The course of geometry is wide, voluminous and multifaceted: it includes many different topics, rules, theorems and useful knowledge. One can imagine that everything in our world consists of simple, even the most complex. Points, lines, planes - all this is in your life. And they succumb to existing laws in the world on the ratio of objects in space. To prove this, we can try to prove the parallelism of lines and planes.
What is a straight line? A straight line is a line that connects two points along the shortest path, not ending and lasting from both sides to infinity. A plane is a surface formed during the kinematic movement of a generatrix of a straight line along a guide. In other words, if any two lines have a point of intersection in space, they can also lie in the same plane. However, how to express the parallelism of planes and lines, if these data are not enough for such a statement?
The main condition for parallelism of the line and the plane is that they do not have common points. Unlike straight lines, which, in the absence of common points, can be not parallel, but diverging, the plane is two-dimensional, which eliminates the concept of diverging lines. If this parallelism condition is not met, then the straight line intersects the given plane at any one point or lies completely in it.
What does the condition of parallelism of a line and a plane show us most clearly? The fact that at any point in space the distance between parallel to the line and the plane will be a constant. If there is even the slightest, in the billionths of a degree, slope, the line will sooner or later cross the plane due to mutual infinity. That is why parallelism of the line and the plane is possible only if this rule is observed, otherwise its main condition - the absence of common points - will not be observed.
What can be added when talking about the parallelism of lines and planes? That if one of the parallel lines belongs to the plane, then the second is either parallel to the plane, or also belongs to it. How to prove it? The parallelism of a straight line and a plane containing a straight line parallel to a given one is proved very simply. Parallel lines do not have common points - therefore, they do not intersect. And if the line does not intersect with the plane at one point, then it is either parallel or lies on the plane. This once again proves the parallelism of the line and the plane without intersection points.
In geometry, there is also a theorem which states that if there are two planes and a straight line is perpendicular to both of them, then the planes are parallel. A similar theorem states that if two lines are perpendicular to any one plane, they will necessarily be parallel to each other. Is the parallelism of lines and planes expressed by these theorems true and provable?
It turns out that it is. A line perpendicular to the plane will always be strictly perpendicular to any line that lies in the plane and also has a point of intersection with another straight line. If a line has similar intersections with several planes and in all cases is perpendicular to it, then all the planes are parallel to each other. A good example is the children's pyramid: its axis will be the desired perpendicular straight line, and the pyramid rings will be planes.
Therefore, to prove the parallelism of the line and the plane is quite easy. This knowledge is obtained by students in the study of the basics of geometry and in many respects determines the further assimilation of the material. If you can correctly use the knowledge obtained at the beginning of training, you can operate with a lot of formulas and skip unnecessary logical connectives between them. The main thing is an understanding of the basics. If it is not there, then the study of geometry can be compared with the construction of a multi-storey building without a foundation. That is why this topic requires close attention and thorough research.