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.
To master the skills of proving mathematics and geometry in junior high school, it is the key to skillfully use and memorize the following principles.
Let's classify it as follows. Practice makes perfect. What principles can you think of to solve geometric proof problems?
First, it is proved that two line segments are equal.
1. The corresponding edges in two congruent triangles are equal.
2. Equiangles and equilateral sides of the same triangle.
3. The bisector of the vertex or the high bisector of the bottom of the isosceles triangle.
4. The opposite sides or diagonals of a parallelogram are equal to two line segments separated by intersection points.
5. The midpoint of the hypotenuse of a right triangle is equal to the distance between the three vertices.
6. Any point on the middle vertical line of a line segment is equal to the distance between two segments of the line segment.
7. The distance from any point on the bisector of an angle to both sides of the angle is equal.
8. A straight line passing through the midpoint of one side of a triangle and parallel to the third side is equal to the line segment formed by the bisection of the second side.
9. In the same circle (or equal circle), the chords opposite to the equal arc or two chords equidistant from the center of the circle or the chords opposite to the equal central angle and circumferential angle are equal.
10. At a point outside the circle, the tangent lengths of the two tangents leading to the circle are equal, or the chord perpendicular to the inner diameter of the circle is equal in two parts divided by the diameter.
1 1. The last two terms (or the first two terms) in the proportional formula are equal.
12. The appearance of the inner (outer) common tangent of two circles, etc.
13. Two line segments equal to the same line segment are equal.
Second, prove that the two angles are equal.
1. The angles corresponding to two congruent triangles are equal.
2. Equiangular corners of the same triangle.
3. In an isosceles triangle, the center line (or height) of the base bisects the vertex angle.
4. The isosceles angle, internal dislocation angle or diagonal of the parallelogram of two parallel lines are equal.
5. The complementary angle (or complementary angle) of the same angle (or equal angle) is equal.
6. In the same circle (or circle), the central angles of a pair of equal chords (or arcs) are equal, the peripheral angles are equal, and the tangent angle is equal to the peripheral angles of a pair of arcs it clamps.
7. A point outside the circle leads to two tangents of the circle, and the connecting line between the center of the circle and this point bisects the included angle of the two tangents.
8. The corresponding angles of similar triangles are equal.
9. The outer angle of the inscribed quadrilateral of a circle is equal to the inner diagonal.
10. Two angles equal to the same angle.
Third, prove that two straight lines are perpendicular to each other.
1. The bisector of the top angle or the median line of the bottom edge of an isosceles triangle is perpendicular to the bottom edge.
2. If the median line of one side of a triangle is equal to half of this side, the angle subtended by this side is a right angle.
In a triangle, if two angles are complementary, the third angle is a right angle.
4. The bisectors of adjacent complementary angles are perpendicular to each other.
If a straight line is perpendicular to one of the parallel lines, it must be perpendicular to the other.
6. When two straight lines intersect at right angles, they are vertical.
7. Use points with the same distance from both ends of the line segment to be located on the middle vertical line of the line segment.
8. Use the inverse theorem of Pythagorean theorem.
9. Use diagonal lines of diamonds to be perpendicular to each other.
10. The diameter of a chord (or arc) bisected by a circle is perpendicular to the chord.
1 1. Use right angles on the semicircle.
Fourth, prove that two straight lines are parallel.
1. Lines perpendicular to the same line are parallel.
2. The congruent angles are equal, and the internal angle is equal to or parallel to two lines with complementary internal angles.
3. The opposite sides of a parallelogram are parallel.
The center line of triangle is parallel to the third side.
5. The center line of the trapezoid is parallel to the two bottom sides.
6. Two lines parallel to the same line are parallel.
7. If the line segments obtained by cutting two sides (or extension lines) of a triangle are proportional, the line is parallel to the third side.
V. Prove the sum and difference of line segments.
1. Do the sum of two line segments and prove that it is equal to the third line segment.
2. Intercept a segment equal to the first segment on the third segment, and prove that the rest is equal to the second segment.
3. Extend the short segment twice, and then prove that it is equal to the long segment.
4. Take the midpoint of the long line segment and prove that half of it is equal to the short line segment.
5. Using some theorems (midline of triangle, right triangle of 30 degrees, midline on hypotenuse of right triangle, center of gravity of triangle, properties of similar triangles, etc.). ).
Sixth, prove that the sum of angles is multiplied by the difference.
1. and prove that the sum, difference, multiplication and division of line segments are the same.
2. Using the definition of angular bisector.
3. The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.