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:

1. 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.
2. 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.

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 with.

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?

Lesson 8 Two angles are called vertical if the sides of one angle are an extension of the sides of the other. THEOREM. Vertical angles are equal. Proof: = = 180 Similar = = = 3 2 = 4 Problem solving: 64, 66 Homework: items 11, 66, 67

Mathematical dictation. 1 option. 1. Complete the sentence: “If angles 1 and 2 are adjacent, then their sum ...” 2. Will the angle adjacent to the 30 degree angle be acute, obtuse or right? 3. The sum of two angles is 180 degrees. Do these angles have to be adjacent? 4. Lines AM and CE intersect at the point O, which lies between them. Did this result in vertical angles? If yes, please name them. 5. What is the angle if the vertical angle with it is 34 degrees? 6. One of the four angles resulting from the intersection of two straight lines is 140 degrees. What are the rest of the angles? 7. Two corners have a common vertex, the first corner is 40 degrees, the second is 140 degrees. Are these corners vertical? Option 2. 1. Complete the sentence: “Two angles are called adjacent if one side is common and the other ...” 2. Will the angle adjacent to the angle of 130 degrees be acute, obtuse or right? 3. The sum of two angles with a common side of 180 degrees. Do these angles have to be adjacent? 4. The student built 2 vertical corners. How many pairs of straight lines did this result in? 5. Two corners have a common vertex, each of these corners is equal to 60 degrees. Do these angles have to be vertical? 6. One of the four angles resulting from the intersection of two straight lines is 80 degrees. What are the rest of the angles? 7. What is the angle if the vertical angle with it is 120 degrees?

Answers. 1. Equal to 180 degrees 2. Obtuse angle 3. No 4. Angles AOC and EOM, AOE and COM degrees and 40 degrees 7. Yes 1. Additional rays 2. Acute angle 3. No 4. One pair 5. No and 100 degrees degrees