Detailed explanation of basic knowledge and common conclusions of mathematics college entrance examination

A, set and simple logic:

First, understand the related concepts in the set.

Characteristics of elements in (1) set: certainty, mutual difference and disorder.

The mutual differences of set elements: such as:,, seeking;

(2) The relationship between sets and elements is represented by symbols.

(3) Symbolic representation of common number sets: natural number set; Positive integer set,; Integer set; Rational number set, real number set.

(4) Representation of sets: enumeration method, description method and Wayne diagram.

Note: distinguish the forms of elements in the set; For example:; ; ; ; ;

;

(5) An empty set refers to a set without any elements. (the difference between and; 0 and the relationship between the three)

An empty set is a subset of any set and a proper subset of any non-empty set.

Note: the condition is that you don't forget the situation during the discussion.

For example, if, evaluate.

Second, the relationship between sets and their operations

(1) symbol ""indicates the relationship between elements and sets, and the relationship between points and lines (surfaces) in solid geometry;

The symbol ""indicates the relationship between sets and the relationship between plane and straight line (plane) in solid geometry.

(2) ; ;

(3) For any set, then:

① ; ; ;

② ; ;

; ;

③ ; ;

(4)① If it is even, then; If it is odd, then;

(2) if divided by 3, then; If 1 is divided by 3, then; If 2 is divided by 3, then;

Third, the calculation of the number of elements in the set:

(1) If there is an element in the set, the number of all different subsets of the set is _ _ _ _ _ _ _ _ _, the number of all proper subset is _ _ _ _ _ _ _, and the number of all non-empty proper subset is.

The formula for calculating the number of elements in (2) is:

(3) The application of Wayne diagram:

Fourth, meet the conditions, meet the conditions,

If; This is a sufficient and unnecessary condition;

If; This is a necessary condition, but not a sufficient condition;

If; Yes, a necessary and sufficient condition;

If; This is neither sufficient nor necessary;

5. Original proposition and negative proposition, negative proposition and negative proposition are the same;

Note: the application of "if, then" in solving problems,

For example, yes conditions.

6. Reduction to absurdity: When "if, then" is difficult to prove, the equivalent proposition "if, then" is proved to be true.

Step: 1, assuming the opposite conclusion holds; 2. Proceed from this assumption, reason and demonstrate, and get the contradiction; 3. Judging from the contradiction that the hypothesis is not established, so as to affirm that the conclusion is correct.

The source of contradiction: 1, which contradicts the conditions of the original proposition; 2. Deduce the proposition that contradicts the hypothesis; 3. Derive an invariant pseudo-proposition.

When the conclusion of the application and the proposition to be proved involves such words as "impossible", "no", "at least", "at most" and "only".

At most one positive number is equal to greater than or less than.

negate

Positive words have at least one arbitrary all and at most n arbitrary two.

negate

Second, function

I. Mapping and function:

The concept of (1) mapping: (2) one-to-one mapping: (3) the concept of function:

Such as: if,; Q: There are two mappings to and two mappings to; There is a function to, and if there is, there is a one-to-one mapping to.

The number of intersection points between the function image and the straight line is.

Second, the three elements of the function:,,.

The judgment method of the same function: ①; (2) (these two points must be met at the same time)

Solution of resolution function (1):

① definition method (patchwork method): ② substitution method: ③ undetermined coefficient method: ④ assignment method:

(2) The solution of functional domain:

1, then; 2 then;

3. Then; 4 If:, then;

⑤ Classify and discuss the universe with parameters;

For example, the domain of a known function is the domain of a solution.

⑥ For practical problems, after finding the resolution function; We must find its domain, and the domain at this time should be determined according to the actual meaning. For example, it is known that the circumference of a sector is 20, the radius is 0, and the sector area is 0, then; The domain is.

(3) The solution of function value domain:

① Matching method: transform it into a quadratic function and evaluate it by using the characteristics of the quadratic function; Often converted into:;

(2) Inverse solution: the value range obtained by inverse solution is expressed by and then obtained by solving inequality; Commonly used to solve, such as:

(4) Substitution method: transforming variables into functions of assignable fields and returning to ideas;

⑤ Triangular Bounded Method: Transform it into a function containing only sine and cosine, and use the boundedness of trigonometric function to find the domain;

⑥ Basic inequality methods: transformation and modeling, such as: using the average inequality formula to find the domain;

⑦ Monotonicity method: The function is monotonous, and the domain can be evaluated according to the monotonicity of the function.

