If in the documents Microsoft Word You have to work not only with text, but sometimes you need to show basic calculations, or insert a certain symbol into the text, then, not finding it on the keyboard, you wonder: how to add it to the document?
This is quite easy to do, since the Word text editor has a special table in which you will definitely find everything you need. In this article we will look at how, using it, you can insert approximately equal amounts into a Word document.
Place the cursor in the place in the document where you will add it. Then go to the “Insert” tab and in the “Symbols” group, click on the button of the same name. Select "Other" from the drop-down list.
A window like this will open. In it, in the “Font” field, select "(plain text)", in the “Set” field – "mathematical operators". Next, find what you need in the list, click on it and then click the “Insert” button.
After the icon has been added to the document, close this window by clicking on the corresponding button in the lower right corner.
If you often have to add various characters to a document that cannot be typed directly from the keyboard, and you have to look for them in the mentioned table, then you can use hotkeys to insert suitable sign into the document.
Find the symbol in the list and click on it with the mouse. Then down in the field "Keyboard shortcut" look at what combination is used for it.
In our case, this is “2248, Alt+X”. First type the number “2248”, and then press “Alt+X”.
I note that not all characters have combinations, but you can assign it yourself by clicking on the button "Keyboard shortcut".
If, as in the example, you need to place the approximate sign immediately after some number, then the combination will be different. In the example it turned out “32248”.
Therefore, after you press “Alt+X”, what you want may not be inserted.
In order to add exactly approximately equal, put a space after the number where it should appear and type the combination “2248”. Then press "Alt+X".
The symbol will be inserted. Now you can put italics in front of the added character and press "Backspace" to remove the space.
This is how, using one of the methods, you can put an icon approximately equal to a Word document.
Rate this article:In this article we will comprehensively analyze one of the basic geometric shapes - an angle. Let's start with auxiliary concepts and definitions that will lead us to the definition of an angle. After this, we present the accepted ways of designating angles. Next, we will look in detail at 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.
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Definition of angle.
The 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, an 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 an angle, we recommend brushing up on the theory from sections and.
So, we will start from the concepts of a point, a 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 by the letter a. Let O be some point of the line a. Point O divides line a into two parts. Each of these parts, together with point O, is called beam, and point O is called the beginning of the ray. You can also hear what the beam is called semidirect.
For brevity and convenience, the following notation for rays was 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 ray CD). Let us 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 divergent rays with a common origin. Each of the rays is called side of the corner, the common origin of the sides of an angle is called vertex of the angle.
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 straight line, then such an angle is called expanded.
We present to your attention a graphic illustration of a rotated angle.
To indicate an angle, use the angle icon "". If the sides of an angle are designated 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 icon, letters corresponding to the sides are written in a row, and the order of writing does not matter (that is, or). If the sides of an angle are designated by two large Latin letters (for example, one side of the angle is OA, and the second side of the angle is OB), then the angle is designated as follows: after the angle icon, three letters are written down that are involved in designating the sides of the angle, and the letter corresponding to the vertex of the angle is is located in the middle (in our case, the angle will be designated as or ). If the vertex of an angle is not the vertex of another angle, then such an angle can be denoted by a letter corresponding to the vertex of the angle (for example, ). Sometimes you can see that the angles in the drawings are marked with numbers (1, 2, etc.), these angles are designated as and so on. For clarity, we present a drawing in which the angles are depicted and indicated.
Any angle divides the plane into two parts. Moreover, if the angle is not turned, then one part of the plane is called inner corner area, and the other - outer corner area. The following image explains which part of the plane corresponds to the internal area of the corner, and which to the external one.
Any of the two parts into which the unfolded angle divides the plane can be considered the interior region of the unfolded angle.
Defining the interior region of an angle brings us to the second definition of an angle.
Definition.
Corner is a geometric figure that is made up of two divergent rays with a common origin and the corresponding internal area of the angle.
It should be noted that the second definition of angle is stricter than the first, since it contains more conditions. However, the first definition of angle should not be dismissed, nor should the first and second definitions of angle be considered separately. Let's clarify this point. When we're talking about about an angle as a geometric figure, then an angle is understood as a figure composed of two rays with a common origin. If there is a need to carry out any actions with this angle (for example, measuring an angle), then the angle should already be understood as two rays with a common beginning and an internal area (otherwise a double situation would arise due to the presence of both internal and external areas of the angle ).
Let us also give definitions of adjacent and vertical angles.
Definition.
Adjacent angles- these are two angles in which one side is common, and the other two form an unfolded angle.
From the definition it follows that adjacent angles complement each other until the angle is turned.
Definition.
Vertical angles- these are two angles in which the sides of one angle are continuations of the sides of the other.
The figure shows vertical angles.
Obviously, two intersecting lines form four pairs adjacent corners and two pairs of vertical corners.
Comparison of angles.
In this paragraph of the article, we will understand the definitions of equal and unequal angles, and also in the case of unequal angles, we will explain which angle is considered larger and which smaller.
Recall that two geometric figures are called equal if they can be combined by overlapping.
