How to find the sum of adjacent angles. Adjacent and vertical angles

Each angle, depending on its size, has its own name:

Angle type	Size in degrees	Example
Spicy	Less than 90°
Straight	Equal to 90°. In a drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other.
Blunt	More than 90° but less than 180°
Expanded	Equal to 180° A straight angle is equal to the sum of two right angles, and a right angle is half of a straight angle.
Convex	More than 180° but less than 360°
Full	Equal to 360°

The two angles are called adjacent, if they have one side in common, and the other two sides form a straight line:

Angles 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 in the case when 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 angles are called vertical, if the sides of one angle complement the sides of the other angle to straight lines:

Angles 1 and 3, as well as angles 2 and 4, are vertical.

Vertical angles are equal.

Let us 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 amounts are equal:

∠1 + ∠2 = ∠3 + ∠2.

In this equality, there is an identical term on the left and right - ∠2. Equality will not be violated if this term on the left and right is omitted. Then we get it.

What is an adjacent angle

Corner is a geometric figure (Fig. 1), formed by two rays OA and OB (sides of the angle), emanating from one point O (vertex of the angle).

ADJACENT CORNERS- two angles whose sum is 180°. Each of these angles complements the other to the full angle.

Adjacent angles- (Agles adjacets) those that have a common top and a common side. Mostly this name refers to angles of which the remaining two sides lie in opposite directions of one straight line drawn through.

Two angles are called adjacent if they have one side in common, and the other sides of these angles are complementary half-lines.

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In Figure 2, angles a1b and a2b are adjacent. They have a common side b, and sides a1, a2 are additional half-lines.

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Figure 3 shows straight line AB, point C is located between points A and B. Point D is a point not lying on straight AB. It turns out that angles BCD and ACD are adjacent. They have a common side CD, and sides CA and CB are additional half-lines of straight line AB, since points A, B are separated by the starting point C.

Adjacent angle theorem

Theorem: the sum of adjacent angles is 180°

Proof:
Angles a1b and a2b are adjacent (see Fig. 2) Ray b passes between sides a1 and a2 of the unfolded angle. Therefore, the sum of angles a1b and a2b is equal to the developed angle, that is, 180°. The theorem has been proven.

An angle equal to 90° is called a right angle. From the theorem on the sum of adjacent angles it follows that an angle adjacent to a right angle is also a right angle. An angle less than 90° is called acute, and an angle greater than 90° is called obtuse. Since the sum of adjacent angles is 180°, then the angle adjacent to an acute angle is an obtuse angle. An angle adjacent to an obtuse angle is an acute angle.

Adjacent angles- two angles with a common vertex, one of whose sides is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is 180°.

Definition 1. An angle is a part of a plane bounded by two rays with a common origin.

Definition 1.1. An angle is a figure consisting of a point - the vertex of the angle - and two different half-lines emanating from this point - the sides of the angle.
For example, angle BOC in Fig.1 Let us first consider two intersecting lines. When straight lines intersect, they form angles. There are special cases:

Definition 2. If the sides of an angle are additional half-lines of one straight line, then the angle is called developed.

Definition 3. A right angle is an angle measuring 90 degrees.

Definition 4. An angle less than 90 degrees is called an acute angle.

Definition 5. An angle greater than 90 degrees and less than 180 degrees is called an obtuse angle.
intersecting lines.

Definition 6. Two angles, one side of which is common and the other sides lie on the same straight line, are called adjacent.

Definition 7. Angles whose sides continue each other are called vertical angles.
In Figure 1:
adjacent: 1 and 2; 2 and 3; 3 and 4; 4 and 1
vertical: 1 and 3; 2 and 4
Theorem 1. The sum of adjacent angles is 180 degrees.
For proof, consider in Fig. 4 adjacent angles AOB and BOC. Their sum is the developed angle AOC. Therefore, the sum of these adjacent angles is 180 degrees.

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The connection between mathematics and music

“Thinking about art and science, about their mutual connections and contradictions, I came to the conclusion that mathematics and music are at the extreme poles of the human spirit, that all creative spiritual activity of man is limited and determined by these two antipodes and that everything between them is located.” what humanity has created in the fields of science and art."
G. Neuhaus
It would seem that art is a very abstract area from mathematics. However, the connection between mathematics and music is determined both historically and internally, despite the fact that mathematics is the most abstract of sciences, and music is the most abstract form of art.
Consonance determines the pleasant sound of a string
This musical system was based on two laws that bear the names of two great scientists - Pythagoras and Archytas. These are the laws:
1. Two sounding strings determine consonance if their lengths are related as integers forming the triangular number 10=1+2+3+4, i.e. like 1:2, 2:3, 3:4. Moreover, the smaller the number n in the ratio n:(n+1) (n=1,2,3), the more consonant the resulting interval.
2. The vibration frequency w of the sounding string is inversely proportional to its length l.
w = a:l,
where a is a coefficient characterizing the physical properties of the string.

