From this there are . If two angles are both adjacent and equal, then they are right angles. If one of the adjacent angles is right, that 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, accordingly, will be acute.
An acute angle is one whose degree measure is less than 90 degrees, but greater than 0. An obtuse angle has a degree 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 means 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 of these adjacent angles also coincide (in the example, their degree measure will be equal to 130 degrees).
Sources:
- Big Encyclopedic Dictionary - Adjacent angles
- angle 180 degrees
The word "" has different interpretations. In geometry, an angle is a part of a plane bounded by two rays emanating from one point - the vertex. When we talk about straight, acute, and unfolded angles, we mean geometric angles.
Like any figures in geometry, angles can be compared. Equality of angles is determined using movement. It is easy to divide the angle into two equal parts. Dividing into three parts is a little more difficult, but it can still be done using a ruler and compass. By the way, this task seemed quite difficult. Describing that one angle is larger or smaller than another is geometrically simple.
The unit of measurement for angles is 1/180
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 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:
- 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.
- 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 is not difficult as long as the angles have a numerical value. Later, 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 where you need to find adjacent angles.
Getting Started with Angles
Let us be given two arbitrary rays. Let's put them on top of each other. Then
Definition 1
We will call an angle two rays that have the same origin.
Definition 2
The point that is the beginning of the rays within the framework of Definition 3 is called the vertex of this angle.
We will denote the angle by its following three points: the 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).
Let us now determine what the magnitude of the angle is.
To do this, we need to select some kind of “reference” angle, which we will take as a unit. Most often, this angle is the angle that is equal to the $\frac(1)(180)$ part of the unfolded angle. This quantity is called a degree. After choosing such an angle, we compare the angles with it, the value of which needs to be found.
There are 4 types of angles:
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 developed if it is equal to $180^0$.
Definition 6
An angle is called right if it is equal to $90^0$.
In addition to the types of angles described above, we can distinguish types of angles in relation to each other, namely vertical and adjacent angles.
Adjacent angles
Consider the reversed angle $COB$. From its vertex we draw a ray $OA$. This ray will split the original one into two angles. Then
Definition 7
We will call two angles adjacent if one pair of their sides is a developed 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.
Let's look at 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 coincides, the ray $OA$ will divide the unfolded angle by 2, therefore
$∠COA+∠BOA=180^0$
The theorem is proven.
Let's consider solving the problem using this concept.
Example 1
Find angle $C$ from the figure below
By Definition 7 we find that the angles $BDA$ and $ADC$ are adjacent. Therefore, by Theorem 1, we get
$∠BDA+∠ADC=180^0$
$∠ADC=180^0-∠BDA=180〗0-59^0=121^0$
By the theorem on the sum of angles in a triangle, we 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 unfolded angles $AOB$ and $MOC$. Let us align their vertices with each other (that is, we will superimpose the point $O"$ on the point $O$) so that no sides of these angles coincide. Then
Definition 8
We will call two angles vertical if the pairs of their sides are unfolded angles and their values coincide (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.
Let's look at Figure 3. Let's prove, for example, that the angle $MOA$ is equal to the angle $BOC$.
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(Diagram 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 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 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
(since the left side of this equality is also 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 what remains is equal. 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?
How to find an adjacent angle?
Mathematics is the oldest exact science, which is compulsorily studied in schools, colleges, institutes and universities. However, basic knowledge is always laid at school. Sometimes, the child is given quite complex tasks, but the parents are unable to help because they simply forgot some things from mathematics. For example, how to find an adjacent angle based on the size of the main angle, etc. The problem is simple, but can cause difficulties in solving due to ignorance of which angles are called adjacent and how to find them.
Let's take a closer look at the definition and properties of adjacent angles, as well as how to calculate them from the data in the problem.
Definition and properties of adjacent angles
Two rays emanating from one point form a figure called a “plane angle”. In this case, this point is called the vertex of the angle, and the rays are its sides. If you continue one of the rays beyond the starting point in a straight line, then another angle is formed, which is called adjacent. Each angle in this case has two adjacent angles, since the sides of the angle are equivalent. That is, there is always an adjacent angle of 180 degrees.
The main properties of adjacent angles include
- Adjacent angles have a common vertex and one side;
- The sum of adjacent angles is always equal to 180 degrees or the number Pi if the calculation is carried out in radians;
- The sines of adjacent angles are always equal;
- The cosines and tangents of adjacent angles are equal but have opposite signs.
How to find adjacent angles
Usually three variations of problems are given to find the size of adjacent angles
- The value of the main angle is given;
- The ratio of the main and adjacent angle is given;
- The value of the vertical angle is given.
Each version of the problem has its own solution. Let's look at them.
The value of the main angle is given
If the problem specifies the value of the main angle, then finding the adjacent angle is very simple. To do this, just subtract the value of the main angle from 180 degrees, and you will get the value of the adjacent angle. This solution is based on the property of an adjacent angle - the sum of adjacent angles is always equal to 180 degrees.
If the value of the main angle is given in radians and the problem requires finding the adjacent angle in radians, then it is necessary to subtract the value of the main angle from the number Pi, since the value of the full unfolded angle of 180 degrees is equal to the number Pi.
The ratio of the main and adjacent angle is given
The problem may give the ratio of the main and adjacent angles instead of the degrees and radians of the main angle. In this case, the solution will look like a proportion equation:
- We denote the proportion of the main angle as the variable “Y”.
- The fraction related to the adjacent angle is designated as the variable “X”.
- The number of degrees that fall on each proportion will be denoted, for example, by “a”.
- The general formula will look like this - a*X+a*Y=180 or a*(X+Y)=180.
- We find the common factor of the equation “a” using the formula a=180/(X+Y).
- Then we multiply the resulting value of the common factor “a” by the fraction of the angle that needs to be determined.
This way we can find the value of the adjacent angle in degrees. However, if you need to find a value in radians, then you simply need to convert the degrees to radians. To do this, multiply the angle in degrees by Pi and divide everything by 180 degrees. The resulting value will be in radians.
The value of the vertical angle is given
If the problem does not give the value of the main angle, but the value of the vertical angle is given, then the adjacent angle can be calculated using the same formula as in the first paragraph, where the value of the main angle is given.
A vertical angle is an angle that originates from the same point as the main one, but is directed in exactly the opposite direction. This results in a mirror image. This means that the vertical angle is equal in magnitude to the main one. In turn, the adjacent angle of the vertical angle is equal to the adjacent angle of the main angle. Thanks to this, the adjacent angle of the main angle can be calculated. To do this, simply subtract the vertical value from 180 degrees and get the value of the adjacent angle of the main angle in degrees.
If the value is given in radians, then it is necessary to subtract the value of the vertical angle from the number Pi, since the value of the full unfolded angle of 180 degrees is equal to the number Pi.
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