⑧ Number-shape combination: According to the geometric figure of the function, the domain is found by the method of number-shape combination.

Find the range of the following functions: ① (2 methods);

② (2 methods); ③ (2 methods);

Third, the nature of the function:

Monotonicity, parity and periodicity of functions

Monotonicity: Definition: Note that the definition is relative to a specific interval.

The judgment methods are: definition method (difference comparison method and quotient comparison method)

Derivative method (for polynomial function)

Composite function method and mirror image method.

Application: compare sizes, prove inequalities and solve inequalities.

Parity: Definition: Pay attention to whether the interval is symmetrical about the origin, and compare the relationship between f(x) and f(-x). F (x)-f (-x) = 0f (x) = f (-x) f (x) is an even function;

F (x)+f (-x) = 0f (x) =-f (-x) f (x) is odd function.

Discrimination methods: definition method, image method and compound function method.

Application: function value transformation solution.

Periodicity: Definition: If the function f(x) satisfies: f(x+T)=f(x) for any x in the definition domain, then t is the period of the function f(x).

Others: If the function f(x) satisfies any x in the domain: f(x+a) = f (x-a), then 2a is the period of the function f (x).

Application: Find the function value and resolution function in a certain interval.

Fourth, graphic transformation: function image transformation: (key) It is required to master the commonly used basic functions.

Five, the inverse function:

(1) Definition:

(2) Conditions for the existence of the inverse function:

(3) The relationship between the domain and the value domain of reciprocal function:

(4) Steps to find the inverse function: ① Take it as an equation about and solve it. If there are two schemes, pay attention to the choice of the scheme; (2) will also be exchanged; ③ Write the domain of the inverse function (that is, the value domain of).

(5) The relationship between reciprocal images:

(6) The original function and the inverse function have the same monotonicity;

(7) If the original function is odd function, its inverse function is still odd function; The original function is even, so there must be no inverse function.

For example, find the inverse of the following function:

Seven, commonly used elementary functions:

(1) One-variable linear function: if, is increasing function; When, is a decreasing function;

(2) One-variable quadratic function:

General formula: the equation of symmetry axis is; The vertex is;

Two-point type:; The equation of symmetry axis is: the intersection point with the axis is;

Vertex:; The equation of symmetry axis is: the vertex is;

① monotonicity of univariate quadratic function:

When: add functions; Is a decreasing function; When: add functions; Is a decreasing function;

② the problem of finding the maximum value of quadratic function: firstly, the collocation method should be adopted,

I. If the abscissa of the vertex is within a given interval, then

Time: the minimum value is obtained at the vertex and the maximum value is obtained at the endpoint far away from the symmetry axis;

Time: the maximum value is obtained at the vertex, and the minimum value is obtained at the endpoint far from the symmetry axis;

Ii. If the abscissa of the vertex is not within the given interval, then

Time: take the minimum value at the endpoint near the symmetry axis and the maximum value at the endpoint far from the symmetry axis;

Time: the maximum value is obtained at the end point near the symmetry axis and the minimum value is obtained at the end point far from the symmetry axis;

There are three types of questions:

(1) Vertex is fixed and interval is fixed. For example:

(2) Vertex contains parameters (i.e. vertex changes) and the interval is fixed. At this time, it is necessary to discuss when the abscissa of the vertex is in the interval and when it is outside the interval.

(3) The vertex is fixed and the interval is variable, so we should discuss the parameters in the interval.

③ Distribution of real roots of quadratic equation: Let two roots of quadratic equation with real coefficients be: Then:

Fundamental situation

There are two equivalent propositions in the interval, two in the interval and one in or above the interval.

necessary and sufficient condition

Note: If the equation has a real number solution in the closed interval, we can first use the real root distribution in the open interval to get the result and check the endpoints.

(3) Inverse proportional function:

(4) Exponential function:

Exponential algorithm:; ; .