I will also offer you a funny parody about an argument between two mathematicians =)

Geometry around us

Geometry in our life is of no small importance. Due to the fact that when you look around, it will not be difficult to notice that we are surrounded by various geometric shapes. We encounter them everywhere: on the street, in the classroom, at home, in the park, in the gym, in the school cafeteria, basically wherever we are. But the topic of today's lesson is adjacent coals. So let's look around and try to find angles in this environment. If you look closely at the window, you can see that some tree branches form adjacent corners, and in the partitions on the gate you can see many vertical angles. Give your own examples of adjacent angles that you observe in your environment.

Exercise 1.

1. There is a book on the table on a book stand. What angle does it form?
2. But the student is working on a laptop. What angle do you see here?
3. What angle does the photo frame form on the stand?
4. Do you think it is possible for two adjacent angles to be equal?

Task 2.

In front of you is a geometric figure. What kind of figure is this, name it? Now name all the adjacent angles that you can see on this geometric figure.

Task 3.

Here is an image of a drawing and painting. Look at them carefully and tell me what types of fish you see in the picture, and what angles you see in the picture.

Problem solving

1) Given two angles related to each other as 1: 2, and adjacent to them - as 7: 5. You need to find these angles.
2) It is known that one of the adjacent angles is 4 times larger than the other. What are the adjacent angles equal to?
3) It is necessary to find adjacent angles, provided that one of them is 10 degrees greater than the second.

Mathematical dictation to review previously learned material

1) Complete the drawing: straight lines a I b intersect at point A. Mark the smaller of the formed angles with the number 1, and the remaining angles - sequentially with the numbers 2,3,4; the complementary rays of line a are through a1 and a2, and line b is through b1 and b2.
2) Using the completed drawing, enter the necessary meanings and explanations in the gaps in the text:
a) angle 1 and angle .... adjacent because...
b) angle 1 and angle…. vertical because...
c) if angle 1 = 60°, then angle 2 = ..., because...
d) if angle 1 = 60°, then angle 3 = ..., because...

Solve problems:

1. Can the sum of 3 angles formed by the intersection of 2 straight lines equal 100°? 370°?
2. In the figure, find all pairs of adjacent angles. And now the vertical angles. Name these angles.

3. You need to find an angle when it is three times larger than its adjacent one.
4. Two straight lines intersected each other. As a result of this intersection, four corners were formed. Determine the value of any of them, provided that:

a) the sum of 2 angles out of four is 84°;
b) the difference between 2 angles is 45°;
c) one angle is 4 times less than the second;
d) the sum of three of these angles is 290°.

Lesson summary

1. name the angles that are formed when 2 straight lines intersect?
2. Name all possible pairs of angles in the figure and determine their type.

Homework:

1. Find the ratio of the degree measures of adjacent angles when one of them is 54° greater than the second.
2. Find the angles that are formed when 2 straight lines intersect, provided that one of the angles is equal to the sum of 2 other angles adjacent to it.
3. It is necessary to find adjacent angles when the bisector of one of them forms an angle with the side of the second that is 60° greater than the second angle.
4. The difference between 2 adjacent angles is equal to a third of the sum of these two angles. Determine the values of 2 adjacent angles.
5. The difference and sum of 2 adjacent angles are in the ratio 1:5 respectively. Find adjacent angles.
6. The difference between two adjacent ones is 25% of their sum. How do the values of 2 adjacent angles relate? Determine the values of 2 adjacent angles.

Questions:

What is an angle?
What types of angles are there?
What is the property of adjacent angles?

Subjects > Mathematics > Mathematics 7th grade

How to find an adjacent angle?

Mathematics is the oldest exact science, which is compulsorily studied in schools, colleges, institutes and universities. However, basic knowledge is always laid at school. Sometimes, the child is given quite complex tasks, but the parents are unable to help, because they simply forgot some things from mathematics. For example, how to find an adjacent angle based on the size of the main angle, etc. The problem is simple, but can cause difficulties in solving due to ignorance of which angles are called adjacent and how to find them.

Let's take a closer look at the definition and properties of adjacent angles, as well as how to calculate them from the data in the problem.

Definition and properties of adjacent angles

Two rays emanating from one point form a figure called a “plane angle”. In this case, this point is called the vertex of the angle, and the rays are its sides. If you continue one of the rays beyond the starting point in a straight line, then another angle is formed, which is called adjacent. Each angle in this case has two adjacent angles, since the sides of the angle are equivalent. That is, there is always an adjacent angle of 180 degrees.

