What is an angle definition. Straight, obtuse, acute and developed angle

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

Injection- 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

Injection- 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 inner region of the 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 6

The 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 arc designations 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.

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This article will consider one of the main geometric shapes - the angle. After a general introduction to this concept, we will focus on a particular type of such a figure. The straight angle is an important concept in geometry and will be the focus of this article.

Introduction to the concept of a geometric angle

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

Also, before proceeding to the definition of the angle itself, you need to remember several equally important objects in geometry - this is a point, a straight line on a plane, and the plane itself. A straight line is the simplest geometric figure, which has neither beginning nor end. A plane is a surface that has two dimensions. Well, a ray (or a half-line) in geometry is a part of a straight line that has a beginning, but no end.

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

Types of angles and geometry

We know that angles can be quite different. And therefore, a little classification will be given below, which will help to better understand the types of angles and their main features. So, there are several types of angles in geometry:

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

The concept of a straight angle

Now let's look at the developed 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 below. This means that we can say with confidence that one of its sides is, in fact, a continuation of the other.

It is worth remembering the fact that such an angle can always be divided using a ray that comes out of its vertex. As a result, we get two angles, which in geometry are called adjacent.

Also, the developed 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 strictly in half. As for the straight 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 a straightened angle): 2 = 90˚.

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

Flat Corner Properties

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

1. The sides of a straight angle are antiparallel and form a straight line.
2. The value of the developed angle is always 180˚.
3. Two adjacent angles together always make a straight angle.
4. The full angle, which is 360˚, consists of two deployed ones and is equal to their sum.
5. Half a straightened angle is a right angle.

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

Problems with straight corners

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

1. What is a straight angle if its sides form a vertical line?
2. Will two angles be adjacent if the magnitude of the first is 72˚ and the other is 118˚?
3. If a full angle consists of two straight angles, how many right angles does it have?
4. A straight angle is divided by a beam into two such angles that their degree measures are related as 1:4. Calculate the resulting angles.

Solutions and answers:

1. No matter how the straight angle is located, it is always by definition equal to 180˚.
2. Adjacent corners have one common side. Therefore, to calculate the size of the angle that they put together, you just need to add the value of their degree measures. So, 72 +118 = 190. But by definition, a straight angle is 180˚, which means that two given angles cannot be adjacent.
3. A straight angle contains two right angles. And since there are two deployed ones in the full one, it means that there will be 4 straight lines in it.
4. If we call the desired angles a and b, then let x be the coefficient of proportionality for them, which means that a \u003d x, and accordingly b \u003d 4x. A straight 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 managed to answer all these questions without prompts and without peeking into the answers, then you are ready to move on to the next geometry lesson.

What is an angle?

An angle is a figure formed by two rays coming out of one point (Fig. 160).
The rays that form injection, are called the sides of the angle, and the point from which they exit is called the vertex of the angle.
In Figure 160, the sides of the angle are the rays OA and OB, and its vertex is the point O. This angle is designated as follows: AOB.

When writing an angle in the middle, write a letter denoting its vertex. An angle can also be denoted by a single letter - the name of its vertex.

For example, instead of "angle AOB" they write shorter: "angle O".

Instead of the word "corner" they write a sign.

For example, AOB, O.

In figure 161, points C and D lie inside the angle AOB, points X and Y lie outside this angle, and points M and H - on the sides of the corner.

Like all geometric shapes, angles are compared using an overlay.

If one angle can be superimposed on another so that they coincide, then these angles are equal.

For example, in Figure 162 ABC = MNK.

From the top of the SOK angle (Fig. 163) a beam OR was drawn. He splits the SOC angle into two angles - COP and ROCK. Each of these angles is less than the ROC angle.

Written by: COP< COK и POK < COK.

Straight and angled

Two complementary to each other beam form a folded corner. The sides of this angle together form a straight line on which lies the top of the expanded angle (Fig. 164).

The hour and minute hands of the clock form a developed angle at 6 o'clock (Fig. 165).

Let's bend a piece of paper in half twice, and then unfold it (Fig. 166).

The fold lines form 4 equal angles. Each of these angles is equal to half of the straightened angle. Such angles are called right angles.

