What are called adjacent angles. Adjacent and vertical corners

Angles in which one side is common, and the other sides lie on the same straight line (in the figure, angles 1 and 2 are adjacent). Rice. to Art. Adjacent corners... Great Soviet Encyclopedia

ADJACENT CORNERS- angles that have a common vertex and one common side, and two other sides of them lie on the same straight line ... Great Polytechnic Encyclopedia

See Angle... Big Encyclopedic Dictionary

ADJACENT ANGLES, two angles whose sum is 180°. Each of these corners complements the other to a full angle... Scientific and technical encyclopedic dictionary

See Angle. * * * ADJACENT CORNERS ADJACENT CORNERS, see Corner (see CORNER) … encyclopedic Dictionary

- (Angles adjacent) those that have a common vertex and a common side. Mostly, this name refers to such S. angles, of which the other two sides lie in opposite directions of one straight line drawn through the vertex ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

See Angle... Natural science. encyclopedic Dictionary

The two lines intersect, creating a pair of vertical angles. One pair consists of angles A and B, the other of C and D. In geometry, two angles are called vertical if they are created by the intersection of two ... Wikipedia

A pair of complementary angles that complement each other up to 90 degrees A complementary angle is a pair of angles that complement each other up to 90 degrees. If two complementary angles are adjacent (that is, they have a common vertex and are separated only ... ... Wikipedia

A pair of complementary angles that complement each other up to 90 degrees Complementary angles are a pair of angles that complement each other up to 90 degrees. If two additional angles are c ... Wikipedia

Books

• About Proof in Geometry, Fetisov A.I. This book will be produced in accordance with your order using Print-on-Demand technology. Once, at the very beginning of the school year, I happened to overhear a conversation between two girls. The oldest one…
• A comprehensive notebook for knowledge control. Geometry. 7th grade. Federal State Educational Standard, Babenko Svetlana Pavlovna, Markova Irina Sergeevna. The manual presents control and measuring materials (KMI) in geometry for conducting current, thematic and final quality control of knowledge of students in grade 7. The contents of the guide…

1. Adjacent corners.

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

Two angles that have one side in 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 straight line (not lying on a given straight line), then we get adjacent angles.

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

Adjacent corners 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 value of one of the adjacent angles, we can find the value of the other adjacent angle.

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

180° - 54° = l26°.

2. Vertical angles.

If we extend the sides of an 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 extensions 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.

∠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 get the same result: the vertical angles are equal to each other.

However, to make sure that the 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 property 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 180°, and its right side is also 180°).

This equality includes the same angle from.

If we subtract equally from equal values, then it will remain equally. 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 the same side of the 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 drawing 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

What is an adjacent angle

Injection- this is a geometric figure (Fig. 1), formed by two rays OA and OB (corner sides), emanating from one point O (corner apex).

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

Adjacent corners- (Agles adjacets) those that have a common top and a common side. Predominantly, this name refers to such angles, of which the other 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.

rice. 2

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

rice. 3

Figure 3 shows line AB, point C is located between points A and B. Point D is a point not lying on line 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 line AB, since points A, B are separated by the initial point C.

Adjacent angle theorem

Theorem: sum of adjacent angles is 180°

Proof:
Angles a1b and a2b are adjacent (see Fig. 2) Beam b passes between sides a1 and a2 of a straightened angle. Therefore, the sum of the angles a1b and a2b is equal to the straight angle, i.e. 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 the 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 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°.

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, the BOS angle in Fig. 1 Consider first two intersecting lines. When they intersect, lines form angles. There are special cases:

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

Definition 3. A right angle is an angle of 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 extend each other are called vertical angles.
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 Fig. 4 adjacent corners AOB and BOS. Their sum is the developed angle AOC. Therefore, the sum of these adjacent angles is 180 degrees.

rice. 4

Relationship 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 these two antipodes limit and determine all the creative spiritual activity of a person, and that everything is placed between them, what mankind has created in the field 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 conditioned both historically and internally, despite the fact that mathematics is the most abstract of the sciences, and music is the most abstract art form.
Consonance determines the sound of a string that is pleasing to the ear.
This musical system was based on two laws, which 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 a triangular number 10=1+2+3+4, i.e. like 1:2, 2:3, 3:4. Moreover, the smaller the number n in relation to n:(n+1) (n=1,2,3), the more consonant the resulting interval.
2. The oscillation frequency w of a 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 your attention a funny parody about a dispute between two mathematicians =)

Geometry around us

Geometry plays an important role in our life. 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, in principle, wherever we are. But the topic of today's lesson is adjacent coals. So let's look around and try to find corners in this environment. If you look carefully out the window, you can see that some branches of the tree form adjacent corners, and you can see many vertical corners in the partitions on the gate. Give your examples of adjacent angles that you see in the 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 is the angle of the photo frame 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 is this figure, 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 a painting. Look at them carefully and say what types of catch you see in the picture, and what angles in the picture.

