Geometric figure angle: definition of angle, measurement of angles, notations and examples. Straight, obtuse, acute and developed angle Which angle is called developed?

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.

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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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

A branch of mathematics that deals with the study of the properties of various figures (points, lines, angles, two-dimensional and three-dimensional objects), their sizes and relative positions. For ease of teaching, geometry is divided into planimetry and stereometry. IN… … Collier's Encyclopedia

1) A closed broken line, namely: if different points, no consecutive three of them, lie on the same straight line, then the set of segments is called. polygon (see Fig. 1). M. can be spatial or flat (below... ... Mathematical Encyclopedia

across- ▲ at a maximum angle, transverse oblique angle. across at right angles. . right angle angle of maximum deviation; an angle equal to its adjacent one; quarter turn. perpendicular. perpendicular at right angles. perpendicular...... Ideographic Dictionary of the Russian Language

degree- a, m. 1) A unit of measurement for a plane angle, equal to 1/90 of a right angle or, respectively, 1/360 of a circle. An angle of 90 degrees is called a right angle. The rotated angle is 180 degrees. 2) A unit of measurement for a temperature interval having... ... Popular dictionary of the Russian language

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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:

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 that are less than 90 degrees in size.
3. Obtuse angle. Here there can be all angles ranging from 90 to 180 degrees.
4. 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:

1. The sides of the rotated angle are antiparallel and form a straight line.
2. The rotated angle is always 180˚.
3. Two adjacent angles together always form a straight angle.
4. A full angle, which is 360˚, consists of two unfolded ones and is equal to their sum.
5. 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.

1. What is the magnitude of a straight angle if its sides form a vertical line?
2. Will two angles be adjacent if the first is 72˚ and the other is 118˚?
3. If a complete angle consists of two reverse angles, then how many right angles does it have?
4. 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:

1. No matter how the rotated angle is located, it is always, by definition, equal to 180˚.
2. 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.
3. A straight angle contains two right angles. And since the complete one has two unfolded ones, it means there will be 4 straight lines.
4. 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.