A right angle is half a straightened angle.

drawing triangle

To construct a right angle, use the drawing triangle(Fig. 167). To construct a right angle, one of the sides of which is the ray OL, it is necessary:

a) arrange the drawing triangle so that the vertex of its right angle coincides with the point O, and one of the sides goes along the ray OA;

b) draw a ray OB along the second side of the triangle.

As a result, we get a right angle AOB.

Questions to the topic

1.What is an angle?
2. What angle is called deployed?
3. What angles are called equal?
4. What angle is called right?
5. How is a right angle built using a drawing triangle?

We already know that any angle divides the plane into two parts. But, if at an angle both sides lie on the same straight line, then such an angle is called deployed. That is, at a developed angle, one side of it is a continuation of its other side of the angle.

Now let's look at the figure, which just shows the developed angle O.

If we take and draw a ray from the vertex of a straight angle, then it will divide this straight angle into two more angles, which will have one common side, and the other two angles will form a straight line. That is, from one unfolded corner, we got two adjacent ones.

If we take a straight angle and draw a bisector, then this bisector will divide the straight angle into two right angles.

And, in the event that we draw an arbitrary ray from the vertex of the developed angle, which is not a bisector, then such a ray will divide the expanded angle into two angles, one of which will be acute and the other obtuse.

Flat Corner Properties

The expanded angle has the following properties:

First, the sides of a straight angle are antiparallel and form a straight line;
secondly, the developed angle is 180°;
thirdly, two adjacent angles form a straight angle;
fourthly, the developed angle is half of the full angle;
fifthly, the full angle will be equal to the sum of two developed angles;
sixth, half of the straightened angle is a right angle.

Angle measurement

To measure any angle, a protractor is most often used for these purposes, in which the unit of measurement is one degree. When measuring angles, it should be remembered that any angle has its own specific degree measure, and naturally this measure is greater than zero. And the developed angle, as we already know, is equal to 180 degrees.

That is, if we take any plane of a circle and divide it by radii into 360 equal parts, then 1/360 of this circle will be an angular degree. As you already know, a degree is indicated by a certain icon, which looks like this: "°".

Now we also know that one degree 1° = 1/360 of a circle. If the angle is equal to the plane of the circle and is 360 degrees, then such an angle is full.

And now we take and divide the plane of the circle with the help of two radii lying on one straight line into two equal parts. Then in this case, the plane of the semicircle will be half the full angle, that is, 360: 2 = 180 °. We have received an angle that is equal to the half-plane of the circle and has 180 °. This is the twisted angle.

Practical task

1613. Name the angles shown in Figure 168. Write down their designations.

1614. Draw four rays: OA, OB, OS and OD. Write down the names of the six angles whose sides are these rays. Into how many parts do these rays divide plane?

1615. Indicate which points in Figure 169 lie inside the angle KOM. Which points lie outside this angle? Which points are on the OK side and which are on the OM side?

1616. Draw an angle MOD and draw a ray OT inside it. Name and label the angles into which this ray divides angle MOD.

1617. The minute hand in 10 minutes turned to the angle AOB, in the next 10 minutes - to the angle BOC, and in another 15 minutes - to the angle COD. Compare the angles AOB and BOC, BOC and COD, AOC and AOB, AOC and COD (Fig. 170).

1618. Use the drawing triangle to draw 4 right angles in different positions.

1619. Using the drawing triangle, find right angles in figure 171. Write down their designations.

1620. Point out the right angles in the classroom.

a) 0.09 200; b) 208 0.4; c) 130 0.1 + 80 0.1.

1629. How many percent of 400 is the number 200; 100; 4; 40; 80; 400; 600?

1630. Find the missing number:

a) 2 5 3 b) 2 3 5
13 6 12 1
2 3? 42?

1631. Draw a square whose side is equal to the length of 10 cells of the notebook. Let this square represent a field. Rye occupies 12% of the field, oats - 8%, wheat - 64%, and the rest of the field is occupied by buckwheat. Show in the picture the part of the field occupied by each crop. What percentage of the field is buckwheat?

1632. During the school year, Petya used up 40% of the notebooks purchased at the beginning of the year, and he had 30 notebooks left. How many notebooks were bought for Petya at the beginning of the school year?

1633. Bronze is an alloy of tin and copper. What percentage of the alloy is copper in a piece of bronze, consisting of 6 kg of tin and 34 kg of copper?