Exponential function: y =(a>；; O, a≠ 1), the constant intersection point of the image (0, 1), and the monotonicity is related to the value of A. In solving problems, A is often graded as A >；; 1 and 0

(5) Logarithmic function:

Exponential algorithm:; ; ;

Logarithmic function: y =(a>；; O, a≠ 1) images have a constant intersection (1, 0), and the monotonicity is related to the value of A. In solving problems, A is often graded as A >；; 1 and 0

Note: the mirror image of (1) and is;

(2) The basic method to compare two exponents or logarithms is to construct corresponding exponents or logarithms. If the base is different, it will be converted into an exponent or logarithm with the same base, and it should also be compared with 1 or 0.

(3) The domain of the known function is and the range of the solution is.

The range of the known function is, and the range of the solution is.

Sixth, the image:

Domain:; Scope:; Parity: monotonicity: increasing function; Is a subtraction function.

Seven. Supplementary content:

Some specific special function models corresponding to the properties of abstract functions;

① proportional function

② ; ;

③ ; ;

④ ;

Third, derivative products

1. Derivation rule:

(c)/=0, where c is a constant. That is, the derivative value of the constant is 0.

(xn)/= nxn- 1 Especially: (x)/=1(x-1)/= ()/=-x-2 (f (x) g (x))/= f/(x) g. 1f(x))/= k？ 6? 1f/(x)

2. Geometric and physical meanings of derivatives:

K = f/(x0) represents the slope of the tangent of point P(x0, f(x0)) on curve y=f(x).

V = s/(t) represents the instantaneous speed. A=v/(t) stands for acceleration.

3. The application of derivative:

① Find the slope of the tangent.

② The relationship between the derivative of function and monotonicity.

One is the relationship with increasing function.

Can be introduced as an increasing function, but not necessarily vice versa. For example, the function is monotonically increasing, but ∴ is a necessary and sufficient condition for increasing function.

Second, the relationship with increasing functions.

If you take the root as the dividing point, because the regulation, that is, the dividing point, is cut off, at this time, it is necessary to add functions. Appropriate time is a necessary and sufficient condition for adding functions.

Third, the relationship with increasing functions.

For the incremental function, it can definitely be introduced, but the reverse is not necessarily true, because it is or. When a function is constant in a certain interval, it is constant and the function is not monotonous. ∴ is a necessary and sufficient condition for increasing function.

Monotonicity of function is an important property of function, and it is also the focus of high school research. We must grasp the above three relations and judge the monotonicity of the function with derivatives. Therefore, in order to solve the endpoint problem of monotonous interval, all new textbooks use open interval as monotonous interval, which avoids discussing the above problems and simplifies the problems. However, in practical application, we will also encounter endpoint discussion problems, so we should handle them carefully.

(1) The process of solving the four monotone intervals of the domain is known. (2) finding the derivative; (3) solving inequalities; The part of the solution set in the definition domain is an increasing interval; (4) solving inequalities; The solution set in the definition domain is a decreasing interval.

When we use derivatives to judge the monotonicity of a function, we must make clear the following three relationships in order to accurately judge the monotonicity of the function. Let's take increasing function as an example to make a simple analysis. The prerequisite is that the function is differentiable in a certain interval.

③ Find the extreme value and the maximum value.

Note: Extreme value ≠ maximum value. The maximum value of the function f(x) in the interval [a, b] is the maximum value and the maximum value in f(a) and f(b). The minimum value is the smallest sum of f(a) and f(b).

You can't get F/(x0) = 0. When x=x0, the function has an extreme value.

However, when x=x0, the function has an extreme value f/(x0) = 0.

Judging extreme value needs to explain the monotonicity of function.

4. The standard of derivative products:

(1) representation function (more accurate and subtle than the elementary method);

(2) the connection with tangent in geometry (the tangent of plane curve can be studied by derivative method);

(3) Application problems (elementary methods often require high skills, while derivative methods are relatively simple) and other derivative problems about sub-polynomials are difficult types.

2. There are many problems about the maximum value of function characteristics, so it is necessary to discuss them specially. The derivative method is faster and simpler than the elementary method.

3. The mixed problem of derivative and analytic geometry or function image is an important type, and it is also a direction of comprehensive ability in college entrance examination, which should be paid attention to.

Fourth, inequality.

First, the basic properties of inequality:

Note: (1) special value method is a method to judge whether inequality propositions are true, especially for those that are not true.

(2) Pay attention to several attributes of textbooks, and pay special attention to:

(1) If ab>, then 0. That is, when the signs on both sides of the inequality are the same, the two sides of the inequality take the reciprocal, and the direction of the inequality will change.

