Angle Angle in Geometry
In this article we will consider one of the main geometric shapes - the angle. After a general introduction to this concept, we will focus on the particular type of such a figure. The developed angle is an important concept of geometry, which will be the main topic of this article.
Introduction to the concept of a geometric angle
In geometry, there are a number of objects that form the basis of all science. The angle just to treat them and is determined using the concept of the beam, so let's start with it.
Also, before starting to determine the angle itself, you need to recall a few equally important objects in geometry - this is a point, straight on the plane and the plane itself. Straight line is called the simplest geometric shape, which has no beginning or end. A plane is a surface that has two dimensions. Well, a ray (or a semi-direct) in geometry is a part of a straight line that has a beginning, but no end.
Using these concepts, we can make a statement that the angle is a geometric figure that lies completely in a certain plane and consists of two mismatched rays with a common beginning.Such rays are called sides of the angle, and the common origin of the sides is its vertex.
Types of angles and geometry
We know that the angles can be quite different. And because a little lower will be given a small classification that will help better understand the types of angles and their main features. So, there are several types of angles in geometry:
- Right angle. It is characterized by a magnitude of 90 degrees, which means that its sides are always perpendicular to each other.
- Sharp corner. These corners include all their representatives, having a size less than 90 degrees.
- Obtuse angle. All angles with a magnitude of 90 to 180 degrees can also be here.
- Angle It has a size strictly 180 degrees and externally its sides are one straight.
The concept of unwrapped angle
Now let's look at the expanded angle in more detail. This is the case when both sides lie on one straight line, which can be clearly seen in the figure a little lower. It means that we can say with confidence that a developed angle has one of its sides as a matter of fact, as a continuation of the other.
It is worth remembering the fact that such an angle can always be divided with the help of a beam that comes out of its top.As a result, we obtain two angles, which are called adjacent in geometry.
Also the developed angle has several features. In order to talk about the first of them, you need to remember the concept of "bisector of the corner." Recall that this is a beam that divides any angle strictly in half. As for the unfolded angle, its bisectrix divides it in such a way that two right angles of 90 degrees are formed. It is very easy to calculate mathematically: 180˚ (degree of the unfolded angle): 2 = 90˚.
If we divide the unfolded angle by a completely arbitrary ray, then as a result we always get two angles, one of which will be sharp, and the other - blunt.
Properties of the expanded corners
It will be convenient to consider this angle, bringing together all its main properties, which we did in this list:
- The sides of the unfolded angle are antiparallel and are straight.
- The magnitude of the unfolded angle is always 180.
- Two adjacent corners together always form an unfolded angle.
- The full angle, which is 360˚, consists of two deployed and is equal to their sum.
- Half the unfolded angle is a right angle.
So, knowing all these characteristics of this type of angles, we can use them to solve a number of geometric problems.
Tasks with expanded corners
In order to understand whether you have understood the concept of a developed angle, try to answer the following several questions.
- What is equal to the unfolded angle if its sides make up the vertical line?
- Will the two corners be contiguous if the magnitude of the first is 72˚, and the other is 118˚?
- If the full angle consists of two unfolded, then how many right angles in it?
- The expanded angle was divided by a beam into two such angles, that their degree measures refer to 1: 4. Calculate the angles obtained.
Solutions and Answers:
- No matter how the unfolded angle is located, it is always equal to 180˚ by definition.
- Adjacent angles have one common side. Therefore, in order to calculate the size of the angle that they put together, you just need to add the value of their degree measures. So, 72 +118 = 190. But by definition, the unfolded angle is 180˚, which means that these two angles cannot be adjacent.
- The developed angle holds two right angles. And since there are two deployed ones in full, it means there will be 4 straight lines in it.
- If we call the desired angles a and b, then let x be the coefficient of proportionality for them, which means that a = x, and accordingly b = 4x. The developed angle in degrees is 180˚. And according to its propertiesthat the degree measure of the angle is always equal to the sum of the degree measures of the angles into which it is divided by any arbitrary ray that passes between its sides, we can conclude that x + 4x = 180˚, which means 5x = 180. From here we find: x = a = 36˚ and b = 4х = 144˚. Answer: 36˚ and 144˚.
If you managed to answer all these questions without prompting and not looking at the answers, then you are ready to move on to the next lesson in geometry.