The main properties of adjacent angles include

Adjacent angles have a common vertex and one side;
The sum of adjacent angles is always equal to 180 degrees or Pi if the calculation is carried out in radians;
The sines of adjacent angles are always equal;
The cosines and tangents of adjacent angles are equal but have opposite signs.

How to find adjacent angles

Usually three variations of problems are given to find the magnitude of adjacent angles

The value of the main angle is given;
The ratio of the main and adjacent angle is given;
The value of the vertical angle is given.

Each version of the problem has its own solution. Let's look at them.

The value of the main angle is given

If the problem specifies the value of the main angle, then finding the adjacent angle is very simple. To do this, just subtract the value of the main angle from 180 degrees, and you will get the value of the adjacent angle. This solution is based on the property of an adjacent angle - the sum of adjacent angles is always equal to 180 degrees.

If the value of the main angle is given in radians and the problem requires finding the adjacent angle in radians, then it is necessary to subtract the value of the main angle from the number Pi, since the value of the full unfolded angle of 180 degrees is equal to the number Pi.

The ratio of the main and adjacent angle is given

The problem may give the ratio of the main and adjacent angles instead of the degrees and radians of the main angle. In this case, the solution will look like a proportion equation:

We denote the proportion of the main angle as the variable “Y”.
The share related to the adjacent angle is denoted as the variable “X”.
The number of degrees that fall on each proportion will be denoted, for example, by “a”.
The general formula will look like this - a*X+a*Y=180 or a*(X+Y)=180.
We find the common factor of the equation “a” using the formula a=180/(X+Y).
Then we multiply the resulting value of the common factor “a” by the fraction of the angle that needs to be determined.

This way we can find the value of the adjacent angle in degrees. However, if you need to find a value in radians, then you simply need to convert the degrees to radians. To do this, multiply the angle in degrees by Pi and divide everything by 180 degrees. The resulting value will be in radians.

The value of the vertical angle is given

If the problem does not give the value of the main angle, but the value of the vertical angle is given, then the adjacent angle can be calculated using the same formula as in the first paragraph, where the value of the main angle is given.

A vertical angle is an angle that originates from the same point as the main one, but is directed in exactly the opposite direction. This results in a mirror image. This means that the vertical angle is equal in magnitude to the main one. In turn, the adjacent angle of the vertical angle is equal to the adjacent angle of the main angle. Thanks to this, the adjacent angle of the main angle can be calculated. To do this, simply subtract the vertical value from 180 degrees and get the value of the adjacent angle of the main angle in degrees.

If the value is given in radians, then it is necessary to subtract the value of the vertical angle from the number Pi, since the value of the full unfolded angle of 180 degrees is equal to the number Pi.

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1. Adjacent angles.

If we extend the side of any angle beyond its vertex, we get two angles (Fig. 72): ∠ABC and ∠CBD, in which one side BC is common, and the other two, AB and BD, form a straight line.

Two angles in which one side is common and the other two form a straight line are called adjacent angles.

Adjacent angles can also be obtained in this way: if we draw a ray from some point on a line (not lying on a given line), we will obtain adjacent angles.

For example, ∠ADF and ∠FDB are adjacent angles (Fig. 73).

Adjacent angles can have a wide variety of positions (Fig. 74).

Adjacent angles add up to a straight angle, so the sum of two adjacent angles is 180°

Hence, a right angle can be defined as an angle equal to its adjacent angle.

Knowing the size of one of the adjacent angles, we can find the size of the other angle adjacent to it.

For example, if one of the adjacent angles is 54°, then the second angle will be equal to:

180° - 54° = l26°.

2. Vertical angles.

If we extend the sides of the angle beyond its vertex, we get vertical angles. In Figure 75, the angles EOF and AOC are vertical; angles AOE and COF are also vertical.

Two angles are called vertical if the sides of one angle are continuations of the sides of the other angle.

Let ∠1 = \(\frac(7)(8)\) ⋅ 90°(Fig. 76). ∠2 adjacent to it will be equal to 180° - \(\frac(7)(8)\) ⋅ 90°, i.e. 1\(\frac(1)(8)\) ⋅ 90°.

In the same way, you can calculate what ∠3 and ∠4 are equal to.

∠3 = 180° - 1\(\frac(1)(8)\) ⋅ 90° = \(\frac(7)(8)\) ⋅ 90°;

∠4 = 180° - \(\frac(7)(8)\) ⋅ 90° = 1\(\frac(1)(8)\) ⋅ 90° (Fig. 77).

We see that ∠1 = ∠3 and ∠2 = ∠4.

You can solve several more of the same problems, and each time you will get the same result: the vertical angles are equal to each other.

However, to make sure that vertical angles are always equal to each other, it is not enough to consider individual numerical examples, since conclusions drawn from particular examples can sometimes be erroneous.

It is necessary to verify the validity of the properties of vertical angles by proof.