② If both sides of the inequality are multiplied by an algebraic expression at the same time, pay attention to its sign. If the constellation is uncertain, pay attention to the classification discussion.

③ Image method: directly compare the sizes by using the images of correlation functions (exponential function, logarithmic function, quadratic function and trigonometric function).

④ Median method: first compare the algebraic expressions to be compared with "0" and "1", and then compare their sizes.

Second, mean inequality: the arithmetic mean of two numbers is not less than their geometric mean.

If, then (if and only if the equal sign holds)

Basic deformation: ①; ;

② If, then,

Basic applications: ① Scaling and deformation;

② Find the maximum value of the function: Note: ① One is positive, two are definite and three are equal; ② The sum of definite products is small, and the sum of definite products is large.

If (constant), if and only if,;

If (constant), if and only if,;

The common methods are: splitting, gathering and square;

Such as: ① the minimum value of the function.

② If positive numbers are satisfied, then.

Third, absolute inequality:

Note: the conditions for the above equal sign "=" to be established;

Four, commonly used basic inequalities:

(1), and then (if and only if the equal sign holds)

(2) (take the equal sign if and only if); (Equal sign if and only if)

(3) ; ;

Five, the common methods to prove inequality:

(1) comparison method: compare the differences:

Steps of difference comparison:

⑴ Difference: distinguish two numbers (or formulas) with different sizes.

⑵ Deformation: decompose or formulate the difference factor into the complete sum of squares of several numbers (or formulas).

⑶ Symbol with poor judgment: Symbol with poor judgment combined with deformation results and problems.

Note: If it is difficult to distinguish two positive numbers, you can use their square difference to compare the sizes.

(2) comprehensive method: cause and effect.

(3) Analysis method: the reason of fruit holding. Basic steps: get a certificate ... a certificate of fairness ... a certificate of fairness. ...

(4) reduction to absurdity: if it is difficult, it will be reversed.

(5) Scaling method: the inequality side is appropriately enlarged or reduced to prove the problem.

Scaling methods include:

(1) Add or omit some items, such as:

(2) Enlarge (or shrink) the numerator or denominator.

(3) using basic inequalities, such as:

(4) Use common conclusions:

Ⅰ、 ;

Ⅱ、 ; (to a great extent)

Ⅲ、 ; (To a lesser extent)

(6) method of substitution: method of substitution's aim is to reduce the variables in inequality, thus making the problem difficult and simplifying the complex. The commonly used substitutions are trigonometric substitution and algebraic substitution. For example:

Known, can be set;

Known, you can set ();

Known, can be set;

Known, can be set;

(7) Construction method: prove inequality by constructing function, equation, sequence, vector or inequality;

Six, the solution of inequality:

(1) unary linear inequality:

Me: (1) If, then; (2) If, then;

Two. : (1) If, then; (2) If, then;

(2) One-dimensional quadratic inequality: if the quadratic coefficient of one-dimensional quadratic inequality is less than zero, the same solution is transformed into a quadratic coefficient greater than zero; Note: To discuss:

(5) Absolute inequality: if, then; ;

Note: (1). Geometric meaning::; : ;

(2) To solve the absolute value problem, consider removing the absolute value. The method of removing the absolute value is as follows:

(1) Discuss that the absolute value is greater than, equal to and less than zero, and then remove the absolute value; (1) If; 2 if it is; 3 if yes;

(3). Square the two sides to remove the absolute value; It should be noted that both sides of the inequality sign are non-negative.

(4) Inequalities with multiple absolute value symbols can be solved by the method of "discussing by zero points".

(6) Solving the fractional inequality: transforming the general solution into algebraic expression inequality;

⑴ ; ⑵ ;

⑶ ; ⑷ ;

(7) Solution of inequality group: Find the solution set of each inequality in the inequality group, and then find its intersection, which is the solution set of this inequality group. In the intersection, the solution set of each inequality is usually drawn on the same number axis, and their common parts are taken.

(8) Solving inequalities with parameters:

When solving inequalities with parameters, we should first pay attention to whether it is necessary to discuss them in categories. If you encounter the following situations, you generally need to discuss them:

① When two ends of inequality multiply and divide a formula with parameters, we need to discuss the positive, negative and zero properties of this formula.

