An acute angle is an angle whose measure is up to 90 degrees.
A right angle is an angle whose measure is 90 degrees.
An obtuse angle is an angle whose measure is greater than 90 degrees. An acute angle is an angle less than 90°. An obtuse angle is an angle greater than 90° but less than 180°. A right angle is an angle = 90°.
20. What angles are called adjacent? What is their sum?
Adjacent corners- two angles with a common vertex, one of the sides of which is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is 180°. Or
Two angles are called adjacent, if they have one side in common, and the other sides are additional rays. the sum of adjacent angles is 180°. Each of these angles complements the other to a full angle.
21. What angles are called vertical? What property do they have?
Vertical angles - two angles whose sides of one are extensions of the sides of the other. Vertical angles are equal. ( Angles are called vertical formed by intersecting straight lines and not adjacent to each other, that is, they do not have a common side, but the vertical angles have a vertex at one point. Vertical angles are equal to each other).
22. What lines are called perpendicular? Two intersecting lines are called perpendicular(or mutually perpendicular) if they form four right angles. Or Perpendicular lines are lines that intersect at 90 degrees. Or two straight lines that form right angles when they intersect, called perpendicular.
23. Explain what a segment is called a perpendicular drawn from a given point to a given line. What is the base of a perpendicular? is a line segment perpendicular to the given one, which has one of its ends at their intersection point. This end of the segment is called the base of the perpendicular. Perpendicular to this line is a line segment perpendicular to the given one, which has one of its ends at their intersection point. Endpoint of a segment on a given line , is called the base of the perpendicular.
24. What is a theorem and proof of a theorem? In mathematics, a statement whose validity is established by reasoning is called a theorem, and the reasoning itself is called a proof of the theorem.
Theorem- a statement for which there is a proof in the theory under consideration (in other words, a conclusion). Unlike theorems, axioms are called statements that, within the framework of a particular theory, are accepted as true without any evidence or justification. Proof is a statement that explains the theorem. Theorem - a hypothesis that needs to be proven; A hypothesis always needs to be proven. Proof - arguments confirming the validity, correctness of the theorem.
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 outer area), may be located 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.
An angle greater than a right angle and less than a deployed ... Big Encyclopedic Dictionary
OBTUSE ANGLE- (see), greater than its adjacent angle; he is always more right angle, but less than expanded ... Great Polytechnic Encyclopedia
Obtuse angle- STUPID, oh, oh; dumb, dumb, dumb, dumb and dumb. Dictionary Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
obtuse angle- — Topics oil and gas industry EN broad angleobtuse angle … Technical Translator's Handbook
obtuse angle- an angle greater than a right angle and less than a straight angle. * * * OBTAIN ANGLE OBTAIN ANGLE, an angle greater than straight and less than deployed ... encyclopedic Dictionary
OBTUSE ANGLE- an angle greater than a right and less than a deployed ... Natural science. encyclopedic Dictionary
STUPID- STUPID, stupid, stupid; dumb, dumb, dumb. 1. Not sharp enough to scratch or prick easily. Dull knife. Dumb saw. Dull needle. Dull scissors. || Rounded, widening towards the end. The blunt bow of the boat. The blunt end of the egg. Dull protrusion. 2. change… … Explanatory Dictionary of Ushakov
STUPID- STUPID, opposite spicy; thick, bran at the end, or blunt; | thick on the rib, obtuse. Dull awl. Dumb cape. Knives are blunt, even on horseback. chill! You will crumble with a blunt ax, but you will not trim. Scissors are blunt, they only pinch, not cut. Like… … Dahl's Explanatory Dictionary
CORNER- angle, about the angle, on (in) the corner and (mat.) in the corner, m. 1. Part of the plane between two straight lines emanating from one point (mat.). The top of the corner. The sides of the corner. Angle measurement in degrees. Right angle. (90°). Sharp corner. (less than 90°). Obtuse angle.… … Explanatory Dictionary of Ushakov
STUPID- STUPID, oh, oh; dumb, dumb, dumb, dumb and dumb. 1. Insufficiently honed, such that it is difficult to cut, prick the eye. T. knife. T. tool. 2. Not tapering towards the end with an acute angle. T. beak. T. bow of the boat. Shoes with blunt toes. 3. trans. Inexpressive… Explanatory dictionary of Ozhegov
Books
- On Proof in Geometry, A.I. Fetisov, Once, at the very beginning school year I had to overhear a conversation between two girls. The eldest of them moved to the sixth grade, the youngest to the fifth. The girls shared their impressions of the lessons, ... Category: Mathematics Publisher: Book on Demand, Manufacturer:
Each angle, depending on its size, has its own name:
| Angle view | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Straight | Equal to 90°. In the drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other. |
 |
| Stupid | Greater than 90° but less than 180° |  |
| deployed | Equals 180° A straight angle is equal to the sum of two right angles, and a right angle is half the straight angle. |
 |
| Convex | More than 180° but less than 360° |  |
| Full | Equals 360° |  |
The two corners are called related, if they have one side in common, and the other two sides form a straight line:
corners MOP and pon adjacent since the beam OP- the common side, and the other two sides - OM and ON make up a straight line.
The common side of adjacent angles is called oblique to straight, on which the other two sides lie, only if the adjacent angles are not equal to each other. If adjacent angles are equal, then their common side will be perpendicular.
The sum of adjacent angles is 180°.
The two corners are called vertical, if the sides of one angle complement to straight lines the sides of another angle:
Angles 1 and 3, as well as angles 2 and 4, are vertical.
Vertical angles are equal.
Let's prove that the vertical angles are equal:
The sum of ∠1 and ∠2 is a straight angle. And the sum of ∠3 and ∠2 is a straight angle. So these two sums are equal:
∠1 + ∠2 = ∠3 + ∠2.
In this equality, on the left and on the right there is the same term - ∠2. Equality is not violated if this term on the left and on the right is omitted. Then we get.
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 a few at least important objects in geometry, it 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 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.
- 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 resulting 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, 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.
