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Induction of knowledge points of mathematical geometry in senior high school entrance examination
Some suggestions on learning mathematics well.
Eight ways of thinking in mathematics
Induction of knowledge points of mathematical geometry in senior high school entrance examination
1. There is only one straight line between two points.
2. The shortest line segment between two points.
3. The complementary angles of the same angle or equal angle are equal.
4. The complementary angles of the same angle or equal angle are equal.
5. There is one and only one straight line perpendicular to the known straight line.
6. Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7. The axiom of parallelism passes through a point outside a straight line, and one and only one straight line is parallel to this straight line.
8. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other.
9. The same angle is equal and two straight lines are parallel.
10. The internal dislocation angles are equal and the two straight lines are parallel.
1 1. The inner angles on the same side are complementary and the two straight lines are parallel.
12. Two straight lines are parallel and have the same angle.
13. Two straight lines are parallel and the internal dislocation angles are equal.
14. Two straight lines are parallel and complementary.
15. Theorem The sum of two sides of a triangle is greater than the third side.
16. Inference that the difference between two sides of a triangle is smaller than the third side.
17. The sum of the internal angles of the triangle is equal to 180.
18. It is inferred that the two acute angles of 1 right triangle are complementary.
19. Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
20. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
2 1. congruent triangles has equal sides and angles.
22. The axiom of edges and corners has two triangles with equal edges and corners.
23. The angle axiom has two angles and two triangles with equal corresponding sides.
24. It is inferred that there are two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
The 25-sided axiom has two triangles corresponding to three sides.
The axiom of hypotenuse and right-angled side has the coincidence of hypotenuse and right-angled side corresponding to two right-angled triangles.
27. Theorem 1: The distance from a point on the bisector of an angle to both sides of the angle is equal.
28. Theorem 2: Points with equal distances to both sides of an angle are on the bisector of this angle.
29. The bisector of an angle is a collection of all points with equal distance to both sides of the angle.
30. The nature theorem of isosceles triangle The two base angles of isosceles triangle are equal.
3 1. Inference 1: The bisector of the top angle of the isosceles triangle bisects the bottom and is perpendicular to the bottom.
32. The bisector of the top corner of the isosceles triangle, the median line on the bottom and the height coincide.
33. Inference 3: All angles of an equilateral triangle are equal, and each angle is equal to 60 34 isosceles triangle. If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equilateral).
35. Inference 1: A triangle with three equal angles is an equilateral triangle.
Inference 2: An isosceles triangle with an angle equal to 60 is an equilateral triangle.
37. In a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse.
38. The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
39. Theorem The point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment.
40. The inverse theorem and the equidistant point between the two endpoints of a line segment are on the vertical line of this line segment.
4 1. The middle vertical line of a line segment can be regarded as the set of all points with the same distance at both ends of the line segment.
42. Theorem 1: Two figures symmetrical about a straight line are conformal.
43. Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.
Theorem 3: Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
45. Inverse Theorem If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
46. Pythagorean theorem The sum of squares of two right-angled sides A and B of a right-angled triangle is equal to the square of hypotenuse C, that is, AB = C.
47. Inverse Theorem of Pythagorean Theorem If three sides of a triangle are related to A, B and C, then the triangle is a right triangle.
48. The sum of the internal angles of a quadrilateral is equal to 360 degrees.
49. The sum of the external angles of the quadrilateral is equal to 360.
50. Theorem The sum of the internal angles of a polygon and an N-sided polygon is equal to (n-2) × 180.
5 1. It is inferred that the sum of the external angles of any polygon is equal to 360.
52. parallelogram property theorem 1 parallelogram diagonal is equal
53. parallelogram property theorem 2 The opposite sides of a parallelogram are equal
54. It is inferred that the parallel segments sandwiched between two parallel lines are equal.
55. parallelogram property theorem 3 diagonal bisection of parallelogram. 56. parallelogram decision theorem 1 two groups of parallelograms with equal diagonals are parallelograms.
57. parallelogram decision theorem 2 Two groups of quadrilaterals with equal opposite sides are parallelograms.
58. parallelogram decision theorem 3 A quadrilateral whose diagonal is bisected is a parallelogram.
59. parallelogram decision theorem 4 A set of parallelograms whose opposite sides are parallel and equal is a parallelogram.