Problem solving

1) Two angles are given, 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 adjacent angles?
3) It is necessary to find adjacent angles, provided that one of them is 10 degrees greater than the second.

Mathematical dictation for the repetition of previously learned material

1) Draw a picture: lines a I b intersect at point A. Mark the smallest of the formed corners with the number 1, and the remaining angles - sequentially with the numbers 2,3,4; the complementary rays of the line a - through a1 and a2, and the line b - through b1 and b2.
2) Using the completed drawing, enter the necessary values ​​and explanations in the gaps in the text:
a) angle 1 and angle .... related 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 at the intersection of 2 lines equal 100°? 370°?
2. In the figure, find all pairs of adjacent corners. And now the vertical corners. Name these angles.

3. You need to find an angle when it is three times larger than the one adjacent to it.
4. Two lines intersect 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 84 °;
b) the difference of 2 angles of them 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 at the intersection of 2 lines?
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 ° more than the second.
2. Find the angles that are formed when 2 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, which is 60 ° greater than the second angle.
4. The difference of 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 the sum of 2 adjacent angles are related as 1: 5, respectively. Find adjacent corners.
6. The difference between two adjacent ones is 25% of their sum. How are the values ​​of 2 adjacent angles related? Determine the values ​​of 2 adjacent angles.

Questions:

1. What is an angle?
2. What are the types of corners?
3. What is the feature of adjacent corners?

Subjects > Mathematics > Mathematics Grade 7 injection to expanded, that is, equal to 180 °, therefore, to find them, subtract from this the known value of the main angle α₁ \u003d α₂ \u003d 180 ° -α.

From this there are . If two angles are both adjacent and equal at the same time, then they are right angles. If one of the adjacent angles is right, that is, it is 90 degrees, then the other angle is also right. If one of the adjacent angles is acute, then the other will be obtuse. Similarly, if one of the angles is obtuse, then the second, respectively, will be acute.

An acute angle is one whose measure is less than 90 degrees but greater than 0. An obtuse angle has a measure greater than 90 degrees but less than 180.

Another property of adjacent angles is formulated as follows: if two angles are equal, then the angles adjacent to them are also equal. This is that if there are two angles for which the degree measure is the same (for example, it is 50 degrees) and at the same time one of them has an adjacent angle, then the values ​​\u200b\u200bof these adjacent angles also coincide (in the example, their degree measure will be 130 degrees).

Sources:

• Big Encyclopedic Dictionary - Adjacent corners
• 180 degree angle

The word "" has various interpretations. In geometry, an angle is a part of a plane bounded by two rays coming out of one point - a vertex. When it comes to straight, sharp, developed angles, it is geometric angles that are meant.

Like any shape in geometry, angles can be compared. The equality of angles is determined by movement. An angle is easy to divide into two equal parts. Dividing into three parts is a little more difficult, but it can still be done with a ruler and compass. By the way, this task seemed quite difficult. It is geometrically easy to describe that one angle is greater or less than another.

The unit of measure for angles is 1/180

CHAPTER I.

BASIC CONCEPTS.

§eleven. ADJACENT AND VERTICAL ANGLES.

1. Adjacent corners.

If we continue the side of some corner beyond its vertex, we will get two corners (Fig. 72): / A sun and / SVD, in which one side BC is common, and the other two AB and BD form a straight line.

Two angles that have one side in 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 straight line (not lying on a given straight line), then we get adjacent angles.
For example, / ADF and / FDВ - adjacent corners (Fig. 73).

Adjacent corners 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 2d.

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

Knowing the value of one of the adjacent angles, we can find the value of the other adjacent angle.

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 an 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 extensions of the sides of the other angle.

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

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

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

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

However, to make sure that the 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 property 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 equal to 2 d, and its right side is also equal to 2 d).

This equality includes the same angle from.

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

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

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

In the future, when studying geometry, we will constantly have to meet with 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 the same side of a straight line and have a common vertex on this straight 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 top. 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 corners are in drawing 81?

5. Can a pair of adjacent angles consist of two acute angles? from two obtuse corners? 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 value of the angle adjacent to it?

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