Let us be given two angles. Let us 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 with either side of the second corner. Let's align the side of the first angle with that side of the second angle so that the remaining sides of the angles are on the same side of the straight line on which the combined sides of the angles lie. Then, if the other two sides of the angles coincide, then the angles are called equal.
If the other two sides of the angles do not coincide, then the angles are called unequal, and smaller the angle that forms part of another is considered ( 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.
Measuring angles.
Measuring angles is based on comparing the angle being measured with the angle taken as the unit of measurement. The process of measuring angles looks like this: starting from one of the sides of the angle being measured, its internal area is sequentially filled with single angles, placing them tightly next to each other. At the same time, the number of laid angles is remembered, which gives the measure of the measured angle.
In fact, any angle can be adopted as a unit of measurement for angles. However, there are many generally accepted units of 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- this is an angle equal to one hundred and eightieth of the turned angle.
A degree is denoted by the symbol "", therefore one degree is denoted as .
Thus, in a rotated 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 unfolded angle, and the side of the last unit angle coincides with the other side of the unfolded angle.
When measuring angles, find out how many times a degree (or other unit of measurement of angles) is placed in the angle being measured until the inner area of the angle being measured is completely covered. As we have already seen, in a rotated angle the degree is exactly 180 times. Below are examples of angles in which an angle of one degree fits exactly 30 times (such an angle is a sixth of the unfolded angle) and exactly 90 times (half of the unfolded angle).
To measure angles less than one degree (or other unit of measurement of angles) and in cases where the angle cannot be measured with a whole number of degrees (taken units of measurement), it is necessary to use parts of a degree (parts of taken units of measurement). Certain parts of a degree are given 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 in a degree (3600 seconds). The symbol “” is used to denote minutes, and the symbol “” is used to denote seconds (do not confuse with the derivative and second derivative signs). Then, with the introduced definitions and notations, we have , and the angle in which 17 degrees 3 minutes and 59 seconds fit can be denoted as .
Definition.
Degree measure of angle is a positive number that shows how many times a degree and its parts fit into a given angle.
For example, the degree measure of a developed angle is one hundred eighty, and the degree measure of an angle is equal to 
.
There are special measuring instruments for measuring angles, the most famous of which is the 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 they say: “Angle AOB is equal to one hundred and ten degrees.”
From the definitions of angle and degree measure of angle it follows that in geometry the measure of angle in degrees is expressed 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 is expressed 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 right angle.
From the principle of measuring angles it follows 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 is made up by 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 by the angles AOC, COD and DOB, in this case.
Thus, the sum of adjacent angles is one hundred eighty degrees, since they form a straight angle.
From this statement it follows 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, which implies the equality.
Along with the degree, a convenient unit of measurement for angles is called radian. The radian measure is widely used in trigonometry. Let's define a radian.
Definition.
Angle one radian- This central angle, which corresponds to an arc length equal to the length of the radius of the corresponding circle.
Let's give a graphic 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, the entry 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 is worth noting separately that the magnitude 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 with a center 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. Or you can calculate the arc length of the corresponding central angle, and then divide it by the length of the radius.
For practical purposes, it is useful to know how degree and radian measures relate to each other, since quite a lot of them have to be carried out. This article establishes a connection between degree and radian measures of angle, and provides examples of converting degrees to radians and vice versa.
Designation of angles in the drawing.
In the drawings, for convenience and clarity, corners can be marked with arcs, which are usually drawn in the inner area 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 indicated by a symbol of the form “”, which is depicted in the inner area of the right angle from one side of the angle to the other.
If you have to mark many different angles in a drawing (usually more than three), then when marking angles, in addition to ordinary arcs, it is permissible to use arcs of any kind 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 the drawings. We recommend marking only those angles that are necessary in the process of solution or proof.
Bibliography.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 – 9: textbook for general education institutions.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of secondary school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 in general education institutions.
The angle is the main geometric figure, which we will analyze throughout the entire topic. Definitions, methods of setting, notation and measurement of angle. Let's look at the principles of highlighting corners in drawings. The whole theory is illustrated and has a large number of visual drawings.
Definition 1Corner– a simple important figure in geometry. The angle directly depends on the definition of the ray, which in turn consists of basic concepts points, straight lines and planes. For a thorough study, you need to delve deeper into 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 straight line a on the plane. Let us denote a certain point O on it. A straight line is divided by a point into two parts, each of which has a name Ray, and point O – beginning of the beam.
In other words, the beam or half-straight – 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, point O.
The beam designation is allowed in two variations: one lowercase or two uppercase letters Latin alphabet. When designated 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 determining an angle.
Definition 3
Corner is a figure located in a given plane, formed by two divergent rays that have a common origin. Angle side is a ray vertex– common origin of the sides.
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 expanded.
The picture below shows a rotated corner.
A point on a straight line is the vertex of an angle. Most often it is designated by the point O.
An angle in mathematics is denoted by the sign “∠”. When the sides of an angle are designated by small Latin ones, then for correct definition angles, letters are written in a row according to the sides. If two sides are designated k and h, then the angle is designated ∠ k h or ∠ h k.