② When monotonicity of exponential function and logarithmic function is needed in solving, their bases need to be discussed.

(3) When solving a quadratic inequality with letters, we need to consider the opening direction of the corresponding quadratic function, the conditions of the roots of the corresponding quadratic equation (sometimes we need to analyze △), compare the sizes of the two roots, let the roots be (or more) but contain parameters, and discuss them separately.

Verb (abbreviation for verb) order

This chapter is one of the main contents of the proposition of college entrance examination. We should review it comprehensively and deeply, and focus on solving the following problems on this basis: (1) The proof of arithmetic and geometric series must be proved by definition. It is worth noting that if the sum of the first few items of a series is given, the general items can be written if it is satisfied. (2) The calculation of series is the central content of this chapter. Using the general formula, antecedents, formulas and their properties of arithmetic progression and geometric progression to make clever calculations is the key content of the college entrance examination proposition. (3) When solving the problem of sequence, we often use various mathematical ideas. It is our goal to be good at using various mathematical ideas to solve the problem of sequence. (1) Function Thought: The summation formula of the general term formula of arithmetic geometric progression can be regarded as a function, so some problems of arithmetic geometric progression can be solved as function problems.

(2) The idea of classified discussion: the summation formula of equal proportion series should be divided and summed; When the time is known, it should also be classified;

③ Holistic thinking: When solving the sequence problem, we should pay attention to getting rid of the rigid thinking mode solved by formulas and use integers.

Body and mind solutions.

(4) When solving the related application problems of series, we should carefully analyze and abstract the actual problems into mathematical problems, and then use the knowledge and methods of series to solve them. Solving this kind of application problem is a comprehensive application of mathematical ability, and it is by no means a simple imitation and application. Pay special attention to the items of geometric series related to years.

First, the basic concept:

1, definition and representation of sequence:

2. Items and number of items in the series:

3, finite sequence and infinite sequence:

4, increasing (decreasing), swing, cycle order:

5. The general formula of sequence {an} an:

6. The first n terms of the sequence and the formula Sn:

7. Structure of arithmetic progression, Tolerance D and arithmetic progression:

8. The structure of geometric series, Bi Gong Q and geometric series;

Second, the basic formula:

9. the relationship between the general term an and the first n terms and Sn of a general sequence: an=

10, the general formula of arithmetic progression: an = a 1+(n-1) Dan = AK+(n-k) d (where a1is the first term and AK is the known k term), when d≠0.

1 1, the first n terms of arithmetic progression and its formula: Sn= Sn= Sn=

When d≠0, Sn is a quadratic form about n, and the constant term is 0; When d=0 (a 1≠0), Sn=na 1 is a proportional formula about n.

12, the general formula of geometric series: an = a1qn-1an = akqn-k.

(where a 1 is the first term, ak is the known k term, and an≠0).

13, the first n terms of geometric series and their formulas: when q= 1, Sn=n a 1 (this is a direct ratio formula about n);

When q≠ 1, Sn= Sn=

Third, the conclusion about arithmetic and geometric series.

Is arithmetic progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 14 or arithmetic progression?

15, arithmetic progression {an}, if m+n=p+q, then

16, geometric series {an}, if m+n=p+q, then

Geometric progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 17 is still geometric progression.

18, the sum and difference of two arithmetic progression {an} and {bn} series {an+bn} is still arithmetic progression.

19, a sequence consisting of the product, quotient and reciprocal of two geometric series {an} and {bn}

{an bn},,, or geometric series.

20. arithmetic progression {an} Any equidistant series is still arithmetic progression.

2 1, the series of any equidistant term of geometric progression {an} is still geometric progression.

22. How to make three numbers equal: A-D, A, A+D; How to make four numbers equal: A-3D, A-D, A+D, A+3D?

23. How to make three numbers equal: A/Q, A, AQ;

Wrong method of four numbers being equal: a/q3, a/q, aq, aq3 (Why? )

24.{an} is arithmetic progression, then (c>0) is a geometric series.

25 、{ bn }(bn & gt； 0) is a geometric series, then {logcbn} (c >; 0 and c 1) are arithmetic progression.

