In the process of studying a geometry course, the concepts of “angle”, “vertical angles”, “ adjacent angles” are found quite often. Understanding each of the terms will help you understand the problem and solve it correctly. What are adjacent angles and how to determine them?
Adjacent angles - 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 straight line. All three rays come out from the same point. A common half-line is simultaneously a side of both one and the other angle.
Adjacent angles - basic properties
1. Based on the formulation of adjacent angles, it is easy to notice that the sum of such angles always forms a reverse angle, the degree measure of which is 180°:
- If μ and η are adjacent angles, then μ + η = 180°.
- Knowing the magnitude of one of the adjacent angles (for example, μ), you can easily calculate the degree measure of the second angle (η) using the expression η = 180° – μ.
2. This property angles allows us to draw the following conclusion: an angle that is adjacent right angle, will also be direct.
3. Considering 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 angles - examples
Example 1
Given a triangle with vertices M, P, Q – ΔMPQ. Find the angles adjacent to the angles ∠QMP, ∠MPQ, ∠PQM.
- Let's extend each side of the triangle with a straight line.
- Knowing that adjacent angles complement each other up to a reversed angle, we find out that:
adjacent to the angle ∠QMP is ∠LMP,
adjacent to the angle ∠MPQ is ∠SPQ,
adjacent to the angle ∠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 to it ∠η = 180° – 35° = 145°.
Example 3
Determine the values of adjacent angles if it is known that the degree measure of one of them is three times greater than the degree measure of the other angle.
- Let us denote the magnitude of one (smaller) angle by – ∠μ = λ.
- Then, according to the conditions 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°.
This means that the first angle is ∠μ = λ = 45°, and the second angle is ∠η = 3λ = 135°.
The ability to use terminology, as well as knowledge of the basic properties of adjacent angles, will help you solve many geometric problems.
In this lesson we will look at and understand the concept of adjacent angles. Let's consider a theorem that concerns them. Let us introduce the concept of “vertical angles”. Let's look at some supporting facts about 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 look at several problems on this topic.
Let's start our lesson with the concept of “adjacent angles”. Figure 1 shows the developed angle ∠AOC and the ray OB, which divides this angle into 2 angles.
Rice. 1. Angle ∠AOC
Let's consider the angles ∠AOB and ∠BOC. It is quite obvious that they have common side VO, and the parties AO and OS are opposite. Rays OA and OS complement each other, which means they lie on the same straight line. 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 adjacent.
Theorem 1: The sum of adjacent angles is 180 o.
Rice. 2. Drawing for Theorem 1
∠MOL + ∠LON = 180 o. This statement is true, since the ray OL divides the unfolded 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 degrees.
Consider the intersection of two lines. The figure shows the intersection of two lines at point O.
Rice. 3. Vertical angles ∠ВОА and ∠СOD
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. Consider the rotated angle ∠AOC. ∠AOB = ∠AOC - ∠BOC = 180 o - β. Let's consider the rotated angle ∠BOD. ∠COD = ∠BОD - ∠BOC = 180 o - β.
From these considerations we conclude that ∠AOB = ∠COD = α. Similarly, ∠AOD = ∠BOS = β.
Corollary 1: The angle between the bisectors of adjacent angles is 90°.
Rice. 4. Drawing for Corollary 1
Since OL is the bisector of the angle ∠BOA, then the angle ∠LOB = , similarly to ∠BOA = . ∠LOK = ∠LOB + ∠BOK = + = 
. The sum of angles α + β is equal to 180°, since these angles are adjacent.
Corollary 2: The angle between the bisectors of vertical angles is 180°.
Rice. 5. Drawing for Corollary 2
KO is the bisector ∠AOB, LO is the bisector ∠COD. Obviously, ∠KOL = ∠KOB + ∠BOC + ∠COL = o. The sum of angles α + β is equal to 180°, since these angles are adjacent.
Let's look at 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 = 180 o.
This means β = 69 o.
This type of problem exploits the sum of adjacent angles theorem.
One of the adjacent angles is a right angle, what is the other angle (acute, obtuse or right)?
If one of the angles is right, and the sum of the two angles is 180°, then the other angle is also right. This problem 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.
Two equal angles are given. 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 correct.
- Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Education.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. Geometry 7. 5th ed. - M.: Enlightenment.
- \Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzova, S.B. Kadomtsev, V.V. Prasolov, edited by V.A. Sadovnichigo. - M.: Education, 2010.
- Measurement of segments ().
- General lesson on geometry in 7th grade ().
- Straight line, segment ().
- 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. Sadovnichigo. - M.: Education, 2010.
- Find two adjacent angles if one is 4 times the other.
- Given the angle. Construct adjacent and vertical angles for it. How many such angles can be constructed?
- * In which case are more pairs of vertical angles obtained: when three straight lines intersect at one point or at three points?
The known value of the main angle α₁ = α₂ = 180°-α.
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 of a developed angle. The magnitude of the angle is a number indicating how much the angle chosen as the unit of measurement fits into the figure in question.
