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Adjacent and vertical angles. Perpendicular lines. Types of angles Adjacent angles add up to

    Two angles placed on the same straight line and having the same vertex are called adjacent.

    Otherwise, if the sum of two angles on one straight line is equal to 180 degrees and they have one side in common, then these are adjacent angles.

    1 adjacent angle + 1 adjacent angle = 180 degrees.

    Adjacent angles are two angles in which one side is common, and the other two sides generally form a straight line.

    The sum of two adjacent angles is always 180 degrees. For example, if one angle is 60 degrees, then the second will necessarily be equal to 120 degrees (180-60).

    Angles AOC and BOC are adjacent angles because all conditions for the characteristics of adjacent angles are met:

    1.OS - common side of two corners

    2.AO - side of the corner AOS, OB - side of the corner BOS. Together these sides form a straight line AOB.

    3. There are two angles and their sum is 180 degrees.

    Remembering the school geometry course, we can say the following about adjacent angles:

    adjacent angles have one side in common, and the other two sides belong to the same straight line, that is, they are on the same straight line. If according to the figure, then the angles SOB and BOA are adjacent angles, the sum of which is always equal to 180, since they divide a straight angle, and a straight angle is always equal to 180.

    Adjacent angles are an easy concept in geometry. Adjacent angles, an angle plus an angle, add up to 180 degrees.

    Two adjacent angles will be one unfolded angle.

    There are several more properties. With adjacent angles, problems are easy to solve and theorems to prove.

    Adjacent angles are formed by drawing a ray from an arbitrary point on a straight line. Then this arbitrary point turns out to be the vertex of the angle, the ray turns out to be the common side of adjacent angles, and the straight line from which the ray is drawn turns out to be the two remaining sides of adjacent angles. Adjacent angles can be the same in the case of a perpendicular, or different in the case of an inclined beam. It is easy to understand that the sum of adjacent angles is equal to 180 degrees or simply a straight line. In another way, this angle can be explained by a simple example - you first walked in one direction in a straight line, then changed your mind, decided to go back and, turning 180 degrees, set off along the same straight line in the opposite direction.

    So what is an adjacent angle? Definition:

    Two angles with a common vertex and one common side are called adjacent, and the other two sides of these angles lie on the same straight line.

    And a short video lesson that sensibly shows about adjacent angles, vertical angles, plus about perpendicular lines, which are a special case of adjacent and vertical angles

    Adjacent angles are angles in which one side is common and the other is one line.

    Adjacent angles are angles that depend on each other. That is, if the common side is slightly rotated, then one angle will decrease by several degrees and automatically the second angle will increase by the same number of degrees. This property of adjacent angles allows one to solve various problems in Geometry and carry out proofs of various theorems.

    The total sum of adjacent angles is always 180 degrees.

    From the geometry course, (as far as I remember in the 6th grade), two angles are called adjacent, in which one side is common, and the other sides are additional rays, the sum of adjacent angles is 180. Each of the two adjacent angles complements the other to an expanded angle. Example of adjacent angles:

    Adjacent angles are 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 one hundred and eighty degrees. In general, all this is very easy to find in Google or a geometry textbook.

    Two angles are called adjacent if they have a common vertex and one side, and the other two sides form a straight line. The sum of adjacent angles is 180 degrees.

    In the figure, angles AOB and BOC are adjacent.

    Adjacent angles are those that have a common vertex, one common side, and the other sides are continuations of each other and form an extended angle. A remarkable property of adjacent angles is that the sum of these angles is always equal to 180 degrees.

    Angles with a common vertex and one common side in geometry are called adjacent

    The sum of adjacent angles is 180 degrees

    It should be noted that adjacent angles have equal sines

    To learn more about adjacent angles, read here

Good afternoon Last time we started to look at the question: “How to understand 7th grade geometry?” and touched on several basic definitions, namely, what is

I think this is very important, because in the future, when you study geometry in grades 8, 9 and beyond, you will come across problems with adjacent and vertical angles more and more often. That is why we are once again solving problems with adjacent angles.

Problem 1. Can a pair of adjacent angles consist of two acute angles? Solution: Let's look at the top picture. Here we see that angle a is less than 90°. This angle is called acute. At the same time, angle b is greater than 90° and less than angle c = 180°. This angle is called obtuse. Therefore, if one of the adjacent angles is acute, then the second must be obtuse. And vice versa. The exception is 90° angles. Those. If two adjacent angles are equal to each other, then they are equal to 90°. Therefore, there are no two adjacent acute angles.

Problem 2. One of the adjacent angles is 56 degrees less than the other. Find the values ​​of these angles. Solution: Let the first angle be equal to X, then the second angle is equal to X+56. In total they give 180°. Let's make the equation: X+X+56 = 180 2X = 180 - 56 2X = 124 X=124/2 = 62. Answer: the first angle is 62°, the second is 62+56 = 118°.

Problem 3. What is the angle between the bisectors of adjacent angles? Solution: To solve this problem, we need to introduce one more concept - bisector. A bisector is a ray that passes inside an angle and divides the angle in half. How is such a problem solved? If we look at the figure, we will see that angles AOB and BOC are adjacent. Their sum is 180°. The bisectors OD and OE divide the angles AOB and BOC into equal α and α, as well as β and β. From here we get: α+α+β+β=180, or 2α +2β = 180 Reducing the right and left sides of the equation by 2, we get the final result: α +β = 90. The angle between the bisectors of adjacent angles is ALWAYS 90°.

