? Now that we have decided to regain the concept of parallel lines, where should we explore? I think it should be from how to determine the relationship between two straight lines into parallel lines, although the composition standard of parallel lines has long been known: in a plane, two straight lines do not intersect. However, if I show you a small part of two straight lines and I don't know what the relationship is, how can I quickly tell whether these two straight lines are parallel? Extension? But if these two straight lines are not parallel lines, but they are not far from parallel lines and intersect for hundreds of kilometers, wouldn't it be too troublesome to judge? Therefore, we urgently need a method that can quickly distinguish parallel lines.
Can you just judge by these two straight lines? Cover a section of one of the straight lines with a ruler and translate it upward. If the translation trajectories are completely coincident, it seems that the two straight lines are parallel, but it seems that there will be many errors, such as shaking hands when translating upwards. Isn't that an extremely inaccurate result? How to measure if you don't use a ruler? At this time, you can introduce another straight line and want to cross these two straight lines, thus forming eight angles:
? If these two straight lines are parallel (straight line A and straight line C), then the incident angles of straight line X Y running through these two straight lines are actually equal. In this way, if line A and line C are parallel lines, there will be four angles on line A and line C, and these four angles are equal to the four angles of another line (if lines A and C are not parallel lines, the corresponding angles will not be equal, because although line X Y enters at the same angle, the angles of the two lines themselves are not equal, which will also cause angle errors. The angles corresponding to this picture should be angle 1 and angle 3, angle 6 and angle 8, angle 2 and angle 4, angle 5 and angle 7. From this, it can be concluded that if a set of angles in all corresponding angles are equal, it can be proved that two straight lines are parallel.
? Now you can quickly prove whether two straight lines are parallel, just draw a straight line that intersects the two straight lines and observe whether the angles in the corresponding angles are equal. However, after exploring this problem, another problem has emerged. What kind of properties can be found from the conclusion that these two lines are parallel?
Since another straight line is used to judge whether two straight lines are parallel lines, this straight line and the eight angles it produces must not be lost when the nature is discovered! Because those eight corners are most likely to produce magical properties.
When line A and line C are parallel, eight corresponding angles will be generated, and each corresponding angle has a corresponding angle with the same degree generated on another line, and these two corresponding angles generally appear on the right side or the left side of the penetrated line, so they are called congruent angles. If two straight lines are parallel to A and C, the relative congruence angles always seem to be equal. However, how to prove this conjecture? It seems impossible to prove it? Any proof process needs a base point, and before that, there is no base point to infer this property. The conjecture that isosceles angles are equal is actually an axiom in mathematical theorems, which cannot be proved and is self-evident. However, a casual axiom is unconvincing, after all, there is no rigorous logical reasoning process. Therefore, it is necessary to try to simulate with real graphics in order to basically determine the accuracy of kilometers. After geometric transformation, we find that the congruence angles in all simulations are equal, which proves the reliability of the axiom and obtains the first property: when two lines are cut by another line and are equal to the congruence angles, the two lines are parallel. The use of symbolic language is; Because: the same angle is equal.
? So: two straight lines are parallel.
There are two angles, both on the left or right of the straight line Y, inside the straight lines A and C, which can be called ipsilateral inner angles intuitively. After observing several sets of data, it is found that the sum of the degrees of the inner angle on the same side always seems to be equal to 180 degrees, but this is only a guess. We need to prove this through strict reasoning. Based on the axiom that isosceles angles are equal, it is inferred that the sum of internal angles on the same side is equal to 65440 degrees.
Known ipsilateral internal angle 6+ angle 3= 180 degrees.
Verification: line A B is parallel to line c d.
? Proof: Because: Angle 6+ Angle 3= Angle 6+ Angle 1= 180 degrees. (known)
So: Angle 3= Angle 1 (equivalent substitution)
So: straight line A B and straight line C D are parallel (the same angle is equal).
According to the proof of reasoning, we verified that the sum of internal angles on the same side is equal to 180 degrees, thus determining the second property of judging parallel lines: two straight lines are cut by a third line, and the sum of internal angles on the same side is equal to 180 degrees, so the two straight lines are parallel.
The symbolic language is: because the sum of the internal angles on the same side is equal to 180 degrees.
? So: two straight lines are parallel.