The proof can be carried out as follows (Fig. 78):

∠a +∠c= 180°;

∠b+∠c= 180°;

(since the sum of adjacent angles is 180°).

∠a +∠c = ∠b+∠c

(since the left side of this equality is equal to 180°, and its right side is also equal to 180°).

This equality includes the same angle With.

If we subtract equal amounts from equal quantities, then equal amounts will remain. The result will be: ∠a = ∠b, i.e. the vertical angles are equal to each other.

3. The sum of angles that have a common vertex.

In drawing 79, ∠1, ∠2, ∠3 and ∠4 are located on one side of a line and have a common vertex on this line. In sum, these angles make up a straight angle, i.e.

∠1 + ∠2 + ∠3 + ∠4 = 180°.

In Figure 80, ∠1, ∠2, ∠3, ∠4 and ∠5 have a common vertex. These angles add up to a full angle, i.e. ∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 360°.

Other materials

CHAPTER I.

BASIC CONCEPTS.

§eleven. ADJACENT AND VERTICAL CORNERS.

1. Adjacent angles.

If we extend the side of any angle beyond its vertex, we get two angles (Fig. 72): / And the sun and / SVD, in which one side BC is common, and the other two A and BD form a straight line.

Two angles in which one side is common and the other two form a straight line are called adjacent angles.

Adjacent angles can also be obtained in this way: if we draw a ray from some point on a line (not lying on a given line), we will obtain adjacent angles.
For example, / ADF and / FDВ - adjacent angles (Fig. 73).

Adjacent angles can have a wide variety of positions (Fig. 74).

Adjacent angles add up to a straight angle, so the umma of two adjacent angles is equal 2d.

Hence, a right angle can be defined as an angle equal to its adjacent angle.

Knowing the size of one of the adjacent angles, we can find the size of the other angle adjacent to it.

For example, if one of the adjacent angles is 3/5 d, then the second angle will be equal to:

2d- 3 / 5 d= l 2 / 5 d.

2. Vertical angles.

If we extend the sides of the angle beyond its vertex, we get vertical angles. In drawing 75, the angles EOF and AOC are vertical; angles AOE and COF are also vertical.

Two angles are called vertical if the sides of one angle are continuations of the sides of the other angle.

Let / 1 = 7 / 8 d(Figure 76). Adjacent to it / 2 will be equal to 2 d- 7 / 8 d, i.e. 1 1/8 d.

In the same way you can calculate what they are equal to / 3 and / 4.
/ 3 = 2d - 1 1 / 8 d = 7 / 8 d; / 4 = 2d - 7 / 8 d = 1 1 / 8 d(Diagram 77).

We see that / 1 = / 3 and / 2 = / 4.

You can solve several more of the same problems, and each time you will get the same result: the vertical angles are equal to each other.

It is necessary to verify the validity of the properties of vertical angles by reasoning, by proof.

The proof can be carried out as follows (Fig. 78):

/ a +/ c = 2d;
/ b+/ c = 2d;

(since the sum of adjacent angles is 2 d).

/ a +/ c = / b+/ c

(since the left side of this equality is also equal to 2 d, and its right side is also equal to 2 d).

This equality includes the same angle With.

If we subtract equal amounts from equal quantities, then equal amounts will remain. The result will be: / a = / b, i.e. the vertical angles are equal to each other.

When considering the issue of vertical angles, we first explained which angles are called vertical, i.e. definition vertical angles.

Then we made a judgment (statement) about the equality of the vertical angles and were convinced of the validity of this judgment through proof. Such judgments, the validity of which must be proven, are called theorems. Thus, in this section we gave a definition of vertical angles, and also stated and proved a theorem about their properties.

In the future, when studying geometry, we will constantly have to encounter definitions and proofs of theorems.

3. The sum of angles that have a common vertex.

On the drawing 79 / 1, / 2, / 3 and / 4 are located on one side of a line and have a common vertex on this line. In sum, these angles make up a straight angle, i.e.
/ 1+ / 2+/ 3+ / 4 = 2d.

On the drawing 80 / 1, / 2, / 3, / 4 and / 5 have a common vertex. In sum, these angles make up a full angle, i.e. / 1 + / 2 + / 3 + / 4 + / 5 = 4d.

Exercises.

1. One of the adjacent angles is 0.72 d. Calculate the angle formed by the bisectors of these adjacent angles.

2. Prove that the bisectors of two adjacent angles form a right angle.

3. Prove that if two angles are equal, then their adjacent angles are also equal.

4. How many pairs of adjacent angles are there in the drawing 81?

5. Can a pair of adjacent angles consist of two acute angles? from two obtuse angles? from right and obtuse angles? from a right and acute angle?

6. If one of the adjacent angles is right, then what can be said about the size of the angle adjacent to it?

7. If at the intersection of two straight lines one angle is right, then what can be said about the size of the other three angles?