5. 1 intersection line

5. 1. 1 intersection line

One vertex has a common * * *, one side has a common * * *, and the other side is an extension line opposite to each other. Such two angles are called adjacent complementary angles.

There are four pairs of adjacent complementary angles when two straight lines intersect.

There is a vertex with a common * * *, and both sides of the corner are opposite extension lines. These two angles are called antipodal angles.

Two straight lines intersect and have two opposite angles.

The vertex angles are equal.

5. 1.2

Two straight lines intersect, and one of the four corners is a right angle, so the two straight lines are perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot.

Note: (1) The vertical line is a straight line.

⑵ The four angles formed by two straight lines with vertical relationship are all 90.

(3) Verticality is a special case of intersection.

(4) Vertical symbols: a⊥b, AB⊥CD.

There are countless vertical lines that draw known straight lines.

One and only one straight line is perpendicular to the known straight line.

Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest. Simply put: the vertical line segment is the shortest.

The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.

5.2 parallel lines

0+0 parallel line

In the same plane, if two straight lines have no intersection, then the two straight lines are parallel to each other, which is marked as: a ∨ b.

There are only two relationships between two straight lines in the same plane: intersecting or parallel.

Parallelism axiom: after passing a point outside a straight line, there is one and only one straight line parallel to this straight line.

If both lines are parallel to the third line, then the two lines are also parallel to each other.

Conditions of parallel lines

Two straight lines are cut by a third line. On the same side of two sections, on the same side of the section, such two angles are called congruent angles.

Two straight lines are cut by a third straight line, and between the two cutting lines, on both sides of the cutting line, such two angles are called inscribed angles.

Two straight lines are cut by a third line, and between the two cut lines, on the same side of the cut line, such two angles are called ipsilateral internal angles.

Judgment method of two parallel lines:

Method 1 Two straight lines were cut by the third straight line. If congruent angles are equal, two straight lines are parallel. To put it simply: the same angle is equal and two straight lines are parallel.

Method 2 Two straight lines are cut by a third straight line. If the internal dislocation angles are equal, two straight lines are parallel. To put it simply: the internal dislocation angles are equal and the two straight lines are parallel.

Method 3 Two straight lines are cut by a third straight line. If they are complementary, then these two straight lines are parallel. To put it simply: the internal angles on the same side are complementary and the two straight lines are parallel.

5.3 Properties of parallel lines

Parallel lines have properties:

Property 1 Two parallel lines are cut by a third line, and the congruence angles are equal. To put it simply: two straight lines are parallel and have the same angle.

Property 2 Two parallel lines are cut by a third straight line, and their internal angles are equal. To put it simply: two straight lines are parallel and their internal angles are equal.

Property 3 Two parallel lines are cut by a third straight line and complement each other. Simply put, two straight lines are parallel and complementary.

The length of a line segment perpendicular to and sandwiched between two parallel lines is called the distance between two parallel lines.

A statement that judges a thing is called a proposition.

5.4 Translation

(1) Move a graphic as a whole in a certain direction, and you will get a new graphic with the same shape and size as the original graphic.

⑵ Every point in the new graph is obtained by moving a point in the original graph. These two points are corresponding points, and the line segments connecting each group of corresponding points are parallel and equal.

This movement of graphics is called translation transformation, or translation for short.

Chapter VI Plane Cartesian Coordinate System

6. 1 plane rectangular coordinate system

6. 1. 1 ordered number pair

A number pair consisting of two consecutive numbers A and B is called an ordered number pair.

6. 1.2 plane rectangular coordinate system

Draw two mutually perpendicular number axes with overlapping origins on the plane to form a plane rectangular coordinate system. The horizontal axis is called the X axis or the horizontal axis, and it is customary to take the right as the positive direction; The vertical axis is called the Y axis or the vertical axis takes 2 as the positive direction; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.

Any point on the plane can be represented by an ordered number pair.

After the rectangular coordinate system is established, the coordinate plane is divided into four parts, I, II, III and IV, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant respectively. The points on the coordinate axis do not belong to any quadrant.

6.2 Simple application of coordinate method

6.2. 1 Geographical location is expressed in coordinates.

The process of drawing the distribution plan of some places in the area using the plane rectangular coordinate system is as follows:

(1) Establish a coordinate system, select a suitable reference point as the origin, and determine the positive direction of the X axis and the Y axis;

⑵ Determine the appropriate scale according to specific problems and mark the unit length on the coordinate axis;

(3) Draw these points on the coordinate plane and write down the coordinates of each point and the name of each place.