1634. The lighthouse of Alexandria, built in antiquity, which was called one of the seven wonders of the world, is 1.7 times higher than the towers of the Moscow Kremlin, but lower than the building of Moscow University by 119 m. Find the height of each of these structures if the towers of the Moscow Kremlin are 49 m lower Lighthouse of Alexandria.

1635. Find with the help of a microcalculator:

a) 4.5% of 168; c) 28.3% of 569.8;
b) 147.6% of 2500; d) 0.09% of 456,800.

1636. Solve the problem:

1) The area of ​​the garden is 6.4 a. On the first day, 30% of the garden was dug up, and on the second day, 35% of the garden. How many ares are left to dig?

2) Serezha had 4.8 hours of free time. He spent 35% of that time reading a book and 40% watching TV shows. How much time does he have left?

1637. Do the following:

1) ((23,79: 7,8 - 6,8: 17) 3,04 - 2,04) 0,85;
2) (3,42: 0,57 9,5 - 6,6) : ((4,8 - 1,6) (3,1 + 0,05)).

1638 Draw an angle BAC and mark one point each inside the angle, outside the angle, and on the sides of the angle.

1639. Which of the points marked in figure 172 lie inside the angle AMK. Which point lies inside the angle AMB> but outside the angle AMK. Which points lie on the sides of the angle AMK?

1640. Use the drawing triangle to find the right angles in figure 173.

1641. Construct a square with a side of 43 mm. Calculate its perimeter and area.

1642. Find the value of the expression:

a) 14.791: a + 160.961: b, if a = 100, b = 10;
b) 361.62s + 1848: d if c = 100, d = 100.

1643. The worker had to make 450 parts. On the first day, he made 60% of the parts, and the rest on the second. How many parts did worker on the second day?

1644. There were 8,000 books in the library. A year later, their number increased by 2000 books. By what percentage has the number of books in the library increased?

1645. Trucks on the first day covered 24% of the intended path, on the second day - 46% of the path, and on the third - the remaining 450 km. How many kilometers did these trucks travel?

1646. Find how many are:

a) 1% of a ton; c) 5% of 7 tons;
b) 1% of a liter; d) 6% of 80 km.

1647. The mass of a walrus cub is 9 times less than the mass of an adult walrus. What is the mass of an adult walrus if, together with the cub, their mass is 0.9 tons?

1648. During the maneuvers, the commander left 0.3 of all his soldiers to guard the crossing, and divided the rest into 2 detachments to defend two heights. The first detachment had 6 times more soldiers than the second. How many soldiers were in the first detachment if there were 200 soldiers in total?

N.Ya. VILENKIN, V. I. ZHOKHOV, A. S. CHESNOKOV, S. I. SHVARTSBURD, Mathematics Grade 5, Textbook for educational institutions

An angle is a geometric figure, which consists of two different rays emanating from one point. In this case, these rays are called the 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 corner with the vertex at the point O, and the parties k and m.

Points A and C are marked on the sides of the corner. This corner can be designated as the angle AOC. In the middle must be the name of the point at which the corner vertex is located. There are also other designations, the angle O or the angle km. In geometry, instead of the word angle, a special icon is often written.

Revolved and non-revolved angle

If both sides of an angle lie on the same straight line, then such an angle is called deployed angle. That is, one side of the corner is a continuation of the other side of the corner. The figure below shows the angle O.

It should be noted that any angle divides the plane into two parts. If the corner is not expanded, then one of the parts is called the inner region of the corner, and the other is the outer region of this corner. The figure below shows a non-flattened corner and marked the outer and inner areas of this corner.

In the case of a developed angle, any 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. The point may lie outside the corner (in the outer region), may be on one of its sides, or may lie inside the corner (in the inner region).

In the figure below, point A lies outside corner O, point B lies on one side of the corner, and point C lies inside the corner.

Angle measurement

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.

Angle measure

The angle in is measured in degrees (degree, minute, second), in revolutions - the ratio of the length of the arc s to the circumference L, in radians - the ratio of the length of the arc s to the radius r; historically, the hail measure for measuring angles was also used; at present, it is almost never used.

1 turn = 2π radians = 360° = 400 degrees.

In nautical terminology, angles are indicated by points.

Corner types

Adjacent angles are acute (a) and obtuse (b). Reversed angle (c)

In addition, the angle between smooth curves at the tangent point is considered: by definition, its value is equal to the angle between the tangents to the curves.

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