1, using axiom 3, two planes have a common point, so the two planes have a common intersection.
2. Draw the plane parallelism theorem.
3.P belongs to plane αP belongs to plane β, so p belongs to α∪β= l (two intersecting lines).
Extended data:
Methods and points for attention in learning solid geometry;
First of all, we should establish the concept of space and improve the imagination of space.
It is a leap from understanding plane graphics to understanding three-dimensional graphics, and there must be a process. It is a good method for some students to make some geometric models of space and observe them repeatedly, which is conducive to establishing the concept of space.
Some students will observe and ponder some three-dimensional figures in their spare time, judge the relationship between lines, lines and planes, and explore various angles and vertical lines, which is also a good way to establish the concept of space. In addition, using graphs to represent concepts and theorems, and "proving" theorems and constructing graphs of theorems in your mind are also very helpful to establish spatial concepts.
Second, we should master basic knowledge and skills.
We should express concepts, theorems and formulas in three forms: graphics, words and symbols, and constantly review what we have learned before. This is because the contents of solid geometry are closely related. The former is the foundation of the latter, which not only consolidates the former, but also develops and popularizes the former.
When solving problems, write specifications. For example, when a parallelogram ABCD is used to represent a plane, it can be written as a plane AC, but the word plane cannot be omitted; To write the basis for solving problems, whether it is a calculation problem or a proof problem, it should be like this, and you can't take it for granted or rely entirely on intuition.
Written proof questions, written known and verified, drawing; When using the theorem, it must be clear that the conditions that the topic satisfies the theorem correspond to each other. It's impossible to know without writing it. Learn to use diagrams (drawing, decomposition diagram, transformation diagram) to help solve problems; It is necessary to master the basic methods of finding various angles and distances and the basic methods of reasoning and proof-analysis, synthesis and reduction to absurdity.
Third, we should constantly improve our abilities in all aspects.
Propose a proposition by connecting with practice, observing models or analogizing plane geometry conclusions; Don't easily affirm or deny the proposition, but use several special cases to test it. It is best to give a negative example and definitely prove it. The content of Euler formula is given in the form of research topic, from which we can experience and create mathematical knowledge. We should constantly structure and systematize what we have learned.
The so-called structure refers to the understanding and organization of knowledge from the whole to the part and from top to bottom, and to understand the ideas and methods implied in it. The so-called systematization is to gather parallel problems, vertical problems, angle problems, distance problems, uniqueness problems and other similar problems, compare their similarities and differences, and form an overall understanding of them.
Firmly grasp some concepts that can control the overall situation and the whole organization, and use these concepts to control the connection between the known knowledge that is occasionally touched or not yet aware of the obvious relationship, so as to improve the overall concept.
Pay attention to accumulating strategies to solve problems. For example, the problem of solid geometry is transformed into a plane problem, or the problem of finding the distance from a point to a plane, or the problem of finding the distance from a straight line to a plane, and then it is transformed into the problem of finding the distance from a point to a plane; Or become a volume problem.
We should constantly improve the level of analyzing and solving problems: on the one hand, from the known to the unknown, on the other hand, from the unknown to the known, we should seek the connection point of positive and negative knowledge-an internal or definite mathematical relationship.
We should constantly improve the level of reflective cognition, actively reflect on our own learning activities, from experience to automation, from sensibility to rationality, deepen our understanding of theory and improve our ability and creativity in solving problems.
Precautions.
First, based on textbooks, lay a solid foundation.
Lines and planes are the basis of solid geometry. A shortcut to learn this part well is to learn theorem proving, especially the proof of some key theorems.
For example, the three vertical theorems. The content of the theorem is very simple, that is, the explanation of the relationship between lines, lines and surfaces. However, the proof of the theorem is generally more complicated or even abstract in schools. Mastering this theorem has the following three advantages:
(1) Grasp the content of the theorem thoroughly, and make clear what the function of the theorem is, how to use it in those places and how to use it.
(2) Cultivate spatial imagination.
(3) Draw some enlightenment from solving problems.
When learning these contents, you can build a graphic framework with pens, rulers, books and the like to help improve your spatial imagination. It also laid a good foundation for later study.
Second, cultivate spatial imagination.
