If a dihedral angle is required, we must first find the intersection line between faces, and then start from a point on the intersection line as the perpendicular of the intersection line. The angle with a small angle is the dihedral angle.
There are two methods to find dihedral angle, one is based on complementary angle theorem directly, and the other is based on vector, which can be found well according to the formula.
Grasping vectors is an important tool in solid geometry.
Such as the distance from a point to a straight line, master the direction vector of the straight line.
Find the plane angle of dihedral angle instead of dihedral angle. The plane angle of dihedral angle is equal to the size of dihedral angle. Specifically, you can, for example, find the normal vector of a plane first, and then the angle between the normal vectors of two planes is the size of the dihedral angle.
To ask for an angle, we must first determine the plane angle and triangle. Sine and cosine theorems and triangle definitions are commonly used. If the cosine value is negative, different planes and line planes are at acute angles. Distance can be summarized as follows: distance is mostly vertical segment, calculated by triangle. Sine cosine theorem and Pythagorean theorem are often used. If the vertical line is not good, use equal product and equal height to convert it. Only by constantly summing up can we continue to rise.
The study of solid geometry mainly lies in developing students' logical thinking ability and spatial imagination ability on the basis of cultivating spatial abstraction ability. Solid geometry is a difficult point in middle school mathematics, and students generally reflect that "geometry is more difficult to learn than algebra". But many students who learn this part well think it is very simple.
I'm just here to discuss learning methods from a big perspective.
First, the improvement of spatial imagination.
When you start learning, look at simple solid geometry topics first, and don't start with difficult problems. Draw some three-dimensional geometric figures by yourself, such as exercises in textbooks and exercises in tutorials. Draw it yourself without looking at the original picture. It is a good thing that the number drawn may be different from the given number. By contrast, that number is easier to solve the problem.
Second, the cultivation of logical thinking ability.
To cultivate logical thinking ability, we must first firmly grasp the basic knowledge of mathematics, and then master the necessary logical knowledge and logical thinking.
1. Strengthen the understanding of basic concepts.
Mathematical concept is one of the two components of mathematical knowledge system, and understanding and mastering mathematical concept is the key to learning mathematics well and improving mathematical ability.
To understand the basic concepts, we must first think more. For example, two straight lines are not in the same plane, which is a simple definition. How can it not be in the same plane? First of all, we leave the straight line on the same plane from this plane, or draw with two pens, which gives us an intuitive concept of straight lines on different planes, and then we want to know how to ensure that two straight lines are not on the same plane mathematically, and those conditions can ensure that two straight lines are not on the same plane. If you think about it more, you will know that as long as the straight lines are not parallel and intersect, they are different planes. For the nonparallel condition, we already know how to ensure the non-intersection in plane geometry, and whether it can be proved by means of imaginary extension lines. If not, then put one of the straight lines on a plane and see if the other straight line is parallel to this plane, so that we can easily grasp the concept of non-planar straight line.
This is particularly prominent in the study of the "simple geometry" part of solid geometry. This chapter involves a large number of basic concepts, grasping the rationality and rigor of concepts, and distinguishing similar and confusing concepts. Such as: regular tetrahedron and regular triangular pyramid, cuboid and straight parallelepiped, axial section and straight section, sphere and sphere.
2. Strengthen the understanding of mathematical propositions and learn to use mathematical propositions flexibly to solve problems.
The understanding and application of mathematical axioms and theorems are prominent in the proof and calculation of topics. We should avoid imprecise logical reasoning, unsubstantiated statements when using theorems, axioms and laws, or replacing rigorous scientific argumentation with subjective assumptions, unreasonable writing format, unclear levels, improper use of mathematical symbol language and unaccustomed.
(1) attaches importance to the proof of the theorem itself. We know that the proof idea of theorem itself is exemplary and typical, which embodies the cultivation of basic logical reasoning knowledge and basic proof idea, as well as the cultivation of standardized writing format. We should not only analyze the conditions and conclusions of the theorem, but also master the content, thinking method, application scope and expression form of the theorem. Especially after entering high school, it involves some new ideas and proof methods, such as the example of solid geometry in the new textbook: "A straight line passing through a point outside the plane and a straight line not passing through the point inside the plane are all straight lines outside the plane." This theorem is proved by reduction to absurdity, so the proof idea of reduction to absurdity is needed.
(2) Using theorems to improve the ability of analyzing and solving problems. This is often reflected in the fact that after encountering a geometry problem, we don't know where to start. For exercises, we must first know what to do (what is the required conclusion) and what conditions can meet the requirements, so that we can find the conditions step by step. Of course, it depends on the specific situation and needs more exercise. I object to asking questions, but necessary exercises are essential.