1, circle angle

The angle whose vertex is on the circle and whose two sides intersect the circle is called the circumferential angle.

2. The theorem of circle angle

An arc subtends a circumferential angle equal to half the central angle it subtends.

Inference 1: the circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

Inference 2: the circumferential angle of a semicircle (or diameter) is a right angle; A chord with a circumferential angle of 90 is a diameter.

Inference 3: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

Second, some basic formulas

Sine, cosine and tangent formulas of triple angle

sin3α=3sinα-4sin^3(α)

cos3α=4cos^3(α)-3cosα

tan3α=[3tanα-tan^3(α)]/[ 1-3tan^2(α)]

Three, binary linear equations

1, binary linear equation

An integral equation with two unknowns and the highest degree of the unknowns is 1 is called a binary linear equation.

2, the solution of binary linear equation

The values of a pair of unknowns that make the values of the left and right sides of the binary linear equation equal are called the solutions of the binary linear equation.

3. Binary linear equation

Two (or more) binary linear equations are combined into one binary linear equation group. General form: (not all zeros)

4, the solution of binary linear equations

The values of two unknowns that make the left and right sides of two equations of binary linear equations equal are called the solutions of binary linear equations.

5, the solution of binary linear equations

Fourth, the basic idea: "elimination"

Solution: (1) substitution method (2) addition and subtraction method (3) binary linear equations.

6, ternary linear equation

Let the whole equation have three unknowns, and the terms of the unknowns are all 1.

Five, column equation (group) to solve application problems

Note: Never memorize the questions and their solutions, but analyze the specific problems. Generally speaking, the following steps should be followed:

1. Examination: Find out the meaning of the problem and the known and unknown quantities in the problem, and find out the equivalent relationship that can express all the meanings of the application problem.

2. Set the unknowns: select one or several suitable unknowns to express by letters, and express the related unknowns by algebraic expressions containing unknowns according to the quantitative relationship of the topics.

3. List equations (groups): List equations (groups) according to equivalence relation.

4. Solving equations (groups): The process can be omitted, but attention should be paid to skills and methods.

5. Test: First check whether the listed equations (groups) are correct, and then check whether the solutions of the obtained equations meet the meaning of the questions.

6. Answer: Don't forget the name of the company.

7. Solution of Fractional Equation

① General solution: denominator removal method, that is, both sides of the equation are multiplied by the simplest common denominator.

② Special solution: method of substitution.

(2) Root test: Because in the process of naming, when the range of unknown values is expanded, it is possible to increase roots. Therefore, the root test is an essential step to solve the fractional equation. Generally, the simplest common denominator is used to replace the value of the root of the whole equation to see if the result is zero, so the simplest root of the common denominator is the root increase of the original equation and must be discarded.

Note: Generally, method of substitution is considered before denominator method in solving fractional equations.

Sixth, the angle in the intersection line

When two straight lines intersect, four angles can be obtained. Among the four angles formed by the intersection of two straight lines, two angles with common vertices but no common edges are called antipodal angles. Among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are called temporary complementary angles.

Complementary angles and equal top angles.

Straight lines AB, CD and EF intersect (or two straight lines AB and CD are cut by the third straight line EF) to form eight angles. Among them, two angles ∠ 1 and ∠5 are above AB and CD respectively, and on the same side of EF, a diagonal line with the same position like this is called congruence angle; ∠3 and∠ 5 are both between AB and CD, opposite EF, and the two corners in this position are called inner corners; ∠3 and∠ 6 are between straight lines AB and CD, and the sides are on the same side of EF. The two angles in this position are called ipsilateral internal angles.

Seven, the nature of the line segment

1, axiom of line segment: Of all the straight lines connecting two points, the line segment is the shortest. It can also be simply said that the line segment between two points is the shortest.

2. The length of the line segment connecting two points is called the distance between these two points.

3. The distance between the midpoint and both ends of the line segment is equal.

4. The relationship between line segments and their lengths is consistent.

5. The property theorem and inverse theorem of the vertical line in the line segment.

A straight line perpendicular to a line segment and bisecting the line segment is the midline of the line segment. The property theorem of the median vertical line: the distance between the point on the median vertical line of the line segment and the two endpoints of the line segment is equal. Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.