Problem 4. Find adjacent angles if their degree measures are in the ratio 4:11. Solution: Let the first angle be 4X, Then the second is 11X. In total they give 180°. We make up the equation: 4X+11X=180 15X = 180 X = 180/15 X=12 4X=4*12 = 48, 11X=11*12 = 132. Answer: the first angle is 48°, the second is 132°.

Problem 5. One of the adjacent angles is 33 degrees greater than half of the second adjacent angle. Find these angles. Solution: Let half the angle be equal to X, then take the entire angle to be 2X. The one adjacent to it is equal to X at 33°. We make up the equation: 2X + X + 33 = 180 3X = 180 - 33 3X = 147 X = 147/3 = 49. Answer: the first angle is 49*2 = 98°, the second is 49+33 = 82°. This concludes the problems with adjacent angles. Next time we will solve problems with vertical angles. See you again!

Two angles are called adjacent if they have one side in common, and the other sides of these angles are complementary rays. In Figure 20, angles AOB and BOC are adjacent.

The sum of adjacent angles is 180°

Theorem 1. The sum of adjacent angles is 180°.

Proof. Beam OB (see Fig. 1) passes between the sides of the unfolded angle. That's why ∠ AOB + ∠ BOS = 180°.

From Theorem 1 it follows that if two angles are equal, then their adjacent angles are equal.

Vertical angles are equal

Two angles are called vertical if the sides of one angle are complementary rays of the sides of the other. The angles AOB and COD, BOD and AOC, formed at the intersection of two straight lines, are vertical (Fig. 2).

Theorem 2. Vertical angles are equal.

Proof. Let's consider the vertical angles AOB and COD (see Fig. 2). Angle BOD is adjacent to each of angles AOB and COD. By Theorem 1 ∠ AOB + ∠ BOD = 180°, ∠ COD + ∠ BOD = 180°.

From this we conclude that ∠ AOB = ∠ COD.

Corollary 1. An angle adjacent to a right angle is a right angle.

Consider two intersecting straight lines AC and BD (Fig. 3). They form four corners. If one of them is straight (angle 1 in Fig. 3), then the remaining angles are also right (angles 1 and 2, 1 and 4 are adjacent, angles 1 and 3 are vertical). In this case, they say that these lines intersect at right angles and are called perpendicular (or mutually perpendicular). The perpendicularity of lines AC and BD is denoted as follows: AC ⊥ BD.

A perpendicular bisector to a segment is a line perpendicular to this segment and passing through its midpoint.

AN - perpendicular to a line

Consider a straight line a and a point A not lying on it (Fig. 4). Let's connect point A with a segment to point H with straight line a. The segment AN is called a perpendicular drawn from point A to line a if lines AN and a are perpendicular. Point H is called the base of the perpendicular.

Drawing square

The following theorem is true.

Theorem 3. From any point not lying on a line, it is possible to draw a perpendicular to this line, and, moreover, only one.

To draw a perpendicular from a point to a straight line in a drawing, use a drawing square (Fig. 5).

Comment. The formulation of the theorem usually consists of two parts. One part talks about what is given. This part is called the condition of the theorem. The other part talks about what needs to be proven. This part is called the conclusion of the theorem. For example, the condition of Theorem 2 is that the angles are vertical; conclusion - these angles are equal.

Any theorem can be expressed in detail in words so that its condition begins with the word “if” and its conclusion with the word “then”. For example, Theorem 2 can be stated in detail as follows: “If two angles are vertical, then they are equal.”

Example 1. One of the adjacent angles is 44°. What is the other equal to?

Solution. Let us denote the degree measure of another angle by x, then according to Theorem 1.
44° + x = 180°.
Solving the resulting equation, we find that x = 136°. Therefore, the other angle is 136°.

Example 2. Let the angle COD in Figure 21 be 45°. What are the angles AOB and AOC?

Solution. Angles COD and AOB are vertical, therefore, by Theorem 1.2 they are equal, i.e. ∠ AOB = 45°. Angle AOC is adjacent to angle COD, which means according to Theorem 1.
∠ AOC = 180° - ∠ COD = 180° - 45° = 135°.

Example 3. Find adjacent angles if one of them is 3 times larger than the other.

Solution. Let us denote the degree measure of the smaller angle by x. Then the degree measure of the larger angle will be 3x. Since the sum of adjacent angles is equal to 180° (Theorem 1), then x + 3x = 180°, whence x = 45°.
This means that adjacent angles are 45° and 135°.

Example 4. The sum of two vertical angles is 100°. Find the size of each of the four angles.

Solution. Let Figure 2 meet the conditions of the problem. The vertical angles COD to AOB are equal (Theorem 2), which means that their degree measures are also equal. Therefore, ∠ COD = ∠ AOB = 50° (their sum according to the condition is 100°). Angle BOD (also angle AOC) is adjacent to angle COD, and therefore, by Theorem 1
∠ BOD = ∠ AOC = 180° - 50° = 130°.

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 lies between them. 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 a 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 smaller 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:

  1. What is an angle?
  2. What types of angles are there?
  3. 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 the number 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:

  1. We denote the proportion of the main angle as the variable “Y”.
  2. The fraction related to the adjacent angle is denoted as the variable “X”.
  3. The number of degrees that fall on each proportion will be denoted, for example, by “a”.
  4. The general formula will look like this - a*X+a*Y=180 or a*(X+Y)=180.
  5. We find the common factor of the equation “a” using the formula a=180/(X+Y).
  6. 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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