26. In the arithmetic series:

(1) If the number of items is, then

(2) If the quantity is,

27. In geometric series:

(1) If the number of items is, then

(2) If the number is 0,

Four, the common methods of sequence summation: formula method, split item elimination method, dislocation subtraction, reverse addition, etc. The key is to find the general item structure of the sequence.

28. Find the sum of series by grouping method: for example, an=2n+3n.

29. Sum by dislocation subtraction: for example, an=(2n- 1)2n.

30. Sum by split term method: for example, an= 1/n(n+ 1).

3 1, sum by addition in reverse order: for example, an=

32. The method of finding the maximum and minimum term of series {an}:

① an+ 1-an = ... For example, an= -2n2+29n-3.

② (An>0) as a =

③ an=f(n) Study the increase and decrease of function f(n), such as an=

33. In arithmetic progression, the problem about the maximum value of Sn is often solved by the adjacent term sign change method:

(1) When >: 0, d < When 0, the number of items m meets the maximum value.

(2) When

We should pay attention to the application of the transformation idea when solving the maximum problem of the sequence with absolute value.

Six, the plane vector

1. Basic concepts:

Definition of vector, modulus of vector, zero vector, unit vector, opposite vector, * * * line vector, equal vector.

2. Algebraic operations of addition and subtraction:

( 1) .

(2) if a b= (). And B = (), AB = ().

Geometric representation of vector addition and subtraction: parallelogram rule and triangle rule.

Take the vector =, = as the adjacent side to make a parallelogram ABCD, then the vectors of the two diagonals are =+,=-and =-

And there are ||||-|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Vector addition has the following laws:+=+(commutative law); +( +c)=(+)+c (law of association);

+0= +(- )=0.

3. Product of real number and vector: The product of real number and vector is a vector.

( 1)| |=| | | |;

(2) When > 0, it is in the same direction; When < 0, the opposite; When =0, = 0.

(3) If = (), then = ().

Necessary and sufficient conditions for two vector lines;

The necessary and sufficient condition for the straight line between (1) vector b and non-zero vector * * * is that there is only one real number, so b =.

(2) If = () and b = (), then ‖ b 。

The basic theorem of plane vector;

If e 1 and e2 are two nonlinear vectors on the same plane, there is only one pair of real numbers for any vector on this plane, so = e 1+ e2. ..

4. The ratio of P-divided directed line segments:

Let P 1 and P2 be two points on a straight line, and point P is any point in the world different from P 1 and P2, then there is a real number that makes =, which is called the ratio of point P to directed line segment.

When point p is on the line segment, > 0; When point P is on the extension line of line segment or, < 0;

Formula of vernal equinox coordinates: if =；; The coordinates of are (), () and () respectively; Then (≦- 1), the midpoint coordinate formula:.

5. Quantity product of vectors:

(1). Vector angle:

Given that two nonzero vectors and b make =, =b, then ∠AOB= () is called the included angle between the vector and b.

(2). Quantity product of two vectors:

If two nonzero vectors and b are known and their included angle is, then b = |||| b | cos.

Where | b | cos is called the projection of vector b in the direction.

(3) Properties of the product of vector numbers:

If = () and b = (), then e = e = || cos (e is the unit vector);

⊥ b b = 0 (,b is a non-zero vector); | |= ;

cos = =。

(4) Vector product algorithm:

b = b()b =(b)=(b)； (+b) c= c+b c。

6. Main ideas and methods:

This chapter mainly sets up the viewpoint of number-shape transformation and combination, handles geometric problems with algebraic operation, especially the relative position relationship of vectors, correctly uses the basic theorems of * * * line vector and plane vector to calculate the modulus of vectors, the distance between two points and the included angle of vectors, and judges whether the two vectors are vertical or not. Because vectors are new tools, they are often combined with trigonometric functions, sequences, inequalities and solutions. And it is the intersection of knowledge.

Seven, solid geometry

The basic properties of 1. plane: If you master three axioms and inferences, you will explain the problems of * * * points, * * lines and * * * planes.

Able to draw by tilt measurement.

2. The positional relationship between two straight lines in space: the concepts of parallelism, intersection and nonplanarity;

Will find the angle formed by straight lines in different planes and the distance between straight lines in different planes; Generally, two straight lines are proved to be non-planar straight lines by reduction to absurdity.

3. Lines and planes

① positional relationship: parallel, straight line in the plane, and straight line intersects with the plane.