1 adjacent vertical corners. Adjacent corners

Introduction to corners

Let us be given two arbitrary rays. Let's put them on top of each other. Then

Definition 1

An angle is a name given to two rays that have the same origin.

Definition 2

The point, which is the beginning of the rays within the framework of Definition 3, is called the vertex of this angle.

An angle will be denoted by its following three points: a vertex, a point on one of the rays, and a point on the other ray, and the vertex of the angle is written in the middle of its designation (Fig. 1).

Now let's define what the value of the angle is.

To do this, you need to choose some kind of "reference" angle, which we will take as a unit. Most often, such an angle is an angle that is equal to $\frac(1)(180)$ of a part of a straight angle. This value is called a degree. After choosing such an angle, we compare the angles with it, the value of which must be found.

There are 4 types of corners:

Definition 3

An angle is called acute if it is less than $90^0$.

Definition 4

An angle is called obtuse if it is greater than $90^0$.

Definition 5

An angle is called straight if it is equal to $180^0$.

Definition 6

An angle is called a right angle if it is equal to $90^0$.

In addition to such types of angles, which are described above, it is possible to distinguish types of angles in relation to each other, namely vertical and adjacent angles.

Adjacent corners

Consider a straight angle $COB$. Draw a ray $OA$ from its vertex. This ray will divide the original one into two angles. Then

Definition 7

Two angles will be called adjacent if one pair of their sides is a straight angle, and the other pair coincides (Fig. 2).

AT this case the angles $COA$ and $BOA$ are adjacent.

Theorem 1

The sum of adjacent angles is $180^0$.

Proof.

Consider Figure 2.

By definition 7, the angle $COB$ in it will be equal to $180^0$. Since the second pair of sides of adjacent angles coincide, then the ray $OA$ will divide the straight angle by 2, therefore

$∠COA+∠BOA=180^0$

The theorem has been proven.

Consider the solution of the problem using this concept.

Example 1

Find the angle $C$ from the figure below

By Definition 7, we get that the angles $BDA$ and $ADC$ are adjacent. Therefore, by Theorem 1, we obtain

$∠BDA+∠ADC=180^0$

$∠ADC=180^0-∠BDA=180〗0-59^0=121^0$

By the theorem on the sum of angles in a triangle, we will have

$∠A+∠ADC+∠C=180^0$

$∠C=180^0-∠A-∠ADC=180^0-19^0-121^0=40^0$

Answer: $40^0$.

Vertical angles

Consider the developed angles $AOB$ and $MOC$. Let's match their vertices with each other (that is, put the point $O"$ on the point $O$) so that none of the sides of these angles coincide. Then

Definition 8

Two angles will be called vertical if the pairs of their sides are straight angles and their values ​​are the same (Fig. 3).

In this case, the angles $MOA$ and $BOC$ are vertical and the angles $MOB$ and $AOC$ are also vertical.

Theorem 2

Vertical angles are equal to each other.

Proof.

Consider Figure 3. Let's prove, for example, that the angle $MOA$ is equal to the angle $BOC$.

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 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°.

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 means 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

• About Proof in Geometry, Fetisov A.I. This book will be produced in accordance with your order using Print-on-Demand technology. Once, at the very beginning of the school year, I happened to overhear a conversation between two girls. The oldest one…
• A comprehensive notebook for knowledge control. Geometry. 7th grade. Federal State Educational Standard, Babenko Svetlana Pavlovna, Markova Irina Sergeevna. 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…

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. 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 axioms and proved 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.

In this lesson, we will consider and understand for ourselves the concept of adjacent angles. Consider the theorem that concerns them. Let's introduce the concept of "vertical angles". Consider the supporting facts concerning these angles. Next, we formulate and prove two corollaries about the angle between the bisectors of vertical angles. At the end of the lesson, we will consider several problems devoted to this topic.

Let's start our lesson with the concept of "adjacent corners". Figure 1 shows the developed angle ∠AOC and the ray OB, which divides this angle into 2 angles.

