Congruent triangle

Two triangles that can completely coincide are called congruent triangles. A triangle can be translated, folded and rotated to obtain its congruence.

2. What is the nature of congruent triangles?

(1): The sides and angles corresponding to congruent triangles are equal.

(2): The circumference and area of congruent triangles are equal.

(3): The corresponding median line, angular bisector and height line on the corresponding side of congruent triangles are equal respectively.

3. congruent triangles's judgment

Edge: Three edges correspond to the coincidence of two triangles (abbreviated as "SSS")

Angle: Two sides and their included angles are equal. Two triangles are congruent (abbreviated as "SAS").

Corner: Two triangles coincide with two corners 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")

Bevel. Right-angled side: the hypotenuse and a right-angled side correspond to the congruence of two right-angled triangles (abbreviated as "HL").

4. The basic idea of proving the coincidence of two triangles:

Second, the bisector of the angle:

1, the distance from the point on the bisector of the angle is equal to both sides of the angle.

2. (Judgment) 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.

Three, 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 angles have two sides and one of them is equal" are not necessarily the same;

(4): Always pay attention to the conditions implied in the diagram, such as "common angle", "common edge" and "diagonal".

Chapter 12 Axisymmetric

A, axisymmetric graphics

1. Fold the chart along a straight line. If the parts on both sides of a straight line can completely overlap, then this graph is called an axisymmetric graph. This straight line is its axis of symmetry. At this time, we also say that this figure is symmetrical about this straight line (axis).

2. Fold the chart along a straight line. If it can completely coincide with another figure, the two figures are said to be symmetrical about this line. This straight line is called the axis of symmetry. The point that overlaps after folding is the corresponding point, which is called the symmetrical point.

3. The difference and connection between axisymmetric figure and axisymmetric figure.

4. The essence of axial symmetry

① Two figures symmetrical about a straight line are conformal.

(2) If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.

③ The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.

(4) 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.

Second, the vertical line of the line segment

1. A straight line passing through the midpoint of a line segment and perpendicular to this line segment is called the median line of this line segment, also called the median line.

2. The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.

3. The point where the distance between the two ends of a line segment is equal is on the middle vertical line of the line segment.

Third, use coordinates to express the axisymmetric summary:

In the plane rectangular coordinate system, the abscissa of the points about the X axis symmetry is equal, and the ordinate is reciprocal. The abscissa of a point symmetrical about the Y axis is reciprocal, and the ordinate is equal.

The coordinates of the point (x, y) on the axis symmetry of X are __(x, -y) _.

The coordinate of the point where the point (x, y) is symmetrical about y is __(-x, y) _.

2. The perpendicular lines of the three sides of a triangle intersect at a point, and the distance from the point to the three vertices of the triangle is equal.

Fourth, (isosceles triangle) knowledge review

1. Properties of isosceles triangle

① The two base angles of an isosceles triangle are equal. (equilateral and angular)

② The bisector of the top angle of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide. (three in one)

2. Determination of isosceles triangle;

If the two angles of a triangle are equal, then the opposite sides of the two angles are equal. (Equiangular and Equilateral)

Five, (equilateral triangle) knowledge review

1. Properties of equilateral triangles:

Three angles of an equilateral triangle are equal, and each angle is equal to 600.

2. Determination of equilateral triangle;

A triangle with three equal angles is an equilateral triangle.

② An isosceles triangle with an angle of 600 is an equilateral triangle.

3. In a right triangle, if an acute angle is equal to 300, then the right side it faces is equal to half of the hypotenuse.

Chapter 13 Summary of Key Points of Real Number Knowledge

First, the classification of real numbers:

There is a one-to-one correspondence between real numbers and points on the number axis.

The number corresponding to any point on the number axis is always greater than the number corresponding to the point to the left of the point.

3. Countdown and countdown;

4. Absolute value

5. Factors and significant numbers;

6. Scientific symbols

7, square root and arithmetic square root, cube root;

8. The nature of non-negative numbers: If the sum of several non-negative numbers is zero, then all these numbers are equal to zero.

