CHAPTER I.

BASIC CONCEPTS.

§11. 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.

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 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?

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 angles (a 1 b) and (a 2 b) are adjacent. They have side b in common, 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 angle (a 1 b) and angle (a 2 b) be given adjacent angles (see Fig. 31). Ray b passes between sides a 1 and a 2 of a straight angle. Therefore, the sum of the angles (a 1 b) and (a 2 b) is equal to the unfolded angle, i.e. 180°. Q.E.D.

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

From the theorem 2.1 It follows that if two angles are equal, then their adjacent angles 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 = 180° - a 1 b and c 2 d = 180° - c 1 d. Since the angles (a 1 b) and (c 1 d) are equal, we get that a 2 b = 180° - a 1 b = 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 obtuse.

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 an angle adjacent to a right angle is a right angle: x + 90° = 180°, x = 180° - 90°, x = 90°.

Question 6. What angles are called vertical?
Answer. Two angles are called vertical if the sides of one angle are 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 the given vertical angles (Fig. 34). Angle (a 1 b 2) is adjacent to angle (a 1 b 1) and to angle (a 2 b 2). From here, using 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) to 180°, i.e. angles (a 1 b 1) and (a 2 b 2) are equal. Q.E.D.

Question 8. Prove that if, when two lines intersect, one of the angles is right, then the other three angles are also right.
Answer. Suppose that lines AB and CD intersect each other at point O. Suppose that angle AOD is 90°. Since the sum of adjacent angles is 180°, we get that AOC = 180° - AOD = 180° - 90° = 90°. Angle COB is vertical to angle AOD, so they are equal. That is, angle COB = 90°. Angle COA is vertical to angle BOD, so they are equal. That is, angle BOD = 90°. Thus, all angles are equal to 90°, that is, they are all right angles. 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 right angles.
The perpendicularity of lines is indicated by the sign \(\perp\). The entry \(a\perp b\) reads: “Line a is perpendicular to line b.”

Question 10. Prove that through any point on a line you 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 a given point on it. Let us denote by a 1 one of the half-lines of the straight line a with the starting point A (Fig. 38). Let us subtract an angle (a 1 b 1) equal to 90° from the half-line a 1. Then the straight line containing the ray b 1 will be perpendicular to the straight line a.

Let us assume that there is another line, also passing through point A and perpendicular to line a. Let us 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), each equal to 90°, 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 put into a given half-plane. Therefore, there cannot be another line passing through point A and perpendicular to line a. The theorem has been proven.

Question 11. What is perpendicular to a line?
Answer. A perpendicular to a given line is a segment of a line perpendicular to a given line, 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 consists of.
Answer. The proof method we used in Theorem 2.3 is called proof by contradiction. This method of proof is that we first make an assumption opposite to what the theorem states. Then, by reasoning, relying on axioms and proven theorems, we come to a conclusion that contradicts either the conditions of the theorem, or one of the axioms, or a previously proven theorem. On this basis, we conclude that our assumption was incorrect, and therefore the statement of the theorem is true.

Question 13. What is the bisector of an angle?
Answer. The bisector of an angle is the ray that emanates 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 type	Size in degrees	Example
Spicy	Less than 90°
Direct	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.

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 their other two sides lie on the same straight line... Big Polytechnic Encyclopedia

See Angle... Big Encyclopedic Dictionary

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

See Angle. * * * ADJACENT CORNERS ADJACENT CORNERS, see Angle (see ANGLE) ... Encyclopedic Dictionary

- (Angles adjacents) those that have a common vertex and a common side. Mostly this name refers to such C. angles, the other two sides of which 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

Two straight lines intersect to create 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. Complementary angles are a pair of angles that complement each other up to 90 degrees. If two complementary angles are adjacent (i.e. 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 complementary angles are with... Wikipedia

Books

• About proof in geometry, A.I. Fetisov. Once, at the very beginning of the school year, I heard a conversation between two girls. The eldest of them moved to the sixth grade, the youngest to the fifth. The girls shared their impressions of the lessons...
• Geometry. 7th grade. Comprehensive notebook for knowledge control, I. S. Markova, S. P. Babenko. The manual presents control and measurement materials (CMM) in geometry for conducting current, thematic and final quality control of the knowledge of 7th grade students. Contents of the manual...

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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