Course wide geometry, volume and multifaceted: it includes many different themes, rules, theorems, and useful knowledge.One can imagine that everything in our world is made up of simple, even the most complex.Points, lines, planes - it's all there in your life.And they lend themselves to the existing laws in the world ratio of objects in space.To prove it, you can try to prove the parallelism of lines and planes.
What line?Direct - a line that connects two points along the shortest path, not lasting and ending on both sides to infinity.The plane - the surface is formed when forming the kinematic motion of a straight line along the rail.In other words, if the two lines have any intersection point in space, they can lie in one plane.However, how to express the parallelism of planes and straight lines, if the data is not sufficient for such a statement?
main condition of parallel lines and planes - that they do not have common points.In contrast to the lines, which can be in the absence of common points is not parallel but divergent, two-dimensional plane, which eliminates such a thing as divergent lines.If this condition is not met parallel - so this line intersects the plane at some one point or is it completely.
What shows us the condition of parallel lines and planes most clearly?The fact that at any point of the distance between the parallel lines and planes is constant.If there is even the slightest, in the billions of degrees, the slope line, sooner or later cross the plane by mutual infinity.That is why parallel lines and planes is possible only in accordance with this rule, or its main condition - the lack of common points - will not be met.
What can be added, talking about parallel lines and planes?What if one of the parallel lines belongs to a plane or parallel to the second plane, or also belongs to it.How to prove it?Parallel to the line and the plane encompasses the line parallel to this, it proved very easy.Parallel lines do not have common points - therefore, they do not overlap.And if the line does not intersect at one point - so it is parallel to or, or lying on the plane.This proves once again parallel to the line and the plane, with no points of intersection.
In geometry, there is also a theorem, which states that if there are two planes and a straight line perpendicular to both of them, the planes are parallel.A similar theorem states that if two lines are perpendicular to the plane of any one, they will be parallel to each other.Is it true and provable whether the parallel lines and planes, expressed these theorems?
turns out, it is.The line perpendicular to the plane, will always be strictly perpendicular to any straight line, which runs in the plane, and also the other point of intersection of the line.If the line is similar to the intersection of several planes and in all cases it is perpendicular - so all the data plane parallel to each other.A good example is children's pyramid: its axis is perpendicular to the desired line, and the ring of the pyramid - the planes.
So, prove parallel lines and planes fairly easily.This knowledge is obtained by the students in the study of the basics of geometry and largely determine further learning.If you know how to properly use the training received at the beginning of knowledge, which can operate a large number of formulas, and skip the logical link between them.The main thing - is understanding the basics.If it is not - then the study of geometry can be compared to the construction of multi-storey building without a foundation.That is why this subject requires careful attention and thorough investigation.