Second, review plan two.

1. irrational number: infinite acyclic decimal

Chapter 14 Linear Functions

I. Constants and variables:

In the process of a change, the amount of numerical change is called variable; A quantity with a constant value is called a constant;

Second, the concept of function:

Definition of function: Generally speaking, if there are two variables X and Y in a change process, and Y has a unique fixed value corresponding to each fixed value of X, then we say that X is an independent variable and Y is a function of X. 。

Third, the solution of the range of independent variables in the function:

(1). For functions expressed by algebraic expressions, the range of independent variables is all real numbers.

(2) For functions expressed by fractions, the range of independent variables is all real numbers with denominators other than 0.

(3) For functions expressed by radicals, the ranges of independent variables are all real numbers.

For functions with even roots, the range of independent variables is a real number that makes the square root non-negative.

(4) If the analytical formula is synthesized by the above forms, we must first find out the value range of each part, and then find out the common * * * range, which is the value range of the independent variable.

(5) For those related to practical problems, the range of independent variables should make practical problems meaningful.

Definition of function image: Generally speaking, for a function, if each pair of corresponding values of independent variables and functions are taken as the abscissa and ordinate of points respectively, then the graph formed by these points on the coordinate plane is the image of this function.

Fifth, the general steps of drawing function images by tracing point method.

1, list (The values of some independent variables and their corresponding function values are given in the table. )

Note: when listing independent variables, the difference from small to large is the same, and sometimes symmetry is needed.

2. Plot points: (In rectangular coordinate system, the points corresponding to the values in the table are plotted with the values of independent variables as abscissa and corresponding function values as ordinate.

3. Connecting line: (according to the order of abscissa from small to large, connect the traced points with smooth curves).

Six, the function has three forms:

(1) list method (2) image method (3) analytical formula method

Seven, the concept of proportional function and linear function:

Generally, a function in the form of y = kx (where k is a constant and k≠0) is called a proportional function, where k is called a proportional coefficient.

Generally, a function in the form of y = kx+b (where k and b are constants and k≠0) is called a linear function.

When b =0, y=kx+b is y=kx, so the proportional function is a special case of linear function.

Eight, the image and properties of the proportional function:

(1) image: the image of the proportional function y= kx (k is a constant, k≠0) is a straight line passing through the origin, and we call it straight line y= kx.

(2) Properties: when k >; 0, the straight line y= kx rises from left to right through the third quadrant, that is, y increases with the increase of x; When k < 0, the straight line y= kx passes through the second and fourth quadrants and decreases from left to right, that is, y decreases with the increase of x.

Nine, analytic function method:

Undetermined coefficient method: first set the resolution function, and then determine the unknown coefficient in the analytical formula according to the conditions, so as to write the method of this formula in detail.

1. Linear function and linear equation of one variable: When looking at the value of x from the perspective of number, the value of function y= ax+b is 0.

2. Find the solution of ax+b=0(a, B is constant, a≠0), and find the abscissa of the intersection of straight line y= ax+b and X axis from the angle of shape.

3. Linear function and unary linear inequality;

Solve the inequality ax+b > 0 (a, b are constants, a ≠ 0). From a numerical point of view, when x is a value, the value of the function y= ax+b is greater than 0.

4. solve the inequality ax+b > 0 (a, b are constants, a ≠ 0). From the angle of "shape", find out the value range of abscissa corresponding to the part (ray) of the straight line y= ax+b above the X axis.

X. Images and properties of linear and proportional functions

Linear function number

If y=kx+b(k and b are constants, k≠0), then y is called a linear function of x, and when b=0, the linear function y=kx(k≠0) is also called a proportional function.

This portrait is a straight line.

When k > 0, y increases (or decreases) with the increase (or decrease) of x;

When k < 0, y decreases (or increases) with the increase (or decrease) of x 。

The relationship between the position of straight line y=kx+b(k≠0) and the symbols of k and b (1) k > 0，b > 0； (2)k & gt； 0，b < 0；

(3)k & gt； 0，b=0 (4)k 0；

(5)k