Mathematics knowledge arrangement of seventh grade in next semester
Chapter V Intersecting Lines and Parallel Lines
I. Knowledge structure diagram
intersection line
Vertical line of intersection line
Isomorphic angle, internal dislocation angle, ipsilateral internal angle
Parallel lines
Parallel lines and their determination
Determination of parallel lines
Properties of parallel lines
Properties of parallel lines
Propositions and theorems
translate
Two. Definition of knowledge
Adjacent complementary angles: among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are adjacent complementary angles.
Diagonal: Two sides of one angle are opposite extension lines of another angle, and two angles like this are diagonal to each other.
Perpendicular: When two straight lines intersect at right angles, they are said to be perpendicular to each other, and one of them is said to be perpendicular to the other.
Parallel lines: In the same plane, two disjoint lines are called parallel lines.
Conformal angle, internal dislocation angle and ipsilateral internal angle:
Isomorphism angle: ∠10 and ∠ 5. Diagonal lines with the same positional relationship like this are called isomorphism angles.
Internal angles: ∠2 and ∠6 A pair of angles like this is called an internal angle.
The diagonal lines such as ∠ 2 and ∠ 5 are called ipsilateral internal angles.
Proposition: A sentence that judges a thing is called a proposition.
Translation: moving a figure in a certain direction in a plane is called translation translation transformation.
Corresponding point: every point in the new graphic after translation is obtained by moving a point in the original graphic. These two points are called corresponding points.
Three. Theorems and properties
The nature of antipodal angle: antipodal angle is equal.
Nature of vertical line:
Property 1: There is one and only one straight line perpendicular to the known straight line.
Property 2: Of all the line segments connecting a point outside and a point on the line, the vertical line segment is the shortest.
Parallelism axiom: One and only one straight line is parallel to the known straight line through a point outside the straight line.
Parallel axiom inference: If two straight lines are parallel to the third straight line, then the two straight lines are parallel to each other.
Properties of parallel lines:
Property 1: Two straight lines are parallel and equal to the complementary angle.
Property 2: Two straight lines are parallel and the internal angles are equal.
Property 3: Two straight lines are parallel and complementary.
Determination of parallel lines:
Decision 1: Equal angles are equal and two straight lines are parallel.
Decision 2: The internal dislocation angles are equal and the two straight lines are parallel.
Decision 3: The internal angles on the same side are equal and the two straight lines are parallel.
Fourth, the classic example.
Example 1 As shown in the figure, straight lines AB, CD and EF intersect at point O, ∠ AOE = 54, ∠ EOD = 90, and find the degrees of ∠ EOB and ∠ COB.
As shown in Figure 2, AD divides equally ∠ CAE, ∠ B = 350, ∠ DAE = 600, so what is ∠ACB?
An outer angle of a triangle is equal to four times the inner angle adjacent to it, which is equal to its difference.
2 times of an adjacent inner angle, then the degree of each angle of this triangle is ().
450,450,900 pounds
C.250、250、 1300 D.360、720、720
Example 4 is a well-known diagram. The number of times to find ∠ A+∠ B+∠ C+∠ D+∠ E+∠ F.
Example 5 as shown in the figure, AB∥CD, EF intersect AB and CD at G, H,MN⊥AB at G, ∠CHG= 1240, ∠EGM is equal to how many degrees?
Chapter VI Plane Cartesian Coordinate System
I. Knowledge structure diagram
Ordered couple
Cartesian coordinates/Cartesian coordinates
Cartesian coordinates/Cartesian coordinates
Coordinate geographical location.
Simple application of coordinate method
Coordinate translation
Two. Definition of knowledge
Ordered number pair: A number pair consisting of two numbers A and B in sequence is called an ordered number pair, and it is recorded as (a, b).
Plane rectangular coordinate system: On a plane, two mutually perpendicular axes with a common origin form a plane rectangular coordinate system.
Horizontal axis, vertical axis and origin: the horizontal axis is called X axis or horizontal axis; The vertical axis is called Y axis or vertical axis; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
Coordinates: For any point P on the plane, the passing P is perpendicular to the X axis and Y axis respectively, and the vertical foot is on the X axis and Y axis respectively. The corresponding numbers a and b are called the abscissa and ordinate of the point p, respectively.
