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 the 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 talk 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 of adjacent angles and two pairs of vertical angles.
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 an angle and the degree measure of an angle, it follows that in geometry, the measure of an 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 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 like “”, 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 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 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.
Let's start by defining what an angle is. Firstly, it is a geometric figure. Secondly, it is formed by two rays, which are called the sides of the angle. Thirdly, the latter emerge from one point, which is called the vertex of the angle. Based on these features, we can create a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degree value, by location relative to each other and relative to the circle. Let's start with the types of angles according to their magnitude.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - straight, obtuse, acute and straight angles.
Straight
It looks like this:
Its degree measure is always 90 o, in other words, a right angle is an angle of 90 degrees. Only such quadrilaterals as square and rectangle have them.
Blunt
It looks like this:
The degree measure of an obtuse angle is always greater than 90°, but less than 180°. It can be found in quadrilaterals such as a rhombus, an arbitrary parallelogram, and in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It is found in all quadrilaterals except the square and any parallelogram.
Expanded
The unfolded angle looks like this:
It does not occur in polygons, but is no less important than all the others. A straight angle is a geometric figure whose degree measure is always 180º. You can construct adjacent angles on it by drawing one or more rays from its vertex in any direction.
There are several other minor types of angles. They are not studied in schools, but it is necessary to at least know about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The name of the angle itself already indicates its size. Its internal area is 0°, and the sides lie on top of each other as shown in the figure.
2. Oblique
An oblique angle can be a straight angle, an obtuse angle, an acute angle, or a straight angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex angles are zero, straight, obtuse, acute and straight angles. As you already understood, the degree measure of a convex angle is from 0° to 180°.
4. Non-convex
Angles with degree measures from 181° to 359° inclusive are non-convex.
5. Full
A complete angle is 360 degrees.
These are all types of angles according to their magnitude. Now let's look at their types according to their location on the plane relative to each other.
1. Additional
These are two acute angles forming one straight line, i.e. their sum is 90 o.
2. Adjacent
Adjacent angles are formed if a ray is passed through the unfolded angle, or rather through its vertex, in any direction. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two straight lines intersect. Their degree measures are equal.
Now let's move on to the types of angles located relative to the circle. There are only two of them: central and inscribed.
1. Central
A central angle is an angle with its vertex at the center of the circle. Its degree measure is equal to the degree measure of the smaller arc subtended by the sides.
2. Inscribed
An inscribed angle is an angle whose vertex lies on a circle and whose sides intersect it. Its degree measure is equal to half the arc on which it rests.
That's it for the angles. Now you know that in addition to the most famous ones - acute, obtuse, straight and deployed - there are many other types of them in geometry.
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Miscellaneous
Miscellaneous
An angle is a geometric figure that consists of two different rays emanating from one point. In this case, these rays are called sides of the angle. The point that is the beginning of the rays is called the vertex of the angle. In the picture you can see the angle with the vertex at the point ABOUT, and the parties k And m.
Points A and C are marked on the sides of the angle. This angle can be designated as angle AOC. In the middle there must be the name of the point at which the vertex of the angle is located. There are also other designations, angle O or angle km. In geometry, instead of the word angle, a special symbol is often written.
Developed and non-expanded angle
If both sides of an angle lie on the same straight line, then such an angle is called expanded angle. That is, one side of the angle is a continuation of the other side of the angle. The figure below shows the expanded angle O.
It should be noted that any angle divides the plane into two parts. If the angle is not unfolded, then one of the parts is called the internal region of the angle, and the other is called the external region of this angle. The figure below shows an undeveloped angle and marks the outer and inner regions of this angle.
In the case of a developed angle, either of the two parts into which it divides the plane can be considered the outer region of the angle. We can talk about the position of a point relative to an angle. A point can lie outside the corner (in the outer region), can be located on one of its sides, or can lie inside the corner (in the inner region).
In the figure below, point A lies outside angle O, point B lies on one side of the angle, and point C lies inside the angle.
