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 remaining three angles?
injection to expanded, that is, equal to 180 °, therefore, to find them, subtract from this the known value of the main angle α₁ \u003d α₂ \u003d 180 ° -α.From this there are . If two angles are both adjacent and equal at the same time, then they are right angles. If one of the adjacent angles is right, that is, it is 90 degrees, then the other angle is also right. If one of the adjacent angles is acute, then the other will be obtuse. Similarly, if one of the angles is obtuse, then the second, respectively, will be acute.
An acute angle is one whose measure is less than 90 degrees but greater than 0. An obtuse angle has a measure greater than 90 degrees but less than 180.
Another property of adjacent angles is formulated as follows: if two angles are equal, then the angles adjacent to them are also equal. This is that if there are two angles for which the degree measure is the same (for example, it is 50 degrees) and at the same time one of them has an adjacent angle, then the values \u200b\u200bof these adjacent angles also coincide (in the example, their degree measure will be 130 degrees).
Sources:
- Big Encyclopedic Dictionary - Adjacent corners
- 180 degree angle
The word "" has various interpretations. In geometry, an angle is a part of a plane bounded by two rays coming out of one point - a vertex. When it comes to straight, sharp, developed angles, it is geometric angles that are meant.
Like any shape in geometry, angles can be compared. The equality of angles is determined by movement. An angle is easy to divide into two equal parts. Dividing into three parts is a little more difficult, but it can still be done with a ruler and compass. By the way, this task seemed quite difficult. It is geometrically easy to describe that one angle is greater or less than another.
The unit of measure for angles is 1/180
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, with the vertex of the angle 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).
In 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. So ∠ 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?
Decision.
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?
Decision.
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.
Decision.
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.
Decision.
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°.
In the process of studying the geometry course, the concepts of “angle”, “vertical angles”, “adjacent angles” are encountered quite often. Understanding each of the terms will help to understand the task and solve it correctly. What are adjacent angles and how to determine them?
Adjacent corners - definition of the concept
The term "adjacent angles" characterizes two angles formed by a common ray and two additional half-lines lying on the same line. All three beams come from the same point. The common half-line is at the same time the side of both one and the second angle.
Adjacent corners - basic properties
1. Based on the formulation of adjacent angles, it is easy to see that the sum of such angles always forms a straight angle, the degree measure of which is 180 °:
- If μ and η are adjacent angles, then μ + η = 180°.
- Knowing the value of one of the adjacent angles (for example, μ), one can easily calculate the degree measure of the second angle (η) using the expression η = 180° - μ.
2. This property of angles allows us to draw the following conclusion: an angle adjacent to a right angle will also be right.
3. Considering the trigonometric functions (sin, cos, tg, ctg), based on the reduction formulas for adjacent angles μ and η, the following is true:
- sinη = sin(180° - μ) = sinμ,
- cosη = cos(180° - μ) = -cosμ,
- tgη = tg(180° - μ) = -tgμ,
- ctgη = ctg(180° - μ) = -ctgμ.
Adjacent corners - examples
Example 1
Given a triangle with vertices M, P, Q – ΔMPQ. Find the angles adjacent to the angles ∠QMP, ∠MPQ, ∠PQM.
- Let us extend each side of the triangle as a straight line.
- Knowing that adjacent angles complement each other to a straight angle, we find out that:
adjacent to the angle ∠QMP is ∠LMP,
adjacent to the angle ∠MPQ is ∠SPQ,
the adjacent angle for ∠PQM is ∠HQP.
Example 2
The value of one adjacent angle is 35°. What is the degree measure of the second adjacent angle?
- Two adjacent angles add up to 180°.
- If ∠μ = 35°, then adjacent ∠η = 180° – 35° = 145°.
Example 3
Determine the values of adjacent angles, if it is known that the degree measure of one of the bottom is three times greater than the degree measure of the other angle.
- Let us denote the value of one (smaller) angle through – ∠μ = λ.
- Then, according to the condition of the problem, the value of the second angle will be equal to ∠η = 3λ.
- Based on the basic property of adjacent angles, μ + η = 180° follows
λ + 3λ = μ + η = 180°,
λ = 180°/4 = 45°.
So the first one angle is ∠μ = λ = 45°, and the second angle is ∠η = 3λ = 135°.
The ability to appeal to terminology, as well as knowledge of the basic properties of adjacent angles, will help to cope with the solution of many geometric problems.
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