In order to cultivate spatial imagination, some simple models can be made at the beginning of learning to help imagination. For example, a cube or a cuboid. Find the relationship between line, line and face, and face to face in a cube. By observing the position relationship of points, lines and surfaces in the model, I gradually cultivate my imagination and recognition ability of spatial graphics.
Secondly, we should cultivate our own painting ability. You can start with simple figures (such as lines and planes) and simple geometries (such as cubes). The last thing we have to do is to establish a three-dimensional concept, so that we can imagine the spatial graphics and draw them on a plane (such as paper and blackboard), or we can imagine the real shape of the original spatial graphics according to the "three-dimensional" graphics drawn on the plane.
Spatial imagination is not a rambling fantasy, but based on assumptions and geometry, which will give spatial imagination wings.
Third, gradually improve the ability of logical argumentation.
The proof of solid geometry is irreplaceable by any part of mathematics. Therefore, there has been a saying of solid geometry in the college entrance examination over the years. When demonstrating, we must first be rigorous and accurately understand any definition, theorem and inference. The symbolic representation is completely consistent with the theorem, and only when all the conditions of the theorem are met can the relevant conclusions be deduced.
Don't jump to conclusions without complete conditions. Secondly, when demonstrating problems, we should use analytical methods, that is, gradually find the sufficient conditions for the conclusion to be established, get close to what is known, and then write it out in the form of comprehensive method ("deductive method").
Fourth, the application of "transformation" thought.
Personally, I think it is very important to make full use of the mathematical thought of "transformation" to solve the problem of solid geometry and clarify what has changed and what has not changed in the process of transformation. For example:
1. The angle formed by two straight lines of different planes is converted into the included angle of two intersecting straight lines, that is, the parallel lines leading to two straight lines of different planes at any point in space. The angle formed by the diagonal and the plane is converted into the angle formed by the straight line, that is, the angle formed by the projection of the diagonal and the diagonal on the plane.
2. The distance of a straight line in different planes can be converted into the distance between a straight line and a plane parallel to it, or into the distance between two parallel planes, that is, the distance of a straight line in different planes can be converted into the distance between a straight line and a plane and the distance between planes. And surface distance can be transformed into line-surface distance, and then into point-surface distance, and point-surface distance can be transformed into point-line distance.
3. Plane parallelism can be transformed into line parallelism, and line parallelism can also be transformed into line parallelism. Line-to-line parallelism can be obtained by line-to-surface parallelism or surface-to-surface parallelism, and the two can be transformed into each other. Similarly, surface verticality can be transformed into line verticality, and then into line verticality.
4. The Three Vertical Theorem can transform two straight lines on the plane into two straight lines on the space, while the Three Vertical Inverse Theorem can transform two straight lines on the plane.
All of the above are the application of the idea of reduction in mathematical thought, and the problem can be greatly simplified through reduction.
Fifth, summarize the rules and standardize training.
In the process of solving solid geometry problems, there is often obvious regularity. For example, to find the angle, we must first determine the plane angle and triangle. Commonly used are sine and cosine theorem and triangle definition. 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 it is difficult to make vertical lines, use equal product heights for conversion. Only by constantly summing up can we continue to rise.
We should also pay attention to standardized training. The problems reflected in the college entrance examination are very serious. Many candidates are not clear about the three links of writing, proving and seeking, and their expressions are not standardized and rigorous, and the causal relationship is not sufficient. The relationship between elements in graphics is misunderstood, and the symbolic language cannot be used.
This requires us to develop a good habit of answering questions in peacetime, specifically, to work out the questions step by step according to the answer format, steps and reasoning process of the examples in the textbook. The standardization of answering questions is very important in every link of mathematics examination, especially solid geometry, because it pays more attention to logical reasoning.
For students who are about to take the college entrance examination, every point in the exam is very important. Under the principle of "grading step by step", it is obvious to cultivate this normative benefit from every question at ordinary times, and in many cases, the original difficult questions are written down step by step, and the ideas are gradually opened.
Sixth, the application of typical conclusions.
In the usual learning process, some typical propositions that have been proved can be written down as conclusions. Using these conclusions, we can quickly solve some complex problems, especially when solving the choice or fill-in-the-blank questions. Although these conclusions can't be directly applied to some problem solving, they will also help us open our minds and work out the answers.