What is a right angle 2. Straight, obtuse, acute and developed angle. How to mark an acute angle
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
An angle with a measure of 360 degrees is a complete angle.
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.
Look at the picture. (Fig. 1)
Rice. 1. Illustration for example
What geometric shapes are familiar to you?
Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the name of both these figures? This word is an angle (Fig. 2).
Rice. 2. Determining the angle
Today we will learn how to draw a right angle.
The name of this angle already has the word "straight". To correctly depict a right angle, we need a square. (Fig. 3)
Rice. 3. Square
The square itself already has a right angle. (Fig. 4)
Rice. 4. Right angle
He will help us to depict this geometric figure.
To correctly depict the figure, we must attach the square to the plane (1), circle its sides (2), name the vertex of the angle (3) and the rays (4).
1. 
2. 
3. 
4. 
Let's determine if there are straight lines among the available angles (Fig. 5). A square will help us with this.
Rice. 5. Illustration for example
Let's find the right angle of the square and apply it to the existing angles (Fig. 6).
Rice. 6. Illustration for example
We see that the right angle coincided with the PTO angle. This means that the PTO angle is right. Let's do the same operation again. (Fig. 7)
Rice. 7. Illustration for example
We see that the right angle of our square did not coincide with the COD angle. This means that the angle COD is not a right angle. Once again we apply the right angle of the square to the angle AOT. (Fig. 8)
Rice. 8. Illustration for example
We see that the AOT angle is much larger than the right angle. This means that the AOT angle is not a right angle.
In this lesson, we learned how to build a right angle using a square.
The word "angle" gave the name to many things, as well as geometric shapes: a rectangle, a triangle, a square, with which you can draw a right angle.
A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.
Look at the picture. (Fig. 1)
Rice. 1. Illustration for example
What geometric shapes are familiar to you?
Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the name of both these figures? This word is an angle (Fig. 2).
Rice. 2. Determining the angle
Today we will learn how to draw a right angle.
The name of this angle already has the word "straight". To correctly depict a right angle, we need a square. (Fig. 3)
Rice. 3. Square
The square itself already has a right angle. (Fig. 4)
Rice. 4. Right angle
He will help us to depict this geometric figure.
To correctly depict the figure, we must attach the square to the plane (1), circle its sides (2), name the vertex of the angle (3) and the rays (4).
1.
2.
3.
4.
Let's determine if there are straight lines among the available angles (Fig. 5). A square will help us with this.
Rice. 5. Illustration for example
Let's find the right angle of the square and apply it to the existing angles (Fig. 6).
Rice. 6. Illustration for example
We see that the right angle coincided with the PTO angle. This means that the PTO angle is right. Let's do the same operation again. (Fig. 7)
Rice. 7. Illustration for example
We see that the right angle of our square did not coincide with the COD angle. This means that the angle COD is not a right angle. Once again we apply the right angle of the square to the angle AOT. (Fig. 8)
Rice. 8. Illustration for example
We see that the AOT angle is much larger than the right angle. This means that the AOT angle is not a right angle.
In this lesson, we learned how to build a right angle using a square.
The word "angle" gave the name to many things, as well as geometric shapes: a rectangle, a triangle, a square, with which you can draw a right angle.
A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.
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.
Definition 1Corner- 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 of 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
Corner- 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 internal area of \u200b\u200bthe 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 6The 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 arch symbols 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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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. |
 |
| Blunt | 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.
