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
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 with 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:
- 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, having a size less than 90 degrees.
- Obtuse angle. All angles with a value from 90 to 180 degrees can also be here.
- 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:
- The sides of a straight angle are antiparallel and form a straight line.
- The value of the developed angle is always 180˚.
- Two adjacent angles together always make a straight angle.
- The full angle, which is 360˚, consists of two deployed ones and is equal to their sum.
- Half a straightened angle is a right angle.
So, knowing all these characteristics of a given 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.
- What is a straight angle if its sides form a vertical line?
- Will two angles be adjacent if the magnitude of the first is 72˚ and the other is 118˚?
- If a full angle consists of two straight angles, how many right angles does it have?
- A straight angle is divided by a beam into two such angles that their degree measures are related as 1:4. Calculate the obtained angles.
Solutions and answers:
- No matter how the straight angle is located, it is always by definition equal to 180˚.
- Adjacent corners have one common side. Therefore, in order to calculate the size of the angle that they make up 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.
- 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.
- 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.
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 come out of one point, which is called the apex of the corner. Based on these signs, we can make a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degrees, by location relative to each other and relative to the circle. Let's start with the types of angles by their size.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - right, obtuse, acute and developed angle.
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 quadrangles as a square and a rectangle have them.
Blunt
It looks like this:
The degree measure is always greater than 90 degrees, but less than 180 degrees. It can occur in such quadrangles as a rhombus, an arbitrary parallelogram, in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It occurs in all quadrilaterals, except for a square and an arbitrary parallelogram.
deployed
The expanded angle looks like this:
It does not occur in polygons, but it is no less important than all the others. A straight angle is a geometric figure, the degree measure of which is always 180º. You can build on it by drawing one or more rays from its vertex in any direction.
There are several other secondary types of angles. They are not studied in schools, but it is necessary to know at least about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The very name of the angle already speaks of its magnitude. Its interior area is 0 o, and the sides lie on top of each other as shown in the figure.
2. Oblique
Oblique can be straight, and obtuse, and acute, and developed angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex are zero, right, obtuse, acute and developed angles. As you already understood, the degree measure of a convex angle is from 0 o to 180 o.
4. Non-convex
Non-convex are angles with a degree measure from 181 o to 359 o inclusive.
5. Full
A complete angle is 360 degrees.
These are all types of angles according to their size. Now consider their types by location on the plane relative to each other.
1. Additional
These are two acute angles that form one straight line, i.e. their sum is 90 o.
2. Related
Adjacent angles are formed if a ray is drawn in any direction through a deployed, more precisely, through its top. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two 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
The central angle is the one with the 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 one whose vertex lies on the circle and whose sides intersect it. Its degree measure is equal to half of the arc on which it rests.
It's all about the corners. Now you know that in addition to the most famous - sharp, obtuse, straight and deployed - in geometry there are many other types of them.
Angle measure
The angle in is measured in degrees (degree, minute, second), 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 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.
Wikimedia Foundation. 2010 .
See what the "Developed angle" is in other dictionaries:
An angle equal to two right angles. * SCAN of a surface is a figure obtained in a plane with such a combination of points of a given surface with this plane, in which the lengths of the lines remain unchanged. Curve development see Involute ... Big Encyclopedic Dictionary
injection- ▲ direction difference (in space) angle extent of turn from one direction to another; direction difference; part of a full turn (tilt #. form #). incline. inclined. deviation. deviate (the road deviated to the right). ... ...
Injection- Corners: 1 general view; 2 adjacent; 3 adjacent; 4 vertical; 5 deployed; 6 straight, sharp and blunt; 7 between curves; 8 between a straight line and a plane; 9 between intersecting straight lines (not lying in the same plane) straight lines. ANGLE, geometric… … Illustrated Encyclopedic Dictionary
A geometric figure consisting of two different rays emanating from the same point. Rays called sides U., and their common beginning is the vertex U. Let [ BA), [ BC) the sides of the angle, B its vertex, the plane determined by the sides U. The figure divides the plane ... ... Mathematical Encyclopedia
An angle equal to two right angles. * * * REVELATED ANGLE REVELATED ANGLE, an angle equal to two right angles ... encyclopedic Dictionary
A branch of mathematics that studies the properties of various shapes (points, lines, angles, two-dimensional and three-dimensional objects), their size and relative position. For the convenience of teaching, geometry is divided into planimetry and solid geometry. AT… … Collier 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 an angle maximum, oblique angle transverse. across at a right angle. . right angle angle of maximum deflection; 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) The unit of measurement of a flat 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 expanded angle is 180 degrees. 2) A unit of measure for a temperature interval having ... ... Popular dictionary of the Russian language
Schwartz Christoffel's theorem, an important theorem in the theory of functions of a complex variable, bears the name of the German mathematicians Karl Schwartz and Alvin Christoffel. Very important from a practical point of view is the problem of conformal ... ... Wikipedia