5. 1 intersection line
1, adjacent complementary angles and diagonal angles
There are several different angles among the four angles formed by the intersection of two straight lines. Their concepts and properties are as follows:
chart
pinnacle
Marginal relation
Size relation
Vertical diagonal
1and ∠ 2
Have a common vertex
Two sides of ∠ 1 and two sides of ∠2 are opposite extension lines.
Equal vertex angle
Namely ∠ 1=∠2
Adjacent complementary angle
Three and four.
Have a common vertex
∠3 and∠ 4 have a common side * *, and the other side is an opposite extension line.
∠3+∠4= 180
Note: (1) antipodal angles appear in pairs, and antipodal angles are two angles with special positional relationship;
(2) If ∠ α and ∠βare antipodal angles, there must be ∠α = ∠β; On the other hand, if ∠ α = ∠β, then ∠αand ∠βare not necessarily diagonal.
(3) If ∠ α and ∠βare adjacent complementary angles, there must be ∠α+∠β =180; On the other hand, if α+β =180, α and β are not necessarily adjacent complementary angles.
(3) Of the four angles formed by the intersection of two straight lines, each angle has two adjacent complementary angles and only one diagonal.
2. Vertical line
(1) Definition: When one of the four angles formed by the intersection of two straight lines is a right angle, it is said that the two straight lines are perpendicular to each other, one of which is called the perpendicular of the other, and their intersection is called the vertical foot.
The symbolic language is recorded as:
As shown in the figure: AB⊥CD, and the vertical foot is O.
⑵ Vertical property 1: There is one and only one straight line perpendicular to the known straight line (compared with the parallel axiom).
(3) Vertical 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. Abbreviation: the vertical segment is the shortest.
3, vertical drawing:
(1) Draw a point on a straight line perpendicular to the known straight line; ⑵ Draw the perpendicular of the known straight line at a point outside the straight line.
Note: ① Draw the vertical lines of line segments or rays, that is, draw the vertical lines of their straight lines; (2) A point perpendicular to the line segment, the vertical foot can be on the line segment or on the extension line of the line segment.
Drawing method: ① Leaning: Leaning on a right-angled known straight line of a triangular ruler; ② Move: move the triangular ruler so that one point falls on the right angle on the other side; Draw a line along this right angle, and don't draw a line that feels like a line segment.
4. Distance from point to straight line
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.
Remember when to use graphics to remember.
As shown in the figure, the distance between PO⊥AB and P and straight line AB is the length of PO. PO is a vertical line segment. PO is the shortest straight line from point p to straight line AB.
In real life, digging ditches to divert water and drinking Petunia are the applications with the shortest vertical section.
5. How to understand the similar but different concepts of "vertical line", "vertical line segment", "distance between two points" and "distance from point to line"
Analyze their connections and differences.
(1) the difference between vertical line and vertical line segment: vertical line is a straight line, and its length cannot be measured; The vertical line segment is a line segment, and the length can be measured. Connection: It has the same characteristics as * * * perpendicular to a known straight line. (Vertical nature)
⑵ The difference between the distance between two points and the distance between points and straight lines: the distance between two points is between points, and the distance between points and straight lines is between points and straight lines. Connection: all are the lengths of line segments; The distance from a point to a straight line is the distance between two special points (namely, a known point and a vertical foot).
(3) The distance between the line segment and the distance is the length of the line segment, which is a quantity; Line segments are a kind of figure, and they cannot be equal.
5.2 parallel lines
1, the concept of parallel lines:
In the same plane, two lines that do not intersect are called parallel lines, and the lines are parallel to each other, which is called ‖.
2. The positional relationship between two straight lines
In the same plane, there are only two positional relationships between two straight lines: (1) intersection; ⑵ Parallel.
Therefore, when we know that two straight lines do not intersect in the same plane, we can be sure that they are parallel; And vice versa (here we regard two overlapping straight lines as a straight line)
When judging the positional relationship between two straight lines in the same plane, it can be determined according to their common points:
(1) has only one common point and two straight lines intersect;
(2) If there is no common point, then two straight lines are parallel;
(3) Two or more common points, then two straight lines coincide (because two points determine a straight line).
3. The axiom of parallelism-the existence and uniqueness of parallel lines
After passing a point outside the straight line, there is one and only one straight line parallel to this straight line.
4, parallel axiomatic reasoning:
If both lines are parallel to the third line, then the two lines are also parallel to each other.
As shown on the left.
∴‖
Pay attention to writing in symbolic language. Only when two straight lines are parallel to the third straight line can we draw the conclusion that two straight lines are parallel.
5, three-line octagon
Two straight lines are cut into eight angles by the third straight line, forming congruent angles, internal angles and internal angles on the same side.
As shown in the figure, the straight line is cut by a straight line.
①∠ 1 and ∠5 are on the same side of the cutting line and above the cutting line.
It's called the isosceles angle (the same position)
②∣∠5 and ∣ ∣ 3 ∣ 3 ∣ 5
③∣∠5 and ∣ ∣ 3 ∣ ∣ 3 ∣ ∣ 5
④ Three lines and octagons can also be seen in the model. The conformal angle is "A"; The internal dislocation angle is z-shaped; The inner angle of the same side is U-shaped.
6. How to distinguish the three-line octagon?
The key to distinguish congruent angle, internal angle or ipsilateral internal angle is to find the "three lines" that make up these two angles. Sometimes it is necessary to "extract" relevant parts or ignore irrelevant lines, and sometimes it is necessary to complete graphics.
For example:
As shown in the figure, judge the position relationship of the following diagonals: (1)≈ 1 and ∠ 2; ⑸ 1 and?7； (3) < 1 and