Parallel knowledge of junior high school mathematics 1
intersection line
1. Two straight lines have one and only one intersection. (On the other hand, if two straight lines have only one intersection, they intersect. )
Two straight lines intersect, resulting in the concept that adjacent complementary angles and vertex angles are opposite:
Adjacent complementary angles: one side of two angles is * * *, and the other side is an extension line opposite to each other. Adjacent complementary angles are complementary. Attention should be paid to distinguishing the similarities and differences between adjacent complementary angles and complementary angles.
Vertex relative: the vertex of two angles, and the two sides of one angle are the relative extension lines of the other angle. The vertex angles are equal.
Note: ① The complementary angles of the same angle or equal angle are equal; The complementary angles of the same angle or equal angle are equal; Equal angles have equal diagonals. or vice versa, Dallas to the auditorium
(2) When expressing adjacent complementary angles and antipodal angles, we should pay attention to relativity, that is, "interaction", and make clear who is the adjacent complementary angle or antipodal angle. For example:
True or false: because ∠ ABC+∠DBC = 180, ∠DBC is an adjacent complementary angle. ( )
Two equal angles are diagonal to each other. ( )
2. Verticality is a special case where two straight lines intersect. Note: two straight lines are perpendicular to each other, that is, if line A is perpendicular to line B, line B is perpendicular to line A. ..
Vertical foot: The intersection of two perpendicular lines is called vertical foot. When it is vertical, it must be represented by a right-angle symbol.
One and only one straight line is perpendicular to the known straight line. (Note: This point can be on a known straight line or outside a known straight line)
3. Distance from point to straight line.
Vertical line segment: take a point outside the straight line as the vertical line of the known straight line, and the line segment between this point and the vertical foot is called the vertical line segment.
Vertical line and vertical line segment: vertical line is a straight line, vertical line segment is a line segment, which is a part of vertical line.
Shortest vertical line segment: The vertical line segment is the shortest of all the line segments connecting the points outside the line and the points on the line. (or in a right triangle, the hypotenuse is larger than the right. )
Distance from point to straight line: the length from a point outside the straight line to the vertical section of the straight line is called the distance from the point to the straight line. Note: Distance refers to the length of the vertical line segment, not the vertical line segment itself. Therefore, it is wrong to judge without the word "long".
4. Equilibrium angle, internal dislocation angle and ipsilateral internal angle
Three-line hexagon octagon: in a plane, two straight lines are cut by a third straight line, which divides the plane into six parts to form eight angles, including 4 pairs of congruent angles, 2 pairs of internal angles and 2 pairs of internal angles on the same side. Attention: We should know and find out these three angles skillfully: ① Distinguish according to concepts; ② Distinguish by model, that is, congruent angle -F type, internal angle -Z type and ipsilateral internal angle -U type.
Please pay special attention to:
(1) Three internal angles of a triangle are all internal angles of the same side;
② The naming of congruent angle, internal dislocation angle and internal ipsilateral angle is not necessarily based on the premise that two parallel lines are cut by a third line. These two straight lines may not be parallel, but they also have congruent angles, internal dislocation angles and internal ipsilateral angles.
5, geometric counting:
(1) n straight lines intersect on the plane, and * * has n (n–1) antipodal angles. (or written as n 2–n groups)
② N straight lines intersect in the plane, with at most n (n–1)/2 intersections. (or (n 2–n)/2)
(3) Every two straight lines intersect N in the plane, and the plane can be divided into [n(n+ 1)/2]+ 1 faces at most.
(4) When any three of the n points on the plane are not * * * straight lines, a * * can be made into n (n–1)/2 straight lines.
Review:
I between n points on a straight line, a * * has n (n–1)/2 line segments;
Ii. if n rays are drawn from a point, a * * has n (n–1)/2 angles.
Knowledge of parallel lines in junior high school mathematics II
Parallel lines
On the same plane, if there is no common point (that is, intersection point), two straight lines are parallel. Note: parallel lines will never intersect.
1. Parallelism axiom: When crossing a point outside a straight line, one and only one straight line is parallel to the known straight line. (Note: This point is outside the straight line)
Inference: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other. (or transitivity of parallel lines)
2. Draw parallel lines: with the help of triangles and rulers. Be specific. This basic drawing method must be mastered and practiced more. )
3. Determination of parallel lines:
(1) Same angle, two straight lines are parallel;
② The internal dislocation angles are equal and the two straight lines are parallel;
③ The internal angles on the same side are complementary, and the two straight lines are parallel.
Note: Look at the angle first, and then judge whether two straight lines are parallel, provided that the angles are equal/complementary.
An important conclusion: in the same plane, two straight lines perpendicular to the same straight line are parallel to each other.
4, the nature of parallel lines:
(1) Two straight lines are parallel and the same angle is equal;
② Two straight lines are parallel and the internal dislocation angles are equal;
③ The two straight lines are parallel and complementary.
Note: the above properties can only be obtained if two straight lines are parallel, provided that the straight lines are parallel.
A conclusion: the distance between parallel lines is equal everywhere. For example, it is used to explain that the opposite sides of a rectangle (including a rectangle and a square) are equal, and the diagonal of a trapezoid divides the trapezoid into two triangles with the same area as the upper bottom or the same area as the lower bottom. (Because the upper bottom of the trapezoid is parallel to the lower bottom, and the heights between parallel lines are equal, there is a triangle with equal bottom and equal height. )
The most difficult thing in this chapter is how to use the judgment or properties of parallel lines to make preliminary reasoning of analytic geometry. On the basis of mastering the basic knowledge points, learn logical reasoning, which should be clear and concise. ※.
5. Proposition
A statement that judges a thing is called a proposition. A proposition consists of two parts: a topic and a conclusion, which can be written in the form of "If …… then ……".
For example, "It may rain tomorrow." This sentence _ _ _ _ proposition, and "It's very hot today, and it may rain tomorrow." This sentence is a proposition. (Fill in "Yes" or "No")
(1) proposition can be divided into true proposition and false proposition. A true proposition refers to a proposition (or correct proposition) in which the topic is established and the conclusion is also established. Pseudo-proposition refers to a proposition (or pseudo-proposition) in which the topic is established, but the conclusion is not necessarily established or not established at all.
② Inverse Proposition: A new proposition is formed after the title and conclusion of a proposition are exchanged, which is called the inverse proposition of the original proposition.
Note: The original proposition is true, but its inverse proposition is not necessarily true. Similarly, the original proposition is false, and its inverse proposition is not necessarily false.
Knowledge of parallel lines in junior high school mathematics III
translate
1. Concept: Move the whole graph in a certain direction to get a new graph. This movement of the figure is called translation.
The key to determining translation is to find the direction of translation (not necessarily horizontal or vertical) and the distance from translation. In the case of oblique translation, it is necessary to split the starting position to the final position into horizontal motion and then into up-and-down motion, or split it into up-and-down motion and then into horizontal motion. Of course, if you are translating in a grid diagram, you can use the characteristic that the translation distance of a known point is the diagonal of a rectangle to complete the translation of other vertices accordingly.
2. Features:
① When translation occurs, the shape and size of the new figure are exactly the same as the original figure (that is, the corresponding line segment and the corresponding angle are equal);
② The line segments between corresponding points are parallel to each other (or on the same straight line) and equal, which are all equal to the translation distance.
3. Drawing method: Grasp the translation direction and distance, draw the corresponding points of the vertices of the original graph by using the property that the connecting lines between the corresponding points (generally referring to the vertices of the graph) are parallel and equal, and then connect them in turn to form a new graph after translation.
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