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, in the study of 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 remaining three angles?

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.

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 two other sides of them lie on the same straight line ... Great Polytechnic Encyclopedia

See Angle... Big Encyclopedic Dictionary

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

See Angle. * * * ADJACENT CORNERS ADJACENT CORNERS, see Corner (see CORNER) … encyclopedic Dictionary

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

The two lines intersect, creating 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 A complementary angle is a pair of angles that complement each other up to 90 degrees. If two complementary angles are adjacent (that is, 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 additional angles are c ... Wikipedia

Books

• On proof in geometry, Fetisov A.I. Once, at the very beginning school year I had to overhear a conversation between two girls. The eldest of them moved to the sixth grade, the youngest - to the fifth. The girls shared their impressions about the lessons, ...
• Geometry. 7th grade. Complex notebook for knowledge control, I. S. Markova, S. P. Babenko. The manual presents control and measuring materials (KMI) in geometry for conducting current, thematic and final quality control of knowledge of students in grade 7. The contents of the guide…

Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary rays. In figure 20, the angles AOB and BOC are adjacent.

The sum of adjacent angles is 180°

Theorem 1. The sum of adjacent angles is 180°.

Proof. The OB beam (see Fig. 1) passes between the sides of the developed angle. That's why ∠ AOB + ∠ BOC = 180°.

From Theorem 1 it follows that if two angles are equal, then the angles adjacent to them are equal.

Vertical angles are equal

Two angles are called vertical if the sides of one angle are complementary rays of the sides of the other. The angles AOB and COD, BOD and AOC, formed at the intersection of two straight lines, are vertical (Fig. 2).

Theorem 2. Vertical angles are equal.

Proof. Consider the vertical angles AOB and COD (see Fig. 2). Angle BOD is adjacent to each of the angles AOB and COD. By Theorem 1, ∠ AOB + ∠ BOD = 180°, ∠ COD + ∠ BOD = 180°.

Hence we conclude that ∠ AOB = ∠ COD.

Corollary 1. An angle adjacent to a right angle is a right angle.

Consider two intersecting straight lines AC and BD (Fig. 3). They form four corners. If one of them is right (angle 1 in Fig. 3), then the other angles are also right (angles 1 and 2, 1 and 4 are adjacent, angles 1 and 3 are vertical). In this case, these lines are said to intersect at right angles and are called perpendicular (or mutually perpendicular). The perpendicularity of lines AC and BD is denoted as follows: AC ⊥ BD.

The perpendicular bisector of a segment is a line perpendicular to this segment and passing through its midpoint.

AN - perpendicular to the line

Consider a line a and a point A not lying on it (Fig. 4). Connect the point A with a segment to the point H with a straight line a. A segment AH is called a perpendicular drawn from point A to line a if lines AN and a are perpendicular. The point H is called the base of the perpendicular.

Drawing square

The following theorem is true.

Theorem 3. From any point that does not lie on a line, one can draw a perpendicular to this line, and moreover, only one.

To draw a perpendicular from a point to a straight line in the drawing, a drawing square is used (Fig. 5).

Comment. The statement of the theorem usually consists of two parts. One part talks about what is given. This part is called the condition of the theorem. The other part talks about what needs to be proven. This part is called the conclusion of the theorem. For example, the condition of Theorem 2 is vertical angles; conclusion - these angles are equal.

Any theorem can be expressed in detail in words so that its condition will begin with the word “if”, and the conclusion with the word “then”. For example, Theorem 2 can be stated in detail as follows: "If two angles are vertical, then they are equal."

Example 1 One of the adjacent angles is 44°. What is the other equal to?

Solution. Denote the degree measure of another angle by x, then according to Theorem 1.
44° + x = 180°.
Solving the resulting equation, we find that x \u003d 136 °. Therefore, the other angle is 136°.

Example 2 Let the COD angle in Figure 21 be 45°. What are angles AOB and AOC?

Solution. The angles COD and AOB are vertical, therefore, by Theorem 1.2 they are equal, i.e., ∠ AOB = 45°. The angle AOC is adjacent to the angle COD, hence, by Theorem 1.
∠ AOC = 180° - ∠ COD = 180° - 45° = 135°.

Example 3 Find adjacent angles if one of them is 3 times the other.

Solution. Denote the degree measure of the smaller angle by x. Then the degree measure of the larger angle will be Zx. Since the sum of adjacent angles is 180° (Theorem 1), then x + 3x = 180°, whence x = 45°.
So the adjacent angles are 45° and 135°.

Example 4 The sum of two vertical angles is 100°. Find the value of each of the four angles.

Solution. Let figure 2 correspond to the condition of the problem. The vertical angles COD to AOB are equal (Theorem 2), which means that their degree measures are also equal. Therefore, ∠ COD = ∠ AOB = 50° (their sum is 100° by condition). The angle BOD (also the angle AOC) is adjacent to the angle COD, and, therefore, by Theorem 1
∠ BOD = ∠ AOC = 180° - 50° = 130°.

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 - 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. 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.