Quadrant: two coordinate axes divide the plane into four parts, the upper right part is called the first quadrant, and the counterclockwise part is called the second quadrant, the third quadrant and the fourth quadrant. The point on the coordinate axis is not in any quadrant.
Third, the classic example.
Example 1 A robot starts from point O, walks 3 meters due east to reach point A 1, walks 6 meters due north to reach point A2, walks 9 meters due west to reach point A3, walks 12 meters due south to reach point A4, and walks 15 meters due east to reach point A5. If A65438+,
Example 2 is a small flag pattern painted on a square paper. If point A is represented by (0,0) and point B by (0,4), the position of point C can be represented by ().
a 、( 0，3) B 、( 2，3) C 、( 3，2) D 、( 3，0)
Example 3 As shown in Figure 2, according to the position of each point on the coordinate plane, write the coordinates of the following points:
a()，B()，C()。
Example 4: As shown in the figure, translate △ABC with an area of 300px2 to the position of △DEF in the positive direction of the X axis, and the corresponding coordinates are shown in the figure (A and B are constants).
(1), find the coordinates of point d and point e.
(2) Find the area of the quadrilateral.
Example 5 If two points A (3 3,4) and B (-2,4) are straight lines AB, then straight line AB ()
A, through the origin b, parallel to the y axis.
C, parallel to the X axis D, none of the above statements are correct.
Chapter VII Triangle
I. Knowledge structure diagram
edge
The height of the line segment associated with the triangle.
median
internal bisector
The sum of internal angles of triangles and polygons.
The sum of the external angles of triangles and polygons.
Two. Definition of knowledge
Triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.
Trilateral relationship: the sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
Height: Draw a vertical line from the vertex of the triangle to the opposite side. The line segment between the vertex and the vertical foot is called the height of the triangle.
Median line: In a triangle, the line segment connecting the vertex and its relative midpoint is called the median line of the triangle.
Angle bisector: the bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angle bisector of the triangle.
Stability of triangle: the shape of triangle is fixed, and this property of triangle is called stability of triangle.
Polygon: On a plane, a figure composed of end-to-end line segments is called a polygon.
Interior Angle of Polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
Exterior angle of polygon: the angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
Diagonal polygon: The line segment connecting two nonadjacent vertices of a polygon is called diagonal polygon.
Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
Plane mosaic: covering a part of a plane with some non-overlapping polygons is called covering the plane with polygons.
Three. Formulas and attributes
Sum of triangle internal angles: The sum of triangle internal angles is 180.
Properties of the external angle of a triangle:
Attribute 1: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
Property 2: The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
The sum formula of polygon internal angles: the sum of n polygon internal angles is equal to (n-2) 180.
Sum of polygon outer angles: the sum of polygon inner angles is 360.
The number of diagonals of a polygon: (1) Starting from a vertex of an n polygon, you can draw (n-3) diagonals and divide the polygon into (n-2) triangles.
(2)n sides * * * have diagonal lines.
Fourth, the classic example.
Example 1 as shown. It is known that in △ABC, AQ=PQ, PR=PS, PR⊥AB in R,PS⊥AC in S, there are three conclusions: ① As = AR; ②QP∑AR； ③△BRP?△CSP, where ().
(a) All correct (b) Only ① correct (c) Only ① and ② correct (d) Only ① and ③ correct.
Example 2 as shown in the figure, combined with graphics to make the following judgment or reasoning:
① As shown in Figure A, CD ⊥ AB and D are vertical feet, so the distance from point C to AB is equal to the distance between points C and D;
② As shown in Figure B, if AB∨CD, then ∠ B = ∠ D;
③ As shown in Figure C, if ∠ACD=∠CAB, then AD ∨ BC;
④ As shown in Figure D, if ∠ 1 = ∠ 2, ∠ D = 120, then ∠ BCD = 60. The correct number is ().
1 (B)2 (C)3 (D)4
Example 3 Draw △DEF and △DEG(F and g can't coincide) on the square paper as shown in the figure, so △ ABC △ def ≌ deg. Can you explain why they are all equal?