60. Rectangular property theorem 1 All four corners of a rectangle are right angles.
6 1. Rectangle property theorem 2 Diagonal lines of rectangles are equal
62. Rectangular Decision Theorem 1 A quadrilateral with three angles at right angles is a rectangle.
63. Rectangular Decision Theorem 2 A parallelogram with equal diagonals is a rectangle.
64. Diamond property theorem 1 All four sides of a diamond are equal.
65. Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
66. Diamond area = half of diagonal product, that is, S=(a×b)÷2.
67. Diamond Decision Theorem 1: A quadrilateral with four equilateral sides is a diamond.
68. Diamond Decision Theorem 2: Parallelograms with mutually perpendicular diagonals are diamonds.
69. Theorem of Square Properties 1: All four corners of a square are right angles and all four sides are equal.
70. Theorem 2 of Square Properties: The two diagonals of a square are equal and vertically divided, and each diagonal bisects a set of diagonals.
7 1. Theorem 1 is congruent on two centrosymmetric graphs.
72. Theorem 2 About two graphs with symmetrical centers, the connecting lines of symmetrical points all pass through the symmetrical center and are equally divided by the symmetrical center.
73. Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a certain point and is equally divided by the point, then the two graphs are symmetrical about the point.
74. The property theorem of isosceles trapezoid The two angles of the isosceles trapezoid on the same bottom are equal.
75. The two diagonals of an isosceles trapezoid are equal.
76. The isosceles trapezoid judgment theorem A trapezoid with two equilateral angles on the same bottom is an isosceles trapezoid.
77. A trapezoid with equal diagonal lines is an isosceles trapezoid.
78. Theorem of Equal Segment of Parallel Lines If a group of parallel lines have equal segments on a straight line, then the segments on other straight lines are also equal.
79. Inference 1: A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.
80. Inference 2: A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.
The median line theorem of 8 1. triangle is parallel to the third side and equal to half of the third side.
82. The trapezoid midline theorem is parallel to the two bottoms and is equal to half of the sum of the two bottoms L = (A B) ÷ 2S = L× H.
83. Basic properties of (1) ratio If a:b=c:d, then ad=bc, if ad=bc, then A: B = C: D.
84.(2) Combinatorial properties If a/b=c/d, then (A B)/B = (C D)/D.
85.(3) Isometric property If a/b=c/d=…=m/n(b d … n≠0), then (a c … m)/(b d … n) = a/b.
86. Parallel lines are divided into line segments and the proportionality theorem. Three parallel lines cut two straight lines, and the corresponding line segments are proportional.
87. It is inferred that a straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments obtained are proportional.
88. Theorem If the corresponding line segments obtained by cutting two sides of a triangle (or extension lines on both sides) are proportional, then this line is parallel to the third side of the triangle.
89. A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.
90. Theorem A straight line parallel to one side of a triangle intersects the other two sides (or extension lines on both sides), and the triangle is similar to the original triangle.
9 1. similar triangles's decision theorem 1: Two angles are equal and two triangles are similar (ASA).
92. The right triangle is divided into two right triangles according to the height on the hypotenuse, which is similar to the original triangle.
93. Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
94. Decision Theorem 3: Three sides are proportional and two triangles are similar (SSS).
95. Theorem If the hypotenuse and a right-angled side of a right-angled triangle are directly proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.
96. Property Theorem 1: similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.
97. Theorem 2: The ratio of similar triangles perimeter is equal to the similarity ratio.
Theorem 3: similar triangles area ratio is equal to the square of similarity ratio.
99. The sine value of any acute angle is equal to the cosine value of the remaining angles, and the cosine value of any acute angle is equal to the sine value of the remaining angles.
100. The tangent of any acute angle is equal to the cotangent of the remaining angles, and the cotangent of any acute angle is equal to the tangent of the remaining angles.
10 1. A circle is a set of points whose distance from a fixed point is equal to a fixed length.
102. The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
103. The outside of a circle can be regarded as a collection of points whose center distance is greater than the radius.
104. Same circle or equal circle has the same radius.
105. The distance to the fixed point is equal to the trajectory of the fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.
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Some suggestions on learning mathematics well.
1. Interested in learning mathematics. Interest is the best teacher. Do anything, as long as you are interested, you will take the initiative to do it, and you will try your best to do it well. But the key to cultivate students' interest in mathematics is to master the basic knowledge and skills of mathematics first. Some students always want to do difficult problems, and when they see others taking math classes, they also want to go. If these students can't even master the basic knowledge in class, they can only make it up in class, which will not help them, but will make them lose confidence in learning mathematics. I suggest that students can read some famous stories about mathematics and interesting mathematics to enhance their self-confidence in learning.