6.2.2 Coordinate translation.

In the plane rectangular coordinate system, the corresponding point (x+a, y) (or (x-a, y)) can be obtained by translating the point (x, y) to the right (or to the left) by a unit length. The corresponding point (x, y+b) (or (x, y-b)) can be obtained by translating the point (x, y) up (or down) by b unit lengths.

In the plane rectangular coordinate system, if a positive number A is added (or subtracted) to the abscissa of each point of the graph, the corresponding new graph is to translate the original graph to the right (or left) by a unit length; If a positive number A is added (or subtracted) to the ordinate of each point, the corresponding new figure is to translate the original figure up (or down) by a unit length.

Chapter VII Triangle

7. 1 Line segment related to triangle

7. 1. 1 triangle edge

A figure composed of three line segments that are not on the same line end to end is called a triangle. The angle formed by two adjacent sides is called the inner angle of a triangle, which is called the angle of a triangle for short.

A triangle with vertices A, B and C is marked as △ABC and pronounced as "triangle ABC".

The sum of two sides of a triangle is greater than the third side.

7. 1.2 The bisector of the height, midline and angle of a triangle.

7. Stability of1.3 Triangle

The triangle is very stable.

7.2 Angle related to triangle

7.2. 1 triangle inner angle

The sum of the internal angles of a triangle is equal to 180.

7.2.2 External Angle of Triangle

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.

The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

7.3 sum of polygons and their internal angles

7.3. 1 polygon

On the plane, a figure composed of some end-to-end line segments is called a polygon.

The line segment connecting two nonadjacent vertices of a polygon is called the diagonal of the polygon.

Diagonal formula of n polygon:

An equilateral polygon is called a regular polygon.

7.3.2 Sum of interior angles of polygons

The formula for the sum of internal angles of n polygons: 180 (n-2)

The sum of the outer angles of a polygon is equal to 360 degrees.

7.4 Project Learning Mosaic

Chapter VIII Binary Linear Equations

8. 1 binary linear equations

An equation with two unknowns whose exponents are 1 is called a binary linear equation.

Two binary linear equations with the same unknowns are combined into one binary linear equation group.

The values of two unknowns that make the values on both sides of the binary linear equation equal are called the solutions of the binary linear equation.

The common * * * solution of two equations of binary linear equations is called the solution of binary linear equations.

8.2 elimination

Starting from an equation in binary linear equations, an unknown number is expressed by a formula containing another unknown number, and then it is substituted into another equation to realize elimination, and then the solution of this binary linear equations is obtained. This method is called substitution elimination method, or substitution method for short.

When the coefficients of the same unknown in two binary linear equations are opposite or equal, the unknown can be eliminated by adding or subtracting the two sides of the two equations respectively, thus a univariate linear equation system can be obtained. This method is called addition, subtraction and elimination, or addition and subtraction for short.

8.3 Re-explore practical problems and binary linear equations

Chapter 9 Inequality and Unequal Groups

9. 1 inequality

9. 1. 1 inequality and its solution set

The formula for expressing the relationship between size with ""is called inequality.

The value of the unknown quantity that makes the inequality valid is called the solution of the inequality.

The range of unknowns that can make inequality hold is called inequality solution set, which is called solution set for short.

An inequality with an unknown degree of 1 is called one-dimensional linear inequality.

9. Properties of1.2 inequality

Inequality has the following characteristics:

The nature of inequality 1 Add (or subtract) the same number (or formula) on both sides of inequality, and the direction of inequality remains unchanged.

The nature of inequality 2 Both sides of inequality are multiplied by (or divided by) the same positive number, and the direction of inequality remains unchanged.

The nature of inequality 3 Both sides of inequality are multiplied (or divided) by the same negative number, and the direction of inequality changes.

9.2 Practical Problems and One-dimensional Linear Inequalities

To solve a linear equation with one variable, the equation should be gradually transformed into the form of x = a according to its properties; To solve one-dimensional linear inequality, it is necessary to gradually transform inequality into the form of x < a (or x > a) according to the nature of inequality.

9.3 One-dimensional linear inequality system

When these two inequalities are combined together, a unitary linear inequality group is formed.

The common part of the solution set of several inequalities is called the solution set of inequalities composed of them. Solving inequality is to find its solution set.

All kinds of inequality problems can be solved by inequality groups. When solving a system of linear inequalities with one variable. Generally, the solution set of each inequality is found first, and then the common part of these solution sets is found. The number axis can be used to express the solution set of inequality groups intuitively.