Rice. 1. Angle ∠AOC

Consider the angles ∠AOB and ∠BOC. It is quite obvious that they have a common side VO, while the sides AO and OS are opposite. Rays OA and OS complement each other, which means they lie on the same straight line. The angles ∠AOB and ∠BOC are adjacent.

Definition: If two angles have a common side, and the other two sides are complementary rays, then these angles are called related.

Theorem 1: The sum of adjacent angles is 180 o.

Rice. 2. Drawing for Theorem 1

∠MOL + ∠LON = 180o. This statement is true because the ray OL divides the straight angle ∠MON into two adjacent angles. That is, we do not know the degree measures of any of the adjacent angles, but we only know their sum - 180 o.

Consider the intersection of two lines. The figure shows the intersection of two lines at point O.

Rice. 3. Vertical angles ∠BOA and ∠COD

Definition: If the sides of one angle are a continuation of the second angle, then such angles are called vertical. That is why the figure shows two pairs of vertical angles: ∠AOB and ∠COD, as well as ∠AOD and ∠BOC.

Theorem 2: Vertical angles are equal.

Let's use Figure 3. Let's consider the developed angle ∠AOC. ∠AOB \u003d ∠AOC - ∠BOC \u003d 180 o - β. Consider the developed angle ∠BOD. ∠COD = ∠BOD - ∠BOC = 180 o - β.

From these considerations, we conclude that ∠AOB = ∠COD = α. Similarly, ∠AOD = ∠BOC = β.

Corollary 1: The angle between the bisectors of adjacent angles is 90°.

Rice. 4. Drawing for consequence 1

Since OL is the bisector of the angle ∠BOA, then the angle ∠LOB = , similarly to ∠BOK = . ∠LOK = ∠LOB + ∠BOK = + = . The sum of the angles α + β is equal to 180 o, since these angles are adjacent.

Corollary 2: The angle between the bisectors of the vertical angles is 180°.

Rice. 5. Drawing for consequence 2

KO is the bisector of ∠AOB, LO is the bisector of ∠COD. Obviously, ∠KOL = ∠KOB + ∠BOC + ∠COL = o . The sum of the angles α + β is equal to 180 o, since these angles are adjacent.

Let's consider some tasks:

Find the angle adjacent to ∠AOC if ∠AOC = 111 o.

Let's make a drawing for the task:

Rice. 6. Drawing for example 1

Since ∠AOC = β and ∠COD = α are adjacent angles, then α + β = 180 o. That is, 111 o + β \u003d 180 o.

Hence, β = 69 o.

This type of problem exploits the adjacent angle sum theorem.

One of the adjacent angles is a right angle, which (acute, obtuse or right) is the other angle?

If one of the angles is right and the sum of the two angles is 180°, then the other angle is also right. This task tests knowledge about the sum of adjacent angles.

Is it true that if adjacent angles are equal, then they are right angles?

Let's make an equation: α + β = 180 o, but since α = β, then β + β = 180 o, which means β = 90 o.

Answer: Yes, the statement is true.

Given two equal angles. Is it true that the angles adjacent to them will also be equal?

Rice. 7. Drawing for example 4

If two angles are equal to α, then their corresponding adjacent angles will be 180 o - α. That is, they will be equal to each other.

Answer: The statement is true.

1. Alexandrov A.D., Werner A.L., Ryzhik V.I. etc. Geometry 7. - M.: Enlightenment.
2. Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. et al. Geometry 7. 5th ed. - M.: Enlightenment.
3. \Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzova, S.B. Kadomtsev, V.V. Prasolov, edited by V.A. Sadovnichy. - M.: Education, 2010.

1. Measurement of segments ().
2. General lesson on geometry in the 7th grade ().
3. Straight line, segment ().

1. No. 13, 14. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzova, S.B. Kadomtsev, V.V. Prasolov, edited by V.A. Sadovnichy. - M.: Education, 2010.
2. Find two adjacent angles if one of them is 4 times the other.
3. Given an angle. Build adjacent and vertical angles for it. How many such corners can be built?
4. * In what case are more pairs of vertical angles obtained: when three lines intersect at one point or at three points?