1 property of adjacent corners. Adjacent and vertical corners. Adjacent corners

Geometry is a very multifaceted science. It develops logic, imagination and intelligence. Of course, due to its complexity and the huge number of theorems and axioms, schoolchildren do not always like it. In addition, there is a need to constantly prove their conclusions using generally accepted standards and rules.

Adjacent and vertical angles are an integral part of geometry. Surely many schoolchildren simply adore them for the reason that their properties are clear and easy to prove.

Formation of corners

Any angle is formed by the intersection of two lines or by drawing two rays from one point. They can be called either one letter or three, which successively designate the points of construction of the corner.

Angles are measured in degrees and can (depending on their value) be called differently. So, there is a right angle, acute, obtuse and deployed. Each of the names corresponds to a certain degree measure or its interval.

An acute angle is an angle whose measure does not exceed 90 degrees.

An obtuse angle is an angle greater than 90 degrees.

An angle is called right when its measure is 90.

In the case when it is formed by one continuous straight line, and its degree measure is 180, it is called deployed.

Angles that have a common side, the second side of which continues each other, are called adjacent. They can be either sharp or blunt. The intersection of the line forms adjacent angles. Their properties are as follows:

The sum of such angles will be equal to 180 degrees (there is a theorem proving this). Therefore, one of them can be easily calculated if the other is known.
It follows from the first point that adjacent angles cannot be formed by two obtuse or two acute angles.

Thanks to these properties, one can always calculate the degree measure of an angle given the value of another angle, or at least the ratio between them.

Vertical angles

Angles whose sides are continuations of each other are called vertical. Any of their varieties can act as such a pair. Vertical angles are always equal to each other.

They are formed when lines intersect. Together with them, adjacent corners are always present. An angle can be both adjacent for one and vertical for the other.

When crossing an arbitrary line, several more types of angles are also considered. Such a line is called a secant, and it forms the corresponding, one-sided and cross-lying angles. They are equal to each other. They can be viewed in light of the properties that vertical and adjacent angles have.

Thus, the topic of corners seems to be quite simple and understandable. All their properties are easy to remember and prove. Solving problems is not difficult as long as the angles correspond to a numerical value. Already further, when the study of sin and cos begins, you will have to memorize many complex formulas, their conclusions and consequences. Until then, you can just enjoy easy puzzles in which you need to find adjacent corners.

In the process of studying the geometry course, the concepts of “angle”, “vertical angles”, “adjacent angles” are encountered quite often. Understanding each of the terms will help to understand the task and solve it correctly. What are adjacent angles and how to determine them?

Adjacent corners - definition of the concept

The term "adjacent angles" characterizes two angles formed by a common ray and two additional half-lines lying on the same line. All three beams come from the same point. The common half-line is at the same time the side of both one and the second angle.

Adjacent corners - basic properties

1. Based on the formulation of adjacent angles, it is easy to see that the sum of such angles always forms a straight angle, the degree measure of which is 180 °:

If μ and η are adjacent angles, then μ + η = 180°.
Knowing the value of one of the adjacent angles (for example, μ), one can easily calculate the degree measure of the second angle (η) using the expression η = 180° - μ.

2. This property of angles allows us to draw the following conclusion: an angle that is adjacent right angle, will also be straight.

3. Considering trigonometric functions(sin, cos, tg, ctg), based on the reduction formulas for adjacent angles μ and η, the following is true:

sinη = sin(180° - μ) = sinμ,
cosη = cos(180° - μ) = -cosμ,
tgη = tg(180° - μ) = -tgμ,
ctgη = ctg(180° - μ) = -ctgμ.

Adjacent corners - examples

Example 1

Given a triangle with vertices M, P, Q – ΔMPQ. Find the angles adjacent to the angles ∠QMP, ∠MPQ, ∠PQM.

Let us extend each side of the triangle as a straight line.
Knowing that adjacent angles complement each other to a straight angle, we find out that:

adjacent to the angle ∠QMP is ∠LMP,

adjacent to the angle ∠MPQ is ∠SPQ,

the adjacent angle for ∠PQM is ∠HQP.

Example 2

The value of one adjacent angle is 35°. What is the degree measure of the second adjacent angle?

Two adjacent angles add up to 180°.
If ∠μ = 35°, then adjacent ∠η = 180° – 35° = 145°.

Example 3

Determine the values of adjacent angles, if it is known that the degree measure of one of the bottom is three times greater than the degree measure of the other angle.

Let us denote the value of one (smaller) angle through – ∠μ = λ.
Then, according to the condition of the problem, the value of the second angle will be equal to ∠η = 3λ.
Based on the basic property of adjacent angles, μ + η = 180° follows

λ + 3λ = μ + η = 180°,

λ = 180°/4 = 45°.

So the first one angle is ∠μ = λ = 45°, and the second angle is ∠η = 3λ = 135°.

The ability to appeal to terminology, as well as knowledge of the basic properties of adjacent angles, will help to cope with the solution of many geometric problems.

Question 1. What angles are called adjacent?
Answer. Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary half-lines.
In figure 31, the corners (a 1 b) and (a 2 b) are adjacent. They have a common side b, and sides a 1 and a 2 are additional half-lines.

