(1) Line segments related to triangles:

(1) A triangle consists of three line segments that are not on the same straight line, and these three line segments are connected end to end. These three line segments are sometimes represented by letters A, B and C respectively. The three points where three line segments intersect are called the vertices of a triangle. If the vertices are represented by a, b and c respectively, the triangle can be represented as △ABC, which is pronounced as "triangle ABC";

(2) Triangle is divided into isosceles triangle, equilateral triangle and equilateral triangle according to the length of three sides;

③ The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is smaller than the third side;

(4) Draw the vertical line of the straight line on the opposite side BC through a vertex A of the triangle, and the vertical foot is d, and the obtained line segment is called the height of the side BC of the triangle;

⑤ Connect the vertex A of the triangle △ABC with the midpoint of its opposite side BC, and the obtained line segment is called the midline on the side BC of △ABC; The intersection of the three midlines of a triangle is called the center of gravity of the triangle.

(2) The relationship between the related angles of triangles: the sum of the internal angles of any triangle is equal to180; The two acute angles of a right triangle are complementary, and vice versa; The angle formed by one side of a triangle and the extension line of the other side is called the outer angle of the triangle, and the outer angle of the triangle is equal to the sum of two inner angles that are not connected with it.

(3) The sum of polygons and their internal angles

(1) On the plane, a closed figure consisting of several line segments connected end to end is called a polygon. If the number of line segments constituting a polygon is n, then this polygon is called n polygon;

② The sum of the internal angles of the N-polygon is equal to (n-2) ×180;

③ The sum of the external angles of the N-polygon is equal to 360.

2. Chapter 12: congruent triangles

(1) Features of congruent triangles:

(1) Two triangles can completely overlap together. Two triangles like this are called congruent triangles;

(2) Two congruent triangles overlap, the overlapping vertices are called corresponding vertices, the overlapping edges are called corresponding edges, and the overlapping angles are called corresponding angles;

③ The corresponding edges of congruent triangles are equal, and the corresponding angles are equal;

(2) congruent triangles's judgment:

(1) congruence of two triangles with three equal sides (called "side" or "SSS" for short);

(2) The superposition of two triangles with equal included angles (referred to as "corner edge" or "SAS" for short);

(3) congruence of two equilateral triangles (referred to as "angle" or "ASA");

(4) congruence of two triangles with two angles and opposite sides of an angle (referred to as "corner edge" or "AAS" for short);

⑤ Two right-angled triangles with equal hypotenuse and right-angled side are congruent (hypotenuse, right-angled side or HL for short).

(3) The nature of the angular bisector:

① The distance from the angle on the bisector to both sides of the angle is equal, which can be proved by the congruence of the triangle;

(2) The point with equal distance from the inside of the angle to both sides of the angle is on the bisector of the angle, which can be proved by the congruence of the triangle;

3. Chapter 13: Axisymmetry

(1) Axisymmetric:

(1) Fold a graph along a straight line, and if it can overlap with another graph, it is said that the two graphs are symmetrical about this straight line, which is called the symmetry axis, and the overlapping point after folding is called the symmetry point;

(2) A straight line passing through the midpoint of a line segment and perpendicular to the line segment is called the midline of the line segment, and the point on the midline of the line segment is equal to the distance between the two end points of the line segment, while the point on the midline of the line segment is equal to the distance between the two end points of the line segment on the plane;

(3) 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.

(2) isosceles triangle and equilateral triangle:

① A triangle with two equal sides is an isosceles triangle; The two base angles of an isosceles triangle are equal (referred to as "equilateral angles" for short); 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 (abbreviated as three lines in one); If the two angles of a triangle are equal, the opposite sides of the two angles are also equal (abbreviated as "equilateral").

② An isosceles triangle with three equal sides is called an equilateral triangle; Three internal angles of an equilateral triangle are equal, and each internal angle is equal to 60; A triangle with three equal angles is an equilateral triangle; An isosceles triangle with an angle of 60 is an equilateral triangle;

(3) In a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse; It can be proved by the properties of equilateral triangles.

Chapter 14: multiplication and factorization of algebraic expressions.

Multiplication of (1) algebraic expressions: the multiplication formula of integers.

(1) same base power multiplication, base constant, exponential addition; Power, constant basis, exponential multiplication; The power of the product is equal to each factor of the product multiplied by the power, and then multiplied by the obtained power;

(2) Multiply the single item with the single item, and multiply them by their coefficients and the same base respectively; For the letter contained only in the monomial, it is used as the factor of the product together with its index;

(3) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial, and then adding the products;

(4) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.

(2) Division of algebraic expressions: integer division formula.

(1) same radix power division, radix constant and exponential subtraction; It can be inferred that any number whose power is not equal to 0 is equal to1;

(2) The monomial division, the coefficient and the same base power are used as a factor of the quotient for division operation, and only the letters contained in the division formula are used as a factor of the quotient, together with their exponents;

(3) Divide the polynomial by the monomial, first divide each term of the polynomial by the monomial, and then add the obtained quotients.

(3) multiplication formula: multiplication formula

① Square difference: the product of the sum of two numbers and the difference between the two numbers is equal to the square difference between the two numbers; And vice versa;

② Complete square: the square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product; or vice versa, Dallas to the auditorium

(3) When adding brackets, if there is a plus sign before the brackets, all items in the brackets remain unchanged; If there is a minus sign in front of the bracket, everything in the bracket will change its sign; The situation is the same after removing the brackets;

5. Chapter 15: Scores

Properties of (1) score:

(1) Like this, if A and B represent two algebraic expressions and B contains letters, then the formula A/B is called a fraction, where A is called a numerator and B is called a denominator;

② The numerator and denominator of the fraction are multiplied (or divided) by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged;

(2) Fraction operation:

(1) fractional multiplication, with the product of molecules as the numerator of the product and the product of denominator as the denominator of the product;

(2) The fraction is divided by the fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor;

③ Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator;

(4) Addition and subtraction of fractions with different denominators, first division, then addition and subtraction of fractions with the same denominator;

(3) Fractional equation: An equation with an unknown denominator like this is called a fractional equation; Fractional equation is transformed into integral equation by general division, and then the solution of this number equation is substituted into the simplest common denominator of fractional equation. If the value of the simplest common denominator is not 0, the solution of the integral equation is the solution of the original fractional equation, otherwise it is not.