The angles of which sides are a continuation of the sides of the other. Vertical and adjacent angles

Lesson 8. Vertical angles. Two angles are called vertical if the sides of one angle are a continuation of the sides of the other. THEOREM. Vertical angles are equal. Proof: = = 180 Similarly = = = 3 2 = 4 Solution of problems: 64, 66 Homework: paragraph 11, 66, 67

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

Geometry is a very multifaceted science. It develops logic, imagination and intelligence. Of course, due to its complexity and huge amount theorems and axioms, schoolchildren do not always like it. In addition, there is a need to constantly prove your 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 intersecting two straight lines or drawing two rays from one point. They can be called either one letter or three, which sequentially designate the points at which the angle is constructed.

Angles are measured in degrees and can (depending on their value) be called differently. So, there is a right angle, acute, obtuse and unfolded. 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 degree measure is 90.

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

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 these angles will be equal to 180 degrees (there is a theorem that proves this). Therefore, one can easily calculate one of them if the other is known.
2. From the first point it follows that adjacent angles cannot be formed by two obtuse or two acute angles.

Thanks to these properties, it is always possible to 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 straight lines intersect. Along with them, adjacent angles are always present. An angle can be simultaneously adjacent for one and vertical for another.

When crossing an arbitrary line, several other types of angles are also considered. Such a line is called a secant line, and it forms 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 angles seems quite simple and understandable. All their properties are easy to remember and prove. Solving problems does not seem difficult as long as the angles correspond numeric value. Later, when the study of sin and cos begins, you will have to memorize a lot complex formulas, their conclusions and consequences. Until then, you can just enjoy easy puzzles where you need to find adjacent angles.

CHAPTER I.

BASIC CONCEPTS.

§eleven. 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(Figure 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 the vertical angles are always equal to each other, it is not enough to consider individual numerical examples, since conclusions drawn on the basis of 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

(as well as left side 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 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?