Each angle has a degree measure, greater than zero. A straight angle is 180 degrees. The degree measure of an angle is considered equal to the sum of the degree measures of the angles into which it is divided by any ray on the plane bounded by its sides.
An angle with a certain degree measure not exceeding 180 can be plotted from any ray into a given plane. Moreover, there will be only one such angle. Measure flat angle, which is part of a half-plane, is considered a degree measure of an angle with similar sides. The measure of the plane of an angle containing a half-plane is the value 360 – α, where α is the degree measure of the complementary plane angle.
The degree measure of an angle makes it possible to move from a geometric description to a numerical one. So, a right angle is an angle equal to 90 degrees, an obtuse angle is an angle less than 180 degrees but greater than 90, an acute angle does not exceed 90 degrees.
In addition to degrees, there is a radian measure of angle. In planimetry, the length is L, the radius is r, and the corresponding central angle is α. Moreover, these parameters are related by the relation α = L/r. This is the basis of the radian measure of angles. If L=r, then the angle α will be equal to one radian. So, the radian measure of an angle is the ratio of the length of an arc drawn with an arbitrary radius and enclosed between the sides of this angle to the radius of the arc. A complete rotation in degrees (360 degrees) corresponds to 2π in radians. One is 57.2958 degrees.
Video on the topic
Sources:
- degree measure of angles formula
Two angles placed on the same straight line and having the same vertex are called adjacent.
Otherwise, if the sum of two angles on one straight line is equal to 180 degrees and they have one side in common, then these are adjacent angles.
1 adjacent angle + 1 adjacent angle = 180 degrees.
Adjacent angles are two angles in which one side is common, and the other two sides generally form a straight line.
The sum of two adjacent angles is always 180 degrees. For example, if one angle is 60 degrees, then the second will necessarily be equal to 120 degrees (180-60).
Angles AOC and BOC are adjacent angles because all conditions for the characteristics of adjacent angles are met:
1.OS - common side of two corners
2.AO - side of the corner AOS, OB - side of the corner BOS. Together these sides form a straight line AOB.
3. There are two angles and their sum is 180 degrees.
Remembering the school geometry course, we can say the following about adjacent angles:
adjacent angles have one side in common, and the other two sides belong to the same straight line, that is, they are on the same straight line. If according to the figure, then the angles SOV and BOA are adjacent angles, the sum of which is always equal to 180, since they divide a straight angle, and a straight angle is always equal to 180.
Adjacent angles are an easy concept in geometry. Adjacent angles, an angle plus an angle, add up to 180 degrees.
Two adjacent angles will be one unfolded angle.
There are several more properties. With adjacent angles, problems are easy to solve and theorems to prove.
Adjacent angles are formed by drawing a ray from an arbitrary point on a straight line. Then this arbitrary point turns out to be the vertex of the angle, the ray is the common side of adjacent angles, and the straight line from which the ray is drawn is the two remaining sides of adjacent angles. Adjacent angles can be the same in the case of a perpendicular, or different in the case of an inclined beam. It is easy to understand that the sum of adjacent angles is equal to 180 degrees or simply a straight line. Another way to explain this angle is simple example- at first you walked in one direction in a straight line, then you changed your mind, decided to go back and, turning 180 degrees, set off along the same straight line in the opposite direction.
So what is an adjacent angle? Definition:
Two angles with a common vertex and one common side are called adjacent, and the other two sides of these angles lie on the same straight line.
AND short video a lesson where it is sensibly shown about adjacent angles, vertical angles, plus about perpendicular lines, which are a special case of adjacent and vertical angles
Adjacent angles are angles in which one side is common, and the other is one line.
Adjacent angles are angles that depend on each other. That is, if the common side is slightly rotated, then one angle will decrease by several degrees and automatically the second angle will increase by the same number of degrees. This property of adjacent angles allows one to solve various problems in Geometry and carry out proofs of various theorems.
The total sum of adjacent angles is always 180 degrees.
From the geometry course, (as far as I remember in the 6th grade), two angles are called adjacent, in which one side is common, and the other sides are additional rays, the sum of adjacent angles is 180. Each of the two adjacent angles complements the other to an expanded angle. Example of adjacent angles:
Adjacent angles are two angles with a common vertex, one of whose sides is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is one hundred and eighty degrees. In general, all this is very easy to find in Google or a geometry textbook.
Two angles are called adjacent if they have a common vertex and one side, and the other two sides form a straight line. The sum of adjacent angles is 180 degrees.
In the figure, angles AOB and BOC are adjacent.
Adjacent angles are those that have a common vertex, one common side, and the other sides are continuations of each other and form an extended angle. A remarkable property of adjacent angles is that the sum of these angles is always equal to 180 degrees.
Angles with a common vertex and one common side in geometry are called adjacent
The sum of adjacent angles is 180 degrees
It should be noted that adjacent angles have equal sines
To learn more about adjacent angles, read here
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 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 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. 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 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?