle is designated as: BOC.
When writing an angle in the middle, write a letter denoting its vertex. An angle can also be denoted by one letter - the name of its vertex, for example: angle O. The word "angle" is replaced by the sign "".
For example: BOS = O
Like all geometric shapes, angles are compared using superposition. If one angle is superimposed on another and they coincide, then these angles are equal.
For example: MRL = AKV
Of all the angles can be distinguished:
1. Acute (the value of such angles is greater than 0 but less than 90).
2. Direct (the value of which is 90).
3. Obtuse (the value of such angles is greater than 90, but less than 180).
4. Expanded (the value of which is 180).
A protractor is used to measure angles.
The protractor scale is located on a semicircle. The center of this semicircle is marked on the protractor with a dash. The strokes of the protractor scale divide the semicircle into 180 shares. The rays drawn from the center of the semicircle through these strokes form 180 angles, each of which is equal to a fraction of the angle unfolded. Such angles are called degrees. Degrees are denoted by a sign. Each division of the protractor scale is equal to 1. In addition to divisions of 1 on the protractor, there are also divisions of 5 and 10.
The protractor is also used to build angles.
Historical reference
Since ancient times, people have faced the need to measure. The concept of a degree and the appearance of the first tools for measuring angles are associated with the development of civilization in ancient Babylon, although the word degree itself is of Latin origin (degree - from Latin gradus - “step, step”).
History has not preserved the name of the scientist who invented the protractor - perhaps in ancient times this tool had a completely different name. The modern name comes from the French word "TRANSPORTER", which means "to carry".
But ancient scientists made measurements not only with a protractor - after all, this tool was inconvenient for measuring on the ground and solving problems of an applied nature. Namely, applied problems were the main subject of interest of ancient geometers. The invention of the first tool that allows you to measure angles on the ground is associated with the name of the ancient Greek scientist Heron of Alexandria (I century BC). He described the diopter tool, which allows you to measure angles on the ground and solve many applied problems.
Thus, we can talk about the emergence of geodesy - a system of sciences about determining the shape and size of the Earth and about measurements on the earth's surface to display it on plans and maps. Geodesy is associated with astronomy, geophysics, astronautics, cartography, etc., and is widely used in the design and construction of structures, navigable canals, and roads.
In the 17th century, the level device was invented, and in the next century, the theodolite was invented by the English mechanic Jesse Ramsden. Today theodolite is a complex instrument. Many works (including construction) require prior consultation with surveyors for measurements using a theodolite.
However, the improvement of tools for measuring angles is not only associated with construction work. Since ancient times, people have traveled, learning about the world around them. Travelers needed to be able to navigate in space. For many centuries, the stars have become the main reference point for travelers. The first instrument of travelers appeared - the astrolabe. Astrolabe (Greek astrolabion, from astron - "star" and labe - "grasping"; lat. astrolabium) is a goniometer that served until the beginning of the 18th century to determine the positions of the stars in the sky.
Sextant is the most advanced instrument for measuring the angular coordinates of celestial bodies of that time. Its invention is attributed to Isaac Newton. The sextant made it possible to measure both the latitude and longitude of the observation point, and with a fairly high accuracy. Note that there are other units for measuring angles.
Artillerymen, on the other hand, have to not only measure angles, but also quickly mentally translate the obtained angular values into linear ones and vice versa. Therefore, measuring angles in degrees and minutes is inconvenient for gunners. Artillerymen came up with a completely different measure of angles. This measure is "thousandth", or, as it is called differently, "division of the goniometer". To get the thousandth, the circle is divided into 6000 parts.
In maritime navigation, it is customary to use the rhumb as the main unit of measurement. The nautical rhumb is defined by the central angle corresponding to an arc equal to 1/32 of a circle. Meteorology has its own rhumb, which is twice the size of the sea.
measure angle means to find its value. The angle value shows how many times the angle selected for the unit of measure fits into the given angle.
The unit of measure for angles is usually degrees. Degree is the angle equal to the part of the straight angle. To indicate degrees in the text, the ° sign is used, which is placed in the upper right corner of the number indicating the number of degrees (for example, 60 °).
Measuring angles with a protractor
A special device is used to measure angles - protractor:
The protractor has two scales - internal and external. The reference point for the inner and outer scales is located on different sides. To get the correct measurement result, the degrees must start from the correct side.
The measurement of angles is carried out as follows: the protractor is placed on the corner so that the top of the corner coincides with the center of the protractor, and one of the sides of the corner passes through the zero division on the scale. Then the other side of the angle will indicate the angle value in degrees:
They say: corner BOC is 60 degrees, angle MON equals 120 degrees and write: ∠ BOC= 60°, ∠ MON= 120°.
For a more accurate measurement of angles, fractions of a degree are used: minutes and seconds. Minute is an angle equal to a fraction of a degree. Second is an angle equal to a fraction of a minute. Minutes are marked with " , and seconds are sign "" . The sign of minutes and seconds is placed in the upper right corner of the number. For example, if the angle has a value of 50 degrees 34 minutes and 19 seconds, then write:
50°34 " 19""
Angle measurement properties
If the ray divides a given angle into two parts (two angles), then the value of this angle is equal to the sum of the values of the two resulting angles.
