First, it is the key to learn geometry well to practice three basic skills and master the concept of geometry.
Junior high school geometry mainly studies the properties of plane graphics, which has a unique language expression. There are generally three kinds of geometric languages: written language, graphic language and symbolic language. The basic skills of three languages have passed, and the basic knowledge of geometry has been solid.
Written language generally uses words to describe the concept or nature of geometry.
The general characteristics are accurate wording and tight expression, which cannot be easily changed. It is the basis of understanding and mastering different geometric figures.
Graphic language expresses the characteristics of geometric figures and studies the properties of geometric figures through understanding and drawing.
The graphic language is intuitive and vivid, which makes the written language more specific and easier to learn. Symbolic language uses a series of concrete symbols to describe the attributes of geometric figures concisely and vividly.
For example, the vertical relationship between two straight lines is represented by ⊥, the parallelism of two straight lines is represented by ∨, and the congruence of two triangles is represented by ∨.
Properties in geometry (including theorems, axioms, etc. ) are generally described in written language, but when demonstrating and solving problems in detail, graphics should be made, marked with letters, and transformed into graphic language and symbolic language to describe. Therefore, we should learn to change these three languages flexibly.
Second, mastering the basic analysis methods of geometric proof is the key to learning geometry well.
How to reason according to the known conditions of the topic and get the content required by the topic requires us to master the common analysis methods of geometric proof. Solving geometric proof problems generally requires mastering the following three analytical methods:
(1) analysis method (also called reverse reasoning method).
The analysis method is to determine the necessary condition of the conclusion based on the verified conclusion, axiom and theorem, and then explore the new necessary condition of this condition from this necessary condition, step by step, until the required condition is derived as a known condition, so as to communicate the relationship between the condition and the conclusion and prove the proposition. This is a method of "taking the fruit as the cause".
The clever use of analytical methods requires us to be familiar with the commonly used theorems to prove conclusions. If you are familiar with these theorems (or axioms), you can analyze the necessary conditions for proving the conclusion in combination with the known conditions, and move closer to the known conditions step by step until the proof is completed.
(2) Comprehensive method (also called forward method).
On the basis of known conditions, axioms and theorems, the synthesis method first explores some relatively direct conclusions. Based on these conclusions, some new conclusions are drawn, which will be deepened step by step, and finally the conclusions to be proved are drawn. This is a "cause and effect" method. Because a condition can often lead to many conclusions, we need to calmly analyze and get the condition we want.
In the study of geometry, we should learn to associate. When a question gives a condition, we should actively reflect all the knowledge related to this condition in our brain, and be good at digging out the known condition implied by a known condition. Of course, to make such a response, we must ask us to memorize these axioms, theorems and properties in our hearts at ordinary times before we can use them freely.
(3) Analytical synthesis method (also called double-headed method).
Because analytic method is easy to find the method to prove the problem, but the compiling process is complicated, and the synthesis method is simple, but it is difficult to find the method to prove the problem, so the two methods are often combined in the proof, that is, the analytic method is used to find the method to prove the problem first, and then the synthesis method is used to compile the proof process.