How to find an adjacent corner?
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Mathematics is the oldest exact science, which is compulsorily studied in schools, colleges, institutes and universities. However, basic knowledge is always laid back in school. Sometimes, the child is given quite complex tasks, and the parents cannot help because they simply forgot some things from mathematics. For example, how to find an adjacent angle by the magnitude of the main angle, etc. The task is simple, but it can cause difficulties when deciding because of ignorance of what angles are called adjacent and how to find them.
Let us consider in more detail 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 beams emanating from one point form a figure called “flat angle”. In this case, this point is called the top of the corner, and the rays are its sides. If we 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 180 degrees or the number of pi, if the calculation is carried out in radians;
- The sinuses of adjacent angles are always equal;
- The cosines and tangents of adjacent angles are equal, but have opposite signs.
Thanks to these properties, it is quite simple to calculate the adjacent angle, knowing some data.
How to find adjacent corners
Usually, three variations of tasks for finding the value of adjacent angles are given.
- Given the magnitude of the main angle;
- Given the ratio of the main and adjacent angle;
- Given the value of the vertical angle.
Each version of the problem has its own solution. Consider them.
Given the magnitude of the main angle
If the problem specifies the magnitude of the main angle, then it is very easy to find the adjacent angle. To do this, it is enough to 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 180 degrees.
If the magnitude of the main angle is given in radians and the problem requires to find an adjacent angle in radians, then it is necessary to subtract the magnitude of the main angle from Pi, since the total unwrapped angle of 180 degrees is equal to Pi.
Given the ratio of the main and adjacent angle
The problem can be given the ratio of the main and adjacent angle instead of degrees and radians of the magnitude of the main angle. In this case, the solution will look like the proportion equation:
- We denote the proportion of the proportion of the main angle, as the variable "Y".
- The share related to the adjacent corner is denoted as the variable "X".
- The number of degrees that fall on each proportion, we denote, for example, "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” by the formula a = 180 / (X + Y).
- Then, the resulting value of the common factor "a" is multiplied by the fraction of the angle that must be determined.
Thus, we can find the value of the adjacent angle in degrees. However, if you need to find the value in radians, then you just need to convert degrees to radians. To do this, multiply the angle in degrees by the number Pi and divide everything by 180 degrees. The resulting value will be in radians.
Given the value of the vertical angle
If the value of the main angle is not given in the problem, 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.
The vertical angle is an angle that comes from the same point as the main one, but at the same time it is directed in the exact opposite direction. This results in a mirror image. This means that the vertical angle is equal to the main angle. In turn, the adjacent angle of the vertical angle is equal to the adjacent corner of the main angle. This makes it possible to calculate the adjacent angle of the main angle. To do this, simply subtract from 180 degrees the vertical value 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 developed angle of 180 degrees is equal to the number Pi.
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