First, the knowledge network structure
Second, the main points of knowledge
1. In the same plane, there are two kinds of positional relationships between two straight lines: intersecting and parallel, and verticality is a special case of intersection.
2. On the same plane, two disjoint straight lines are called parallel lines. If two straight lines have only one common point, they are said to intersect; If two straight lines have no common point, they are said to be parallel.
3. Among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are
The properties of adjacent complementary angles: complementary adjacent complementary angles. As shown in figure 1, they are complementary angles,
Fill the corner with neighbors. += 180 ; += 180 ; += 180 ;
+= 180 。
4. Among the four corners formed by the intersection of two straight lines, two sides of one corner are opposite extension lines of two sides of the other corner, so the two corners are opposite. The nature of antipodal angle: antipodal angle is equal. As shown in figure 1, and they are opposite to each other. =;
=。
5. If one of the angles formed by the intersection of two straight lines is a right angle or 90, the two straight lines are said to be perpendicular to each other.
One of them is called the perpendicular of the other. As shown in Figure 2, when = 90, ⊥.
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 the straight line and a point on the straight line, the vertical line segment is the shortest.
Property 3: As shown in Figure 2, when a ⊥ b = = = 90.
Distance from point to straight line: The length from a point outside a straight line to the vertical section of this straight line is called the distance from point to straight line.
6. The basic characteristics of congruent angle, internal dislocation angle and ipsilateral internal angle:
(1) is on the same side of two straight lines (cut lines) and the same side of the third straight line (cut lines), so that
These two angles are called isosceles angles. In Figure 3, * * * has a pair of isosceles angles; And is an isosceles angle;
And are at the same angle; And are at the same angle; And it's the same angle.
(2) Between two straight lines (secant) and on both sides of the third straight line (secant), such two angles are called inscribed angles. In figure 3; * * has a pair of inner corners; Is the inner corner; And is an inner corner.
(3) Between two straight lines (intersecting lines), both are on the same side of the third straight line (intersecting line), and such two angles are called ipsilateral inner angles. In figure 3; * * has a pair of inner corners on the same side; And are internal angles on the same side; And it is the same inner angle.
7. Parallelism axiom: At a point outside a straight line, only one straight line is parallel to the known straight line.
Inference of the axiom of parallelism: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
Properties of parallel lines:
Property 1: Two straight lines are parallel and equal to the complementary angle. As shown in fig. 4, if a∑b,
Then =; =; =; =。
Property 2: Two straight lines are parallel and the internal dislocation angles are equal. As shown in figure 4, if a∨b, then =; =。
Property 3: Two straight lines are parallel and complementary. As shown in figure 4, if a∨b,+=180;
+= 180 。
Property 4: Two lines parallel to the same line are parallel to each other. If a∨b, a∨c, then ∨.
8. Determination of parallel lines:
Judgment 1: congruent angles are equal and two straight lines are parallel. As shown in figure 5, if =
Or = or = or =, then a ∨ b.
Decision 2: The internal dislocation angles are equal and the two straight lines are parallel. As shown in figure 5, if = or =, then a ∨ b.
Judgment 3: The internal angles on the same side are complementary and the two straight lines are parallel. As shown in figure 5, if+=180;
+= 180, then a ∨ b.
Decision 4: Two straight lines parallel to the same straight line are parallel to each other. If a∨b, a∨c, then ∨.
9. A statement that judges a thing is called a proposition. A proposition consists of a topic and a conclusion, which can be divided into true proposition and false proposition. If the topic is established, then the conclusion must be established, and such a proposition is called a true proposition; If the topic holds, then the conclusion may not hold. Such a proposition is called a false proposition. Prove the correctness of the true proposition by reasoning. Such a true proposition is called a theorem, which can be used as the basis for further reasoning.
10. Translation: A figure moves a certain distance in a certain direction in a plane. This movement of graphics is called translation transformation, or translation for short.
After translation, the shape and size of the new picture are exactly the same as the original picture. Every point in the new graphic after translation is obtained by moving a point in the original graphic. Such two points are called corresponding points.
Translation properties: ① The connecting lines of corresponding points in the two images before and after translation are parallel and equal; ② The corresponding line segments are equal; ③ The corresponding angles are equal.
Chapter VI Real Numbers
Knowledge point classification-real number
1. Classification by definition: 2. Classification by natural symbols:
Note: 0 is neither positive nor negative.
Knowledge point 2 Related concepts of real numbers
1. Inverse
Algebraic meaning of (1): There are only two numbers with different signs, and we say that one of them is opposite to the other. The antonym of 0 is 0.
(2) Geometric meaning: On both sides of the origin on the number axis, two points with the same distance from the origin represent two opposite numbers, or on the number axis, the points corresponding to two opposite numbers are symmetrical about the origin.
(3) The sum of two opposites is equal to 0.a and B are opposites a+b=0.
2. Absolute value | a | ≥ 0.
3. The reciprocal (1)0 has no reciprocal. (2) Two numbers whose product is 1 are reciprocal. A and b are reciprocal.
4. Square root
(1) If the square of a number is equal to a, it is called the square root of a, a positive number has two square roots, and the two square roots are in opposite directions. 0 has a square root, and the square root itself is 0; Negative numbers have no square root. The square root of a (a ≥ 0) is written as.
(2) The positive square root of a positive number is called the arithmetic square root of a, and the arithmetic square root of A (A ≥ 0) is recorded as.
5. Cubic root
If x3=a, then x is called the cube root of a, and positive numbers have positive cube roots; Negative numbers have negative cubic roots; The cube root of zero is zero.
Knowledge point 3 Real number and axis
Definition of number axis: the straight line defining the origin, positive direction and unit length is called number axis, and the three elements of number axis are indispensable.
Comparison of real numbers of knowledge point four
1. For any two points on the number axis, the point on the right represents a larger number.
2. Positive numbers are all greater than 0, negative numbers are all less than 0, and two positive numbers, the greater the absolute value, the greater the positive number; Two negative numbers; The absolute value is large but small.
3. The relative size of irrational numbers:
The operation of knowledge point five real numbers
1. Add
Add two numbers with the same sign, take the same sign, and add the absolute values; Add different symbols with different absolute values of two numbers, take the symbol with the larger absolute value, and subtract the symbol with the smaller absolute value from the larger absolute value; Two opposite numbers add up to 0; When a number is added to 0, it still gets the number.
2. subtraction: subtracting a number is equal to adding the reciprocal of this number.
multiply
Multiply several non-zero real numbers, and the sign of the product is determined by the number of negative factors. When there are even negative factors, the product is positive. When there are odd negative factors, the product is negative. Multiply several numbers, one factor is 0 and the product is 0.
break up
Dividing by a number is equal to multiplying the reciprocal of this number. Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.
5. Multipliers and prescriptions
The meaning of (1)an is the product a of n, any power of positive number is positive, even power of negative number is positive, and odd power of negative number is negative.
(2) Positive numbers and 0 can be squared, but negative numbers cannot be squared; Positive numbers, negative numbers and 0 can all be turned on.
(3) Zero exponent and negative exponent
Six significant figures of knowledge points and scientific notation
1. Valid numbers:
A divisor, from the first non-zero number on the left to the exact number, is called the significant digits of this divisor.
2. Scientific symbols:
The method of counting a number in the form of (1 ≤ < 10, where n is an integer) is called scientific notation.