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.

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.

The 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

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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“Basic concepts of geometry” - Test for the equality of a triangle. Segments. Geometry. Adjacent and vertical angles. Construction of parallel lines. Construction of a triangle. Conclusions. The lines are parallel. Peaks. The simplest geometric shapes. What figure is called a triangle. Equal segments have equal lengths. An angle is a geometric figure that consists of a point and two rays.

“Geometry in tables” - Coordinates of a point and coordinates of a vector in space Dot product of vectors in space Motion Cylinder Cone Sphere and ball Volume of a rectangular parallelepiped Volume of a straight prism and cylinder Volume of an inclined prism Volume of a pyramid Volume of a cone Volume of a sphere and area of ​​a sphere. Geometry tables.

“Geometry 8th grade” - Each statement is based on what has already been proven. Every building has a foundation. The concept of the theorem. An axiom is a statement whose truth is accepted without proof. Every mathematical statement obtained through logical proof is a theorem. So, by going through the theorems, you can get to the axioms.

“Geometry is a science” - Geometry consists of two sections: planimetry and stereometry. What geometric figure was the distinctive sign of the Pythagoreans? What shape did the Pythagoreans think the entire universe had? Answer: 580 – 500 BC era. When did Ancient Greece exist? Introduction. Answer: “Flatness”. The Pythagoreans closely associated the explanation of the structure of the world with geometry.

"Geometric Terms" - Cone. Pyramid. Radius and center. Diagonal. Geometry. Square. Rhombus. Cube Trapezoid. The emergence of geometric terms. Dot. Line. Cylinder. Hypotenuse and leg. Sphere. Prism. From the history of geometric terms.

“What geometry studies” - The word “parallel” comes from the Greek “parallelos” - to walk side by side. History of geometry. The transformations were mainly limited to similarities. L=(P1+P2)/2 L – circumference P1 - perimeter of a large square P2 - perimeter of a small square. Vstraight Geometry in Ancient Greece. Muse of Geometry, Louvre. We will find out where it came from and what geometry used to be.

There are a total of 24 presentations in the topic

What is an angle?

An angle is a figure formed by two rays emanating from one point (Fig. 160).
Rays forming corner, are called the sides of the angle, and the point from which they emerge is the vertex of the angle.
In Figure 160, the sides of the angle are the rays OA and OB, and its vertex is point O. This angle is designated as follows: AOB.

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

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

Instead of the word “angle” the sign is written.

For example, AOB, O.

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

Like all geometric shapes, angles are compared using overlap.

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 vertex of the angle SOK (Fig. 163) a ray OR is drawn. He splits the angle SOK into two angles - COP and ROCK. Each of these angles is less than the angle SOC.

Write: COP< COK и POK < COK.

Straight and straight angle

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

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

Fold a sheet 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 a reverse angle. Such angles are called right angles.

A right angle is half a turned angle.

Drawing triangle

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

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

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

As a result, we obtain a right angle AOB.

Questions to the topic

1.What is an angle?
2.Which angle is called turned?
3.What angles are called equal?
4.What angle is called a right angle?
5.How do you build a right angle using a drawing triangle?

You and I already know that any angle divides the plane into two parts. But, if an angle has both sides lying on the same straight line, then such an angle is called unfolded. That is, in a rotated angle, one side of it is a continuation of the other side of the angle.

Now let's look at the drawing, which exactly shows the unfolded angle O.

If we take and draw a ray from the vertex of the unfolded angle, then it will divide this unfolded 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, if we draw an arbitrary ray from the vertex of the unfolded angle, which is not a bisector, then such a ray will divide the unfolded angle into two angles, one of which will be acute and the other obtuse.

Properties of a rotated angle

A straight angle has the following properties:

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

Measuring angles

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

That is, if you and I take any plane of a circle and divide it by radii into 360 equal parts, then 1/360 of a given 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 complete.

Now we will take and divide the plane of the circle using two radii lying on the same 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 obtained an angle that is equal to the half-plane of a circle and has 180°. This is the turned 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. How many parts do these rays divide into? 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 the angle MOD and draw the ray OT inside it. Name and label the angles into which this ray divides the angle MOD.

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

1618. Using a drawing triangle, draw 4 right angles in different positions.

1619. Using a drawing triangle, find right angles in Figure 171. Write down their designations.

1620. Identify right angles in the classroom.

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

1629. What percentage 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 in 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 figure 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 purchased 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 Alexandria Lighthouse, built in ancient times, which was called one of the seven wonders of the world, is 1.7 times higher than the towers of the Moscow Kremlin, but 119 m lower than the building of Moscow University. Find the height of each of these structures if the towers of the Moscow Kremlin are 49 m lower Alexandria lighthouse.

1635. Use a microcalculator to find:

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 was dug up. How many ares are left to dig up?

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

1637. Follow these steps:

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 the corner BAC and mark one point each inside the corner, outside the corner and on the sides of the corner.

1639. Which of the 172 points marked in the figure 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. Using a drawing triangle, find the right angles in Figure 173.

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

1642. Find the meaning of the expression:

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

1643. A worker had to produce 450 parts. He made 60% of the parts on the first day, and the rest on the second. How many parts did you make? worker on the second day?

1644. The library had 8,000 books. A year later, their number increased by 2000 books. By what percentage did the number of books in the library increase?

1645. The trucks covered 24% of the intended route on the first day, 46% of the route on the second day, and the remaining 450 km on the third. 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 calf is 9 times less than the mass of an adult walrus. What is the mass of an adult walrus if, together with the calf, 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 general education institutions

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.

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.