Question 2. Prove that the sum of adjacent angles is 180°.
Answer. Theorem 2.1. The sum of adjacent angles is 180°.
Proof. Let the angle (a 1 b) and the angle (a 2 b) be given adjacent angles (see Fig. 31). The beam b passes between the sides a 1 and a 2 of the developed angle. Therefore, the sum of the angles (a 1 b) and (a 2 b) is equal to the developed angle, i.e. 180 °. Q.E.D.

Question 3. Prove that if two angles are equal, then the angles adjacent to them are also equal.
Answer.

From the theorem 2.1 It follows that if two angles are equal, then the angles adjacent to them are equal.
Let's say the angles (a 1 b) and (c 1 d) are equal. We need to prove that the angles (a 2 b) and (c 2 d) are also equal.
The sum of adjacent angles is 180°. It follows from this that a 1 b + a 2 b = 180° and c 1 d + c 2 d = 180°. Hence, a 2 b \u003d 180 ° - a 1 b and c 2 d \u003d 180 ° - c 1 d. Since the angles (a 1 b) and (c 1 d) are equal, we get that a 2 b \u003d 180 ° - a 1 b \u003d c 2 d. By the property of transitivity of the equal sign, it follows that a 2 b = c 2 d. Q.E.D.

Question 4. What angle is called right (acute, obtuse)?
Answer. An angle equal to 90° is called a right angle.
An angle less than 90° is called acute angle.
An angle greater than 90° and less than 180° is called an obtuse angle.

Question 5. Prove that an angle adjacent to a right angle is a right angle.
Answer. From the theorem on the sum of adjacent angles it follows that the angle adjacent to a right angle is a right angle: x + 90° = 180°, x= 180° - 90°, x = 90°.

Question 6. What are the vertical angles?
Answer. Two angles are called vertical if the sides of one angle are the complementary half-lines of the sides of the other.

Question 7. Prove that the vertical angles are equal.
Answer. Theorem 2.2. Vertical angles are equal.
Proof. Let (a 1 b 1) and (a 2 b 2) be given vertical angles (Fig. 34). The corner (a 1 b 2) is adjacent to the corner (a 1 b 1) and to the corner (a 2 b 2). From here, by the theorem on the sum of adjacent angles, we conclude that each of the angles (a 1 b 1) and (a 2 b 2) complements the angle (a 1 b 2) up to 180 °, i.e. the angles (a 1 b 1) and (a 2 b 2) are equal. Q.E.D.

Question 8. Prove that if at the intersection of two lines one of the angles is a right angle, then the other three angles are also right.
Answer. Assume that lines AB and CD intersect each other at point O. Assume that angle AOD is 90°. Since the sum of adjacent angles is 180°, we get that AOC = 180°-AOD = 180°- 90°=90°. The COB angle is vertical to the AOD angle, so they are equal. That is, the angle COB = 90°. COA is vertical to BOD, so they are equal. That is, the angle BOD = 90°. Thus, all angles are equal to 90 °, that is, they are all right. Q.E.D.

Question 9. Which lines are called perpendicular? What sign is used to indicate perpendicularity of lines?
Answer. Two lines are called perpendicular if they intersect at a right angle.
The perpendicularity of lines is denoted by \(\perp\). The entry \(a\perp b\) reads: "Line a is perpendicular to line b".

Question 10. Prove that through any point of a line one can draw a line perpendicular to it, and only one.
Answer. Theorem 2.3. Through each line, you can draw a line perpendicular to it, and only one.
Proof. Let a be a given line and A - given point on her. Denote by a 1 one of the half-lines by the straight line a with the starting point A (Fig. 38). Set aside from the half-line a 1 the angle (a 1 b 1) equal to 90 °. Then the line containing the ray b 1 will be perpendicular to the line a.

Assume that there is another line that also passes through the point A and is perpendicular to the line a. Denote by c 1 the half-line of this line lying in the same half-plane with the ray b 1 .
Angles (a 1 b 1) and (a 1 c 1), equal to 90° each, are laid out in one half-plane from the half-line a 1 . But from the half-line a 1, only one angle equal to 90 ° can be set aside in this half-plane. Therefore, there cannot be another line passing through the point A and perpendicular to the line a. The theorem has been proven.

Question 11. What is a perpendicular to a line?
Answer. Perpendicular to a given line 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 basis perpendicular.

Question 12. Explain what proof by contradiction is.
Answer. The method of proof that we used in Theorem 2.3 is called proof by contradiction. This way of proof consists in that we first make an assumption opposite to what is stated by the theorem. Then, by reasoning, relying on axioms and proved theorems, we come to a conclusion that contradicts either the condition of the theorem, or one of the axioms, or the previously proven theorem. On this basis, we conclude that our assumption was wrong, which means that the assertion of the theorem is true.

Question 13. What is an angle bisector?
Answer. The bisector of an angle is a ray that comes from the vertex of the angle, passes between its sides and divides the angle in half.

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 / 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 With.

If we are from equal values subtract equally, 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. The sum of these angles is 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?

What is an adjacent angle

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

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.

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

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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 BOC. Their sum is the developed angle AOC. Therefore, the sum of these adjacent angles is 180 degrees.

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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 physical properties strings.

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: