Angular measure
Angle b is measured in degrees (degrees, minutes, seconds), in revolutions - the ratio of the arc length s to the circumference L, in radians - the ratio of the arc length s to the radius r; Historically, the grad measure of angles was also used; nowadays it is almost never used.
1 revolution = 2π radians = 360° = 400 degrees.
In maritime terminology, angles are designated by rhumbs.
Types of angles
Adjacent angles - acute (a) and obtuse (b). Straight angle (c)
In addition, the angle between smooth curves at the point of tangency is considered: by definition, its value is equal to the angle between the tangents to the curves.
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See what a “full angle” is in other dictionaries:
An angle equal to two right angles. *Surface development is a figure obtained in a plane by combining the points of a given surface with this plane in such a way that the lengths of the lines remain unchanged. Development of a curve, see Involute... Big Encyclopedic Dictionary
corner- ▲ difference direction (in space) angle, length of rotation from one direction to another; direction difference; part of a full turn (# tilt. form #). incline. inclined. deviation. evade (the road deviates to the right).... ...
Corner- Corners: 1 general view; 2 adjacent; 3 adjacent; 4 vertical; 5 expanded; 6 straight, sharp and obtuse; 7 between curves; 8 between a straight line and a plane; 9 between intersecting lines (not lying in the same plane) lines. ANGLE, geometric... ... Illustrated Encyclopedic Dictionary
A geometric figure consisting of two different rays emanating from one point. The rays are called sides U., and their common beginning is the vertex U. Let [ BA),[ BC) the sides of the angle, B its vertex, the plane defined by the sides U. The figure divides the plane... ... Mathematical Encyclopedia
An angle equal to two right angles. * * * DEVELOPED ANGLE DEVELOPED ANGLE, an angle equal to two right angles... encyclopedic Dictionary
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Angular measure
Angle b is measured in degrees (degrees, minutes, seconds), in revolutions - the ratio of the arc length s to the circumference L, in radians - the ratio of the arc length s to the radius r; Historically, the grad measure of angles was also used; nowadays it is almost never used.
1 revolution = 2π radians = 360° = 400 degrees.
In maritime terminology, angles are designated by rhumbs.
Types of angles
Adjacent angles - acute (a) and obtuse (b). Straight angle (c)
In addition, the angle between smooth curves at the point of tangency is considered: by definition, its value is equal to the angle between the tangents to the curves.
Wikimedia Foundation. 2010.
See what “Full angle” is in other dictionaries:
Unlegalized non-system unit. flat angle. 1 P.u.= 2PI rad 6.283 185 rad (see Radian) ... Big Encyclopedic Polytechnic Dictionary
The angle of vertical aiming of the gun barrel when firing, taking into account the pitching angles of the ship. Determined by instruments of the central artillery post. EdwART. Explanatory Naval Dictionary, 2010 ... Marine Dictionary
The horizontal aiming angle of the gun barrel when firing, taking into account the ship's pitching angles. Determined by the location of the central artillery post. EdwART. Explanatory Naval Dictionary, 2010 ... Marine Dictionary
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Full mechanical rotation angle of moving variable resistor system- 52. Total mechanical rotation angle of the moving variable resistor system Full mechanical rotation angle D. Mechanischer Drehwinkel E. Total mechanical rotation F. Course mécanique totale Total rotation angle of the moving variable resistor system ... ... Dictionary-reference book of terms of normative and technical documentation
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Go full force. Jarg. corner. Confess to committing a crime. Baldaev 1, 169. Two are complete, the third is not complete. Novg. Iron. About a small number of people where l. NOSE 2, 76...
Jarg. corner. Approved Everything is fine, things are going well. B., 159; Bykov, 202. /i> Probably from Yiddish or Hebrew, where the word is an assessment of the highest quality. Elistratov 1994, 537 ... Large dictionary of Russian sayings
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.
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 straight line and a plane. 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 of the 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 letters, then to correctly determine the angle, letters are written in a row corresponding 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 angles are indicated in different ways.
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 a measuring device such as 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 of interior angles. 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 transformed into radii O A and O B. By definition, this 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 problem solving. 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's not that 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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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.
This article will discuss one of the basic geometric shapes - an angle. After a general introduction to this concept, we will focus on a specific type of such a figure. Straight angle is an important concept in geometry, which will be the main topic of this article.
Introduction to Geometric Angle
In geometry there are a number of objects that form the basis of all science. The angle refers to them and is defined using the concept of a ray, so let's start with it.
Also, before you begin to determine 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 neither beginning nor end. A plane is a surface that has two dimensions. Well, a ray (or half-line) in geometry is a part of a 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 entirely in a certain plane and consists of two divergent rays with a common origin. Such rays are called sides of an angle, and the common beginning of the sides is its vertex.
Types of angles and geometry
We know that angles can be completely 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:
- Right angle. It is characterized by a value of 90 degrees, which means that its sides are always perpendicular to each other.
- Sharp corner. These angles include all their representatives that are less than 90 degrees in size.
- Obtuse angle. Here there can be all angles ranging from 90 to 180 degrees.
- Unfolded corner. It has a size of strictly 180 degrees and externally its sides form one straight line.
The concept of a straight angle
Now let's look at the rotated angle in more detail. This is the case when both sides lie on the same straight line, which can be clearly seen in the figure a little lower. This means that we can say with confidence that in a reversed angle, 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 emerges from its apex. As a result, we get two angles, which in geometry are called adjacent.
Also, the unfolded angle 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 exactly in half. As for the unfolded angle, its bisector divides it in such a way that two right angles of 90 degrees are formed. This is very easy to calculate mathematically: 180˚ (degree of the rotated angle): 2 = 90˚.
If we divide a rotated 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.
Properties of rotated corners
It will be convenient to consider this angle, bringing together all its main properties, which is what we did in this list:
- The sides of the rotated angle are antiparallel and form a straight line.
- The rotated angle is always 180˚.
- Two adjacent angles together always form a straight angle.
- A full angle, which is 360˚, consists of two unfolded ones and is equal to their sum.
- Half of a straight angle is a right angle.
So, knowing all these characteristics of this type of angles, we can use them to solve a number of geometric problems.
Problems with rotated angles
To see if you have grasped the concept of a straight angle, try answering the following few questions.
- What is the magnitude of a straight angle if its sides form a vertical line?
- Will two angles be adjacent if the first is 72˚ and the other is 118˚?
- If a complete angle consists of two reverse angles, then how many right angles does it have?
- A straight angle is divided by a ray into two angles such that their degree measures are in the ratio 1:4. Calculate the resulting angles.
Solutions and answers:
- No matter how the rotated angle is located, it is always, by definition, equal to 180˚.
- Adjacent angles have one side in common. Therefore, to calculate the size of the angle they make together, you just need to add the value of their degree measures. This means 72 +118 = 190. But by definition, a reversed angle is 180˚, which means that two given angles cannot be adjacent.
- A straight angle contains two right angles. And since the complete one has two unfolded ones, it means there will be 4 straight lines.
- If we call the desired angles a and b, then let x be the coefficient of proportionality for them, which means that a=x, and accordingly b=4x. The rotated 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 divided 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 were able to answer all these questions without prompts and without peeking at the answers, then you are ready to move on to the next geometry lesson.