How to learn plane geometry
The geometry of the second grade is mainly the application of congruence, the properties of quadrilateral and Pythagorean theorem. The third grade mainly studies the knowledge of similarity and circle.
The key to learning geometry is to master the transformation of three languages. Symbolic language, graphic language, written language. We should be able to draw sketches according to written language and symbolic language and convert them into graphic language. When learning geometry, don't memorize theorems and definitions, but understand them. How to understand it? The key is to be able to translate it into graphic language and understand it with pictures. For example, congruent angle can be regarded as the letter f, internal angle can be regarded as z, and internal angle on the same side can be regarded as u, for example, the proportion theorem of parallel lines can be regarded as a shape, and trapezoid can be regarded as an x shape. Verticality can be regarded as+,and two lines perpendicular to the same line in the same plane are parallel, which can be regarded as H, which proves that the sum of internal angles can be divided into n-2 triangles. C is BD, and the symbol language is BC=DC= 1/2BD, or BD=2BC=2DC. The definition of angular bisector can also be similar to analogy, which requires a deep understanding of the language of symbols, graphics and characters. Some problems about angles in geometry are complicated. We can use Greek letters for algebraic operations to avoid too complicated equivalent substitution, such as proving verticality. As long as I calculate that the intersection angle is 90 degrees, the problem will be solved naturally. For example, in triangle ABC, I is the center and the angle BIC=90+ 1/2 Angle A. It is easy to prove such a problem by replacing it with letters. As long as you pay attention to the sum of the bisector and the inner angle of the two angles, it will be easy. For what is the trinity and inverse theorem of isosceles triangle, the nature and judgment of angular bisector, we should be familiar with words and figures. Combining graphic understanding and theorem application is a good learning method. The mutual conversion of characters, symbols and graphic languages is just like Chinese-English translation and English-Chinese translation in English.
There are several geometric problems. Prove the relationship between equal angle, equal segment, vertical, parallel, proportion, segment and. Next, I will talk about how to prove these kinds of problems. We can use the definitions of congruence, similar corresponding angles equal, angular bisector, same arc equal to circumferential angle, same arc equal to tangent angle, property theorem of parallel lines, property theorem of parallelogram and so on. An isosceles triangle has equal sides and equal angles. To prove that line segments are equal, we can use congruence, equiangular equilateral, chords opposite to the same arc are equal, the property of midpoint, tangent length theorem, vertical diameter theorem, parallelogram property, isosceles trapezoid property and so on. It is proved that verticality can be changed by vertical definition, angle calculation, inverse theorem of three lines in one, congruence, similarity method and diameter circle angle equal to 90 degrees. By using the judgment theorem of parallel lines, two lines parallel to the same line are parallel, two lines perpendicular to the same line are parallel in the same plane, and the parallel lines are proportional to each other, which proves the parallelism. It is proved that the proportional formula can be proportional to the corresponding side of similar triangles. Parallel lines are divided into segments in proportion, algebraic calculation, etc. Generally, two methods can be used to prove line segments and relationships, and then combined with congruence proof. Of course, if the geometric method is not easy to do, it can also be calculated by algebra.
Finally, when learning geometry, we should pay attention to clear concepts. For example, the bisector, height and center line of a triangle are all line segments. Why can't the edges and corners be congruent? Why should two straight lines perpendicular to the agreed line be parallel and add the same plane? Without this premise, the three sides intersecting at the vertex of the cube are perpendicular to each other. Parallel lines define why two straight lines on the same plane never intersect. Because there are straight lines in different planes, there is no need to cross them. There is another point outside the intersection line, only one line is parallel to the known line. These preconditions. And the difference between the shortest vertical line segment and the shortest line segment between two points. Because the vertical line refers to the shortest vertical line between a point outside the line and a known line. The shortest line segment between two points refers to the shortest line segment among all the connecting lines between two points. Lines include arcs, irregular lines, etc. The object of the former is a line segment. Therefore, we must flexibly use theorems to define concepts in learning. Don't confuse 30 degrees north latitude with 60 degrees azimuth. 30 degrees north by east means starting from the north and heading east, that is, turning 30 degrees clockwise. When learning three geometric transformations, it is clear that translation, rotation and symmetry do not change the size and shape of graphics. Just change the position. The corresponding edge and the corresponding angle are equal. Translation and parallel corresponding edges, rotation and symmetry do not have this property.
As long as you can translate geometry into three languages, the basic concepts are clear and you are good at summarizing and proving methods. Geometry is actually easy to learn. The rigor of process writing is not a problem as long as you practice more questions. In the last semester of senior one and senior two, you should indicate the reasons, be familiar with theorems and definitions, and don't write the reasons later. Finally, I hope everyone can make friends with geometry, enjoy the beauty of geometry and feel the beauty of mathematics.