Measuring angles
To measure angles there is a device called a protractor. The unit of angle is degree. It should be noted that each angle has a certain degree measure, which is greater than zero.
Depending on the degree measure, angles are divided into several groups.
Let's start by defining what an angle is. Firstly, it is Secondly, it is formed by two rays, which are called the sides of the angle. Thirdly, the latter emerge from one point, which is called the vertex of the angle. Based on these features, we can create a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degree value, by location relative to each other and relative to the circle. Let's start with the types of angles according to their magnitude.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - straight, obtuse, acute and straight angles.
Straight
It looks like this:
Its degree measure is always 90 o, in other words, a right angle is an angle of 90 degrees. Only such quadrilaterals as square and rectangle have them.
Blunt
It looks like this:
The degree measure is always more than 90 o, but less than 180 o. It can be found in quadrilaterals such as a rhombus, an arbitrary parallelogram, and in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It is found in all quadrilaterals except the square and any parallelogram.
Expanded
The unfolded angle looks like this:
It does not occur in polygons, but is no less important than all the others. A straight angle is a geometric figure whose degree measure is always 180º. You can build on it by drawing one or more rays from its top in any direction.
There are several other minor types of angles. They are not studied in schools, but it is necessary to at least know about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The name of the angle itself already indicates its size. Its internal area is 0°, and the sides lie on top of each other as shown in the figure.
2. Oblique
An oblique angle can be a straight angle, an obtuse angle, an acute angle, or a straight angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex angles are zero, straight, obtuse, acute and straight angles. As you already understood, the degree measure of a convex angle is from 0° to 180°.
4. Non-convex
Angles with degree measures from 181° to 359° inclusive are non-convex.
5. Full
A complete angle is 360 degrees.
These are all types of angles according to their magnitude. Now let's look at their types according to their location on the plane relative to each other.
1. Additional
These are two acute angles forming one straight line, i.e. their sum is 90 o.
2. Adjacent
Adjacent angles are formed if a ray is passed through the unfolded angle, or rather through its vertex, in any direction. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two straight lines intersect. Their degree measures are equal.
Now let's move on to the types of angles located relative to the circle. There are only two of them: central and inscribed.
1. Central
A central angle is an angle with its vertex at the center of the circle. Its degree measure is equal to the degree measure of the smaller arc subtended by the sides.
2. Inscribed
An inscribed angle is an angle whose vertex lies on a circle and whose sides intersect it. Its degree measure is equal to half the arc on which it rests.
That's it for the angles. Now you know that in addition to the most famous ones - acute, obtuse, straight and deployed - there are many other types of them in geometry.
Each angle, depending on its size, has its own name:
| Angle type | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Straight | Equal to 90°. In a drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other. |
 |
| Blunt | More than 90° but less than 180° |  |
| Expanded | Equal to 180° A straight angle is equal to the sum of two right angles, and a right angle is half of a straight angle. |
 |
| Convex | More than 180° but less than 360° |  |
| Full | Equal to 360° |  |
The two angles are called adjacent, if they have one side in common, and the other two sides form a straight line:
Angles MOP And PON adjacent, since the beam OP- the common side, and the other two sides - OM And ON make up a straight line.
The common side of adjacent angles is called oblique to straight, on which the other two sides lie, only in the case when adjacent angles are not equal to each other. If adjacent angles are equal, then their common side will be perpendicular.
The sum of adjacent angles is 180°.
The two angles are called vertical, if the sides of one angle complement the sides of the other angle to straight lines:
Angles 1 and 3, as well as angles 2 and 4, are vertical.
Vertical angles are equal.
Let us prove that the vertical angles are equal:
The sum of ∠1 and ∠2 is a straight angle. And the sum of ∠3 and ∠2 is a straight angle. So these two amounts are equal:
∠1 + ∠2 = ∠3 + ∠2.
In this equality, there is an identical term on the left and right - ∠2. Equality will not be violated if this term on the left and right is omitted. Then we get it.