Looking back, I seem to see that the two corresponding angles are not both relative angles, either on the left side or on the right side of the straight line. For example, Angle III, Angle V, Angle VII and Angle I in the above figure all seem to be corresponding angles with equal degrees, but they do not conform to the above properties. Then we may have to invent a new property: Angle III and Angle V, one of which appears on the left side and the other on the right side of the through line, and they appear between straight lines A B and C D at the same time, so let's call it an internal dislocation angle. The other properties of angle 7 and angle 1 are the same as those of angle 3 and angle 5, but they appear outside the straight lines A B and C D, so they are called external dislocation angles. From this, two properties can be obtained: the internal dislocation angle is equal and the external dislocation angle is equal. However, these are just guesses, and we need to prove the nature of our guess by reasoning: the internal angle and the external angle are almost the same in essence, otherwise we will just verify whether the internal angle is equal.
It is known that the isosceles angles are equal, and Angle 5= Angle 3.
Verification: Two lines are parallel.
? Proof: Because: Angle 5= Angle 1 (theorem of intersecting straight lines research: vertex angles are equal), Angle 5= Angle 3 (known).
So: angle 1= angle 3 (equivalent replacement)
?
So: two straight lines are parallel (same angle, two straight lines are parallel)
Through reasoning, we successfully proved the property that the internal dislocation angles are equal, and found the third and fourth properties: two straight lines are cut by another straight line, and if the internal dislocation angles or external dislocation angles are equal, the two straight lines are parallel.
The symbolic language is: because the outer angle or the inner angle are equal.
? So: two straight lines are parallel.
In this way, we have found four methods to judge parallel lines, that is, the relative angles are equal, the internal angles are equal, the external angles are equal, and the sum of the internal angles on the same side is equal to 180 degrees. If the eight angles formed by two straight lines satisfy one of the properties, then the two straight lines must be parallel straight lines.
Through this exploration, we found a method to judge parallel lines. So, what are the properties of parallel lines? It may also be related to the three-line octagon. Since the congruence angle is equal, the two lines are parallel, the internal dislocation angle is equal, and the two lines are parallel and complementary to the lateral internal angle, can it be inferred that the two lines are parallel, equal to the complementary angle, equal to the internal dislocation angle, and parallel and complementary to the lateral internal angle? So, don't we know the nature of parallel lines very smoothly? But this method doesn't seem to work, just as we can't infer that Yao Ming is Yao Ming because he is a man, nor can we say that the dog is a Teddy dog, let alone infer that two straight lines are parallel because the same angle is equal, because it will easily lead to many ambiguities and logical loopholes. It seems that although the nature of parallel lines may be no different from that of parallel lines, they are all: the parallelism of two straight lines is equal to the complementary angle, the parallel internal angle of two straight lines is equal, and the parallel internal angles of two straight lines are complementary. But it still needs to be proved step by step.
Of course, reasoning proof needs a starting point, and this starting point is a self-evident axiom. In the judgment of parallel lines, we say that the same angle is equal and the two lines are parallel, so we might as well take two parallel lines with the same angle as an axiom as the basic point of reasoning. Geometric changes make this axiom more credible. From this, the first property, the first theorem of parallel lines, is obtained: two straight lines are parallel and the same angle is equal.
Then began to prove that our two straight lines are parallel and the internal angles are equal:
Known: AB parallel CD
Verification: Angle 5= Angle 3
Proof: because: AB parallel CD
So: angle 1= angle 3.
Because: angle 1= angle 5 (equal to the vertex angle)
So: Angle 5= Angle 3 (equivalent replacement)
After proof, we successfully obtained the second theorem of parallel lines: two lines are parallel, the internal angles are equal, and the symbol language is; Because: AB is parallel to CD, so: Angle 5= Angle 3.
Next, the proof of the complementation of the internal angles on the same side is given:
Known: AB parallel CD
Verification: Angle 6+ Angle 3= 180 degrees.
Proof: because: angle 1= angle 3.
Angle 1+ Angle 6= 180 degrees.
? So: angle 3+ angle 6= 180 degrees (complementary angles of the same angle are equal).
? Now, we have successfully proved the nature of parallel lines, which is basically consistent with the judgment of parallel lines, that is, the parallel congruent angles of two lines are equal, the parallel internal angles of two lines are equal, and the parallel internal angles of two lines are complementary. Now, whether judging parallel lines or using parallel lines, you can cope with it freely.