Good day! Last time, we began to analyze the question: “How to understand the geometry of grade 7?” and touched upon several basic definitions, namely, what is

I think that 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 once again solve problems with adjacent corners.

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 the angle a is less than 90°. Such an angle is called an acute angle. However, the angle b is greater than 90° and less than the angle c=180°. Such an angle is called obtuse. Therefore, if one of the adjacent angles is acute, then the second one 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 X, then the second angle is X+56. They add up to 180°. We make the equation: X + X + 56 \u003d 180 2X \u003d 180 - 56 2X \u003d 124 X \u003d 124/2 \u003d 62. Answer: the first angle is 62°, the second 62+56 = 118°.

Task 3. What is equal to the angle between bisectors of adjacent angles? Solution: To solve this problem, one more concept must be introduced - the bisector. A bisector is a ray that passes inside an angle and bisects the angle. How is this problem solved. If we look at the figure, we can see that the 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°.

Task 4. Find adjacent angles if their degree measures are related as 4:11. Solution: Let the first angle be 4X, Then the second one is 11X. They add up to 180°. We make 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 more than half of the second adjacent angle. Find those angles. Solution: Let half of the angle be X, then the whole angle will be taken as 2X. Adjacent to it is X at 33 °. We make the equation: 2X + X + 33 \u003d 180 3X \u003d 180 - 33 3X \u003d 147 X \u003d 147/3 \u003d 49. Answer: the first angle is 49*2 = 98°, the second is 49+33 = 82°. This concludes the tasks with adjacent corners. Next time we will solve problems with vertical corners. See you soon!

How to find an adjacent angle?

Mathematics is the oldest exact science, which is mandatory studied in schools, colleges, institutes and universities. However, basic knowledge is always laid down at school. Sometimes, the child is given quite difficult tasks, and the parents are unable to help, because they simply forgot some things from mathematics. For example, how to find an adjacent angle by the value of the main angle, etc. The task is simple, but it can be difficult to solve due to not knowing which angles are called adjacent and how to find them.

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

Definition and properties of adjacent corners

Two rays emanating from the same point form a figure called a "flat angle". In this case, this point is called the vertex of the angle, and the rays are its sides. If one of the rays is continued further than the starting point along 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 corners have a common vertex and one side;
• The sum of adjacent angles is always 180 degrees, or pi if the calculation is 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 corners

Usually three variations of problems are given for finding the value 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 consider them.

Given the value of the main angle

If the value of the main angle is indicated in the problem, then finding the adjacent angle is very simple. To do this, it is enough to subtract the value of the main angle from 180 degrees, and you will get the value of the adjacent angle. This solution comes from the property of an adjacent angle - the sum of adjacent angles is always 180 degrees.

If the value of the main angle is given in radians and in the problem it is required to find 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 angle of 180 degrees is equal to the number Pi.

Given the ratio of the main and adjacent angle

In the problem, the ratio of the main and adjacent angle can be given instead of degrees and radians of the magnitude of the main angle. In this case, the solution will look like an equation of proportion:

1. We denote the proportion of the proportion of the main angle as the variable "Y".
2. The proportion related to the adjacent corner is denoted as the variable "X".
3. The number of degrees that fall on each proportion, we denote, for example, "a".
4. 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" by the formula a=180/(X+Y).
6. Then we multiply the obtained 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 the value in radians, then you just need to convert degrees to radians. To do this, multiply the angle in degrees by pi and divide by 180 degrees. The resulting value will be in radians.

Given the value of the vertical angle

If the value of the main angle is not given in the problem, 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 comes from the same point as the main one, but at the same time it is directed in the exact 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, it is possible to calculate the adjacent angle of the main angle. To do this, simply subtract the value of the vertical 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 angle of 180 degrees is equal to the number Pi.

You can also read our helpful articles and.

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 an 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 be a given point on it. 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. A 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 the axioms and proven 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.

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

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

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

corners MOP and pon adjacent since the beam OP- the common side, and the other two sides - OM and ON make up a straight line.

The common side of adjacent angles is called oblique to straight, on which the other two sides lie, only if the adjacent angles are not equal to each other. If adjacent angles are equal, then their common side will be perpendicular.

The sum of adjacent angles is 180°.

The two corners are called vertical, if the sides of one angle complement to straight lines the sides of another angle:

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

Vertical angles are equal.

Let's prove that the vertical angles are equal:

The sum of ∠1 and ∠2 is a straight angle. And the sum of ∠3 and ∠2 is a straight angle. So these two sums are equal:

∠1 + ∠2 = ∠3 + ∠2.

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

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 adjacent to a right angle will also be right.

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.