When the designation is in capital letters, then, respectively, the sides of the angle are named O A and O B. In this case, the angle has a name made up 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 angles do not have names or letter designations. Below is a picture where different ways angles are indicated.
An angle divides a plane into two parts. If the angle is not turned, then one part of the plane is called 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 developed angle on a plane, any of its parts is considered to be the interior region of the developed angle.
The inner area of the angle is an element that serves for the second definition of the angle.
Definition 5
Angle called a geometric figure consisting of two divergent rays that have a common origin and a corresponding internal angle area.
This definition is more strict 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 emanating from one point. When it is necessary to perform actions with an angle, the definition means the presence of two rays with a common beginning and an internal area.
Definition 6The two angles are called adjacent, if there is a common side, and the other two are additional half-lines or form a straight angle.
The figure shows that adjacent angles complement each other, since they are a continuation of one another.
Definition 7
The two angles are called vertical, if the sides of one are complementary half-lines of the other or are continuations of the sides of the other. The picture below shows an image of vertical angles.
When straight lines intersect, 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. Let's look at which angle is considered larger, 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.
Two angles are given. It is necessary to come to a conclusion whether these angles are equal or not.
It is known that there is an overlap of the vertices of two angles and the sides of the first angle with any other side of the second. That is, if there is a complete coincidence when the angles are superimposed, the sides of the given angles will align completely, the angles equal.
It may be that when superimposed the sides may not align, then the corners unequal, smaller of which consists of another, and more contains a complete different angle. Below are unequal angles that were not aligned when overlaid.
Straight angles are equal.
Measuring angles begins with measuring the side of the angle being measured and its internal area, filling which with unit angles and applying them to each other. It is necessary to count the number of laid angles, they predetermine the measure of the measured angle.
The angle unit can be expressed by any measurable angle. There are generally accepted units of measurement that are used in science and technology. They specialize in other titles.
The concept most often used degree.
Definition 8
One degree called an angle that has one hundred and eightieth part of a straight angle.
The standard designation for a degree is “°”, then one degree is 1°. Therefore, a straight angle consists of 180 such angles of one degree. All available corners are tightly laid to each other and the sides of the previous one are aligned with the next one.
It is known that the number of degrees in an angle is the very measure of the angle. An unfolded angle has 180 stacked angles in its composition. The figure below shows examples where the angle is laid 30 times, that is, one sixth of the unfolded, and 90 times, that is, half.
Minutes and seconds are used to accurately measure angles. They are used when the angle value is not a whole degree designation. These fractions of a degree allow for more accurate calculations.
Definition 9
in a minute called one sixtieth of a degree.
Definition 10
In a second called one sixtieth of a minute.
A degree contains 3600 seconds. Minutes are designated """, and seconds are """. The designation takes place:
1 ° = 60 " = 3600 "" , 1 " = (1 60) ° , 1 " = 60 "" , 1 "" = (1 60) " = (1 3600) ° ,
and the designation for an angle of 17 degrees 3 minutes and 59 seconds is 17 ° 3 "59"".
Definition 11
Let's give an example of the designation of the degree measure of an angle equal to 17 ° 3 "59 "". The entry has another form: 17 + 3 60 + 59 3600 = 17 239 3600.
To accurately measure angles, use this measuring device, like a protractor. When denoting the angle ∠ A O B and its degree measure of 110 degrees, a more convenient notation is used ∠ A O B = 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 rotation angles. The value of angles is always expressed as a real number. Right angle- This 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 angle - (90, 180). Three types of angles are clearly shown below.
Any degree measure of any angle has the same value. A larger angle has a correspondingly larger degree measure than a smaller one. The degree measure of one angle is the sum of all available degree measures internal corners. Below is a figure showing the angle AOB, consisting of 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 everyone adjacent angles are equal to 180 degrees, because they all make up a straight angle.
It follows that any vertical angles are equal. If we consider this as an example, we find that the angles 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 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 denoting the angles of polygons. What is a radian called?
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 dot, with two points on the circle connected and converted into radii O A and O B. By definition given triangle A O B is equilateral, which means the length of the arc A B is equal to the lengths of the radii O B and O A.
The designation of the angle is taken to be “rad”. That is, writing 5 radians is abbreviated as 5 rad. Sometimes you can find a notation called pi. Radians do not depend on the length of a given circle, since the figures have a certain limitation by the angle and its arc with the center located at the vertex of the 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 arc length of the central angle by the length of its radius.
In practice they use converting degrees to radians and radians to degrees for more convenient solution tasks. This article contains information about the connection between the degree measure and the radian, where you can study in detail the conversions from degrees to radians and vice versa.
Drawings are used to visually and conveniently depict arcs and angles. It is not always possible to correctly depict and mark this or that angle, arc or name. Equal angles are designated by the same number of arcs, and unequal angles by a different number. 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. It doesn't have that much important. Below is a picture showing their designation.
Angle symbols should be kept simple so as not to interfere with other meanings. When solving a problem, it is recommended to highlight only the angles necessary for the solution, so as not to clutter 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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