2. Have a correct learning attitude. First of all, it must be clear that learning is for yourself, not for teachers and parents. Therefore, we should concentrate, think positively and speak boldly in class. Secondly, after returning home, you should finish your homework carefully, review what you learned that day in time, and then preview what you will learn tomorrow. In this way, you will learn more easily and understand more deeply.
3. Have the spirit of "perseverance". If you want to improve your academic performance, you should do it step by step. Don't expect to learn everything overnight. Even if the progress is slow, as long as you persist, math learning will be successful! We should also have the spirit of "not ashamed to ask questions" and not be afraid of losing face. In fact, no matter how difficult the knowledge is, as long as you learn and understand it, that is the greatest face!
4. Pay attention to learning skills and methods. Some formulas and laws should not be memorized by rote, but should be understood by analysis and applied flexibly. Special attention should be paid to the study of new knowledge and the analysis of exercises in class. We shouldn't be distracted and mind our own business. Attention must be highly focused and think positively. When you don't understand the topic, you should make a good record in time, discuss it with your classmates after class, and do a good job of filling the vacancy.
5. Have a good habit of observing and reading. As long as we pay attention to mathematics and carefully observe and think, we will find that there is mathematics everywhere in our lives. In addition, students can learn mathematics from many aspects and channels. For example, learn mathematics from newspapers and magazines such as TV, Internet, Math Newspaper for Primary School Students, Math PHS, etc., and constantly expand their knowledge.
6. Have your own opinions. At present, most students encounter some difficult or unclear problems and give up easily without thinking, and some simply listen to the opinions of teachers, parents and books. Even teachers, elders, books and other authorities are not without some mistakes. We should attach importance to authoritative opinions, but it does not mean that we agree without thinking.
7. Learn to generalize and accumulate. Summarize the law of solving problems in time, especially accumulate some classic and special problems. In this way, we can study easily and improve the efficiency and quality of learning.
8. Pay attention to the study of other subjects. Because there is a close relationship between disciplines, it can promote the study of mathematics. For example, learning Chinese well is very helpful to understand the purpose of math problems, and so on.
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Eight ways of thinking in mathematics
1, algebraic thought This is one of the basic mathematical ideas. The unknown X in primary school and a series of letters representing numbers in junior high school are all algebraic ideas and the most basic roots of algebra!
2. The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. "Less is not intuitive, but more is difficult to be nuanced" is a famous saying of Professor Hua, a famous mathematician in China, which highly summarizes the role of the combination of numbers and shapes. There are many problems in junior high school and senior high school involving the combination of numbers and shapes. For example, using geometric figures to label data to solve problems and using functional images to solve problems are all expressions of numbers and shapes.
3. The idea of conversion Throughout junior high school mathematics, the idea of conversion has been running through it. Transforming thinking is to transform an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, changing the difficult to the easy, changing the unknown to the known, and changing the high order to the low order. It is one of the most basic problem-solving ideas and one of the basic thinking methods of mathematics.
4. Corresponding thinking method Correspondence is a way of thinking about the relationship between two set factors. Mathematics in primary schools is generally a one-to-one intuitive chart, which breeds the idea of function. For example, there is a one-to-one correspondence between points (number axes) on a straight line and specific numbers.
5. Hypothesis thinking method Hypothesis is a thinking method that first makes some assumptions about the known conditions or problems in the topic, then calculates according to the known conditions in the topic, makes appropriate adjustments according to the contradiction in quantity, and finally finds the correct answer. Hypothetical thinking is a meaningful imaginative thinking, which can make the problem to be solved more vivid and concrete after mastering it, thus enriching the thinking of solving problems.
6. Comparative thinking method Comparative thinking is one of the commonly used thinking methods in mathematics and a means to promote the development of students' thinking. In the application of teaching scores, teachers are good at guiding students to compare the situation before and after the change of known quantity and unknown quantity, which can help students find solutions quickly.
7. Symbolic thinking method Symbolic thinking is used to describe mathematical content (including letters, numbers, graphics and various specific symbols). For example, in mathematics, all kinds of quantitative relations, quantitative changes and deduction and calculation between quantities all use lowercase letters to represent numbers, and use condensed forms of symbols to express a large amount of information. Such as laws, formulas, etc.
8. Extreme thinking methods Things change from quantitative change to qualitative change. The essence of extreme methods is to achieve qualitative change through the infinite process of quantitative change. When talking about "the area and perimeter of a circle", the idea of limit division of "turning a circle into a square" and "turning a curve into a straight line" is to imagine their limit states on the basis of observing the limit division, which not only enables students to master the formula, but also germinates the limit idea of infinite approximation from the contradictory transformation of curves and straight lines.
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