The first volume of eighth grade mathematics Unit 1 Knowledge 1
Congruent triangle
1. congruent triangles's concept can completely coincide with two triangles called congruent triangles. When two triangles are congruent, overlapping vertices are called corresponding vertices, overlapping edges are called corresponding edges, and overlapping angles are called corresponding angles. An edge is the common edge of two adjacent angles in a triangle, and an included angle is the angle formed by two edges with common endpoints in a triangle. A triangle can be translated, folded and rotated to obtain its congruence.
2. congruent triangles uses the symbol ""to indicate congruence, which is pronounced as "all things are equal". Such as △ ABC△ DEF, which is read as "all triangles ABC are equal to triangle def". Note: When remembering two congruent triangles, the letters representing the corresponding vertices are usually written in the corresponding positions.
3. What is the nature of congruent triangles?
(1) congruent triangles has equal sides and angles.
(2) The circumference and area of congruent triangles are equal.
(3) The corresponding median line, angular bisector and high line on the corresponding side of congruent triangles are equal respectively.
4, learning congruent triangles should pay attention to the following questions:
(1) The different meanings of "corresponding edge" and "opposite edge", "corresponding angle" and "diagonal" should be correctly distinguished;
(2) When two triangles are congruent, the letters representing the corresponding vertices should be written in the corresponding positions;
(3) Two triangles with "three corresponding angles are equal" or "two opposite corners have two sides and one of them is equal" are not necessarily the same;
(4) Always pay attention to the implicit conditions in graphics, such as "corner", "edge" and "diagonal".
5. congruent triangles's judgment edge: three edges correspond to the congruence of two equal triangles (abbreviated as "SSS"). Angle: Two sides and their included angles are equal. Two triangles are congruent (abbreviated as "SAS"). Angle: The intersection of two angles of two triangles and their clamping edges (abbreviated as "ASA"). Corner edge: the opposite side of two angles and one angle corresponds to the congruence of two triangles (abbreviated as "AAS"). Determination of congruence of right-angled triangles: For special right-angled triangles, there is also HL theorem (hypotenuse and right-angled edge theorem), and two right-angled triangles with hypotenuse and a right-angled edge are congruent (can be abbreviated as "hypotenuse, right-angled edge" or "HL").
6. congruent transformation only changes the position of graphics. second, graphic transformation without changing the shape and size is called congruent transformation. Congruent transformations include the following three types:
(1) Translation transformation: the transformation in which a graph moves in parallel along a straight line is called translation transformation.
(2) Symmetric transformation: the graph is folded along a straight line 180, which is called symmetric transformation.
(3) Rotation transformation: rotating a figure around a certain point to another position is called rotation transformation. The basic idea of proving the congruence of two triangles is: generally speaking, the known equilateral or equilateral angles of two triangles are determined according to the topic setting and the graphics, and then the missing conditions are found and proved according to the judgment axiom or theorem. The basic idea is as follows:
A. Two sides are equal, the included angle is equal, or the third side is equal. The former is judged by SAS and the latter by SSS.
B. If two angles are equal, the edges are equal, or the opposite sides of any equal angle are equal, the former is judged by ASA and the latter by AAS.
C. If the diagonal lines of one side and the other side are equal, the other corner is equal and judged by AAS.
D, one side is equal to the adjacent angle of the side, the other side is equal in angle, or the other angle is equal, the former is judged by SAS, and the latter is judged by AAS.
The first volume of eighth grade mathematics Unit 1 Knowledge 2
Angular bisector 1, angular bisector: the ray that divides an angle into two identical angles is called the bisector of the angle;
2. Theorem of the properties of the angular bisector: the distance from the point on the angular bisector to both sides of the angle is equal: ① the point on the angular bisector; ② Distance from point to edge;
3. Theorem for judging the bisector of an angle: the point where the distance from the inside of the angle to both sides of the angle is equal is on the bisector of the angle.
4, the law of the method
(1) bisector of an angle, usually perpendicular to both sides of the angle.
(2) The key to prove that a point is on the bisector of an angle is to prove that the distance between the point and both sides of the angle is equal, that is, to prove that the line segments are equal. The commonly used methods are: using congruent triangles, the nature of the bisector of the angle is equal to the utilization area, but special attention should be paid to the distance from the point to both sides of the angle.
(3) Note: When proving the problem, you can directly apply the property theorem and judgment theorem of the angular bisector without looking for congruent triangles.
How to learn junior high school mathematics well
1, see examples for after-class analysis.
Knowing the examples in class does not mean that you have the ability to solve problems and transfer knowledge. After class, we need to re-examine and analyze the examples from a new angle. Due to the mastery of new knowledge, the expansion of knowledge and the guidance and teaching of teachers, we have different understandings of difficulties when we look at examples and enter a higher level. You will have a deeper understanding of the application of basic knowledge, the choice of analysis and reasoning methods. If you don't look at the examples after class, your thinking will stay at a shallow level, and you can't complete the transformation process from shallow to deep, from the outside to the inside. ?
2. Example of Job Reasoning
Doing problems is the most important and effective way to use knowledge to solve problems and improve ability, and it is also the key to learn mathematics well. When doing homework, we must first identify the examples, that is, what kind of examples this problem belongs to in this chapter; Secondly, we should recall how the teacher solved the problem in class, then analyze several methods to solve the problem, and finally make clear which method is the easiest. If you can't remember clearly or forget the examples you have learned before, you should take time out to read, analyze and remember.
Eighth grade, the first volume of mathematics, unit 1, articles related to knowledge points;
★ Summary of knowledge points in the first volume of eighth grade mathematics published by People's Education Press
★ Summary of knowledge points in the first volume of eighth grade mathematics
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★ Knowledge points of the first volume of mathematics in the second day of junior high school
★ Knowledge points in the first volume of the eighth grade math book
★ The first volume of mathematics knowledge points induction of the second grade teaching edition.
★ Arrangement of knowledge points in the first volume of eighth grade mathematics
★ Summary of knowledge points in the first volume of eighth grade mathematics
★ Knowledge points in the first volume of eighth grade mathematics
★ The first unit of physics in Grade Two is a complete collection of knowledge points.