Example 4 Measure the diameter of the small glass tube on the measuring tool CDE, CD = l0mm, DE = 80mm. What is the length of the caliber AB of the small tube if it points to the 50mm scale on the measuring tool?
In the rectangular coordinate system of Example 5, three points, A (-4,0), B( 1 0) and C (0 0,2), are known. Please design two schemes according to the following requirements: make a straight line that does not coincide with the axis and intersects with both sides of △ABC, so that the cut triangle is similar to △ABC and has an area of △AOC.
Chapter VIII Binary Linear Equations
I. Knowledge structure diagram
Set an unknown number and an equation.
Solution substitution method
Square addition and subtraction
Process (elimination)
group
test
Two. Definition of knowledge
Binary linear equation: There are two unknowns whose exponents are 1. Equations like this are called binary linear equations, and the general form is ax+by=c(a≠0, b≠0).
Binary linear equations: two binary linear equations are combined into one binary linear equation.
Generally speaking, the unknown value that makes the values on both sides of the binary linear equations equal is called the solution of the binary linear equations.
The common solution of two equations of general binary linear equations is called binary linear equations.
Elimination method: the idea of reducing the number of unknowns one by one is called elimination thought.
Substitution elimination method: an unknown number is represented by a formula containing another unknown number, and then it is substituted into another equation to realize elimination, and then the solution of this binary linear equation group is obtained. This method is called substitution elimination method, or substitution method for short.
Addition and subtraction elimination method: when the coefficients of the same unknown in two equations are opposite or equal, the two sides of the two equations can be added or subtracted to eliminate them respectively. This method is called addition, subtraction and elimination, or addition and subtraction for short.
Third, the classic example.
Example 1 Solve the equation by addition, subtraction and elimination, which is obtained from ①× 2-②.
Example 2 If it is a similar item, the value of is ().
a 、=-3、=2 B 、=2、=-3
c 、=-2、=3 D 、=3、=-2
Example 3 Calculation:
Example 4 Wang contracted 25 mu of land and planted eggplant and tomato in greenhouse this spring, which cost 44,000 yuan, including planting eggplant per mu 1.7 million yuan and net profit of 2,400 yuan. Planting tomatoes cost 1800 yuan per mu and earned a net profit of 2600 yuan. How much did Uncle Wang make?
Example 5 It is known that the solutions of binary linear equations about X and Y satisfy the binary linear equations.
Chapter 9 Inequality and Unequal Groups
I. Knowledge structure diagram
practical problem
(including inequality)
mathematical problem
(One-dimensional linear inequality (group))
Set unknowns and column inequalities (group)
solve
no
wait for
style
group
Solutions to mathematical problems
(Solution of Inequality (Group))
The answer to the practical question
test
Two. Definition of knowledge
Inequality: Generally speaking, formulas that express the relationship between size with symbols "< >" ≤ "and ≥" are called inequalities.
The value of the unknown quantity that makes the inequality valid is called the solution of the inequality.
Solution set of inequality: All solutions of an unknown inequality constitute the solution set of this inequality.
One-dimensional linear inequality: the left and right sides of the inequality are algebraic expressions, and there is only one unknown, and the highest order of the unknown is 1. Inequalities like this are called one-dimensional linear inequalities.
One-dimensional linear inequality group: generally, several one-dimensional linear inequalities about the same unknown quantity are combined to form a one-dimensional linear inequality group.
Solution set of linear inequality group: The common part of the solution set of each inequality in linear inequality group is called the solution set of this linear inequality group.
Three. Theorems and properties
The essence of inequality:
The basic property of inequality is 1: add (or subtract) the same number (or formula) on both sides of inequality, and the direction of inequality remains unchanged.
The basic property of inequality 2: both sides of inequality are multiplied (or divided) by the same positive number, and the direction of inequality remains unchanged.
The basic property of inequality 3: when both sides of inequality are multiplied (or divided) by the same negative number, the direction of inequality changes.
Fourth, the classic example.
Example 1 When x, the value of algebra 2-3x is positive.
Example 2 The solution set of the unary linear inequality group is ()
A.-2