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 of angles allows us to draw the following conclusion: an angle that is 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 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.
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, angles AOB and BOC are adjacent.
The sum of adjacent angles is 180°
Theorem 1. The sum of adjacent angles is 180°.
Proof. Beam OB (see Fig. 1) passes between the sides of the unfolded angle. That's why ∠ AOB + ∠ BOS = 180°.
From Theorem 1 it follows that if two angles are equal, then their adjacent angles 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. Let's consider the vertical angles AOB and COD (see Fig. 2). Angle BOD is adjacent to each of angles AOB and COD. By Theorem 1 ∠ AOB + ∠ BOD = 180°, ∠ COD + ∠ BOD = 180°.
From this 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 straight (angle 1 in Fig. 3), then the remaining angles are also right (angles 1 and 2, 1 and 4 are adjacent, angles 1 and 3 are vertical). In this case, they say that these lines intersect at right angles and are called perpendicular (or mutually perpendicular). The perpendicularity of lines AC and BD is denoted as follows: AC ⊥ BD.
A perpendicular bisector to a segment is a line perpendicular to this segment and passing through its midpoint.
AN - perpendicular to a line
Consider a straight line a and a point A not lying on it (Fig. 4). Let's connect point A with a segment to point H with straight line a. The segment AN is called a perpendicular drawn from point A to line a if lines AN and a are perpendicular. Point H is called the base of the perpendicular.
Drawing square
The following theorem is true.
Theorem 3. From any point not lying on a line, it is possible to draw a perpendicular to this line, and, moreover, only one.
To draw a perpendicular from a point to a straight line in a drawing, use a drawing square (Fig. 5).
Comment. The formulation 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 that the angles are vertical; conclusion - these angles are equal.
Any theorem can be expressed in detail in words so that its condition begins with the word “if” and its 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.
Let us 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 = 136°. Therefore, the other angle is 136°.
Example 2. Let the angle COD in Figure 21 be 45°. What are the angles AOB and AOC?
Solution.
Angles COD and AOB are vertical, therefore, by Theorem 1.2 they are equal, i.e. ∠ AOB = 45°. Angle AOC is adjacent to angle COD, which means according to Theorem 1.
∠ AOC = 180° - ∠ COD = 180° - 45° = 135°.
Example 3. Find adjacent angles if one of them is 3 times larger than the other.
Solution.
Let us denote the degree measure of the smaller angle by x. Then the degree measure of the larger angle will be 3x. Since the sum of adjacent angles is equal to 180° (Theorem 1), then x + 3x = 180°, whence x = 45°.
This means that adjacent angles are 45° and 135°.
Example 4. The sum of two vertical angles is 100°. Find the size of each of the four angles.
Solution.
Let Figure 2 meet the conditions 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 according to the condition is 100°). Angle BOD (also angle AOC) is adjacent to angle COD, and therefore, by Theorem 1
∠ BOD = ∠ AOC = 180° - 50° = 130°.
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 are always laid down 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 magnitude 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 denoted 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.
You can also read our useful articles And .
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 a short video lesson that sensibly shows 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.
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're talking about about right, acute, unfolded angles, then it means precisely 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
