First of all, we must examine the questions. After reading a topic, many students haven't figured out what it means. The title requires you to prove that you know nothing, which is very undesirable. We should read out the conditions one by one, what is the use of the given conditions, put a question mark in our mind, and then sit in the corresponding picture, where to find the conclusion and where to find the position in the picture.
Second, remember. The record here has two meanings. The first layer means mark. When reading questions, you should mark each condition in the given chart. If the opposite sides are equal, they are represented by equilateral symbols. The second meaning is to remember that the conditions given by the topic should not only be marked, but also kept in mind, so that you can repeat it without looking at the topic.
Third, we should extend it. Difficult topics often hide some conditions, so we need to be able to extend, so the extension here needs to be accumulated at ordinary times. Usually, the basic knowledge points learned in class are firmly grasped, and some special graphics that are usually trained should also be memorized. When reviewing and memorizing topics, you should think about what conclusions can be drawn from these conditions (just like clicking on the computer and the corresponding menu will pop up immediately), and then mark it next to the graph. Although some conditions may not be used when they are proved, it is such a long time.
Fourth, we must analyze the comprehensive method. Analytical synthesis, that is, reverse reasoning, starts with the conclusion that the topic needs your proof. See whether the conclusion proves that the angles are equal or the sides are equal, and so on. For example, the methods to prove the angle are (1. The vertex angles are equal. 2. The congruent angles in parallel lines are equal, and the internal dislocation angles are equal. 3. Complementary Angle and Complementary Angle Theorem. 4. Definition of angle bisector. 5. isosceles triangle. 6. The corresponding angle of congruent triangles, and so on. Then choose one of the methods according to the meaning of the question, and then consider what conditions this method still lacks, and turn the topic into proof of other conclusions. Usually, the missing conditions will appear in the conditions and topics expanded in the third step. At this time, these conditions are combined to make the proof process very orderly.
Fifth, we must sum up. Many students worked out a problem and breathed a long sigh of relief. It is not advisable to do other things next. We should take a few minutes to look back at the theorems, axioms and definitions used, re-examine this problem, sum up the thinking of solving this problem, and how to start with the same type of problems in the future.
The above are the answers to common proof questions. Of course, some questions are cleverly designed and often require us to add auxiliary lines.
Analyze the known, proof and graph, and explore the idea of proof.
There are three ways to think about proving the problem:
(1) Think positively. For general simple topics, we are all actively thinking and can make them easily, so I won't go into details here.
(2) Reverse thinking. As the name implies, it is thinking in the opposite direction. Using reverse thinking to solve problems can enable students to think about problems from different angles and directions and explore solutions, thus broadening students' thinking of solving problems. This method is recommended for students to master. In junior high school mathematics, reverse thinking is a very important way of thinking, which is more obvious in the proof questions. There are few knowledge points in mathematics, and the key is how to use them. For junior high school geometry proof, the best way is to use reverse thinking. If you are in grade three, you are not good at geometry and have no idea of doing the problem, then you must pay attention to it: from now on, summarize the methods of doing the problem. Students read the stem of a question carefully and don't know where to start. I suggest you start with the conclusion. For example, there can be such a thinking process: prove that two sides are equal, as can be seen from the picture, we only need to prove that two triangles are equal; To prove the congruence of a triangle, we should combine the given conditions to see what conditions need to be proved and how to make auxiliary lines to prove this condition. If you keep thinking like this, you will find a solution to the problem and then write out the process. This is a very useful method. Students must try.
(3) positive and negative combination. For topics that are difficult to separate ideas from conclusions, students can carefully analyze conclusions and known conditions. In junior high school mathematics, known conditions are usually used in the process of solving problems, so we can look for ideas from known conditions, such as giving us the midpoint of a triangle, and we must figure out whether to connect the midline or use midpoint multiplication method. Give us a trapezoid, we should think about whether to be tall, or to translate the waist, or to translate the diagonal, or to supplement the shape, and so on. The combination of positive and negative is invincible.