? (Back to the topic) Because junior high school is different from primary school, it needs rigorous logical reasoning and proof, which is why we need to re-verify parallel lines. So how to judge the parallelism of two lines (or that one line is parallel to the other)? Before that, we must understand the definition of parallel lines. Parallel lines are actually the corresponding relationship between two straight lines that do not intersect in a plane.
Then according to the nature of the straight line, it extends infinitely to both sides, and then you can judge by looking at the intersection? No, no, no. We need more rigorous logical reasoning to prove that in this process, we should draw conclusions based on our previous experience.
? Our conclusion is to prove parallel lines.
? Guess 1:
? First, I randomly draw two arcs on the straight line A with compasses (and mark the center of the circle). Then, at the junction of the straight line and the arc line, that is, point A and point B, I draw arcs in two places with the radius of line segment AB, and get point C just outside this line. Using the origin of this point and the center of the circle, I make a straight line C that passes through these two points, which is the vertical line of A. Since it is vertical, the angle 1 is 90 degrees. Similarly, if it is vertical, the included angle must be 90 degrees. I used a protractor to measure the angle from 2 to 93 degrees, which at least proved that line B and line C are not perpendicular. According to the square conclusion drawn in primary school (because there are two groups of parallel lines in the square), the four internal angles are not only equal, but also 90 degrees. What does 90 degrees mean? All four sides are vertical! Therefore, parallel lines have an indirect vertical relationship. A straight line is perpendicular to one, this one is perpendicular to the other, and then this one is parallel to the original straight line. This is the picture below:
? This time, I completed the verification with the same steps, and found that the straight line C is not only perpendicular to the straight line A, but also perpendicular to the straight line B. And this time, the angle 1 is equal to the angle 2, which is 90 degrees! So this time it can be judged that the straight line A is parallel to the straight line B.
? Guess two:
? In fact, you can get that A is parallel to B without this method. This time, the angle is also used, but it is not a question of vertical line and right angle. The outline is the same as above, but the verification method is different. Using the three-line octagonal model, cut a straight line at two straight lines.
? So how does this model verify the same angle? The four corners reflected in the picture are clues. In fact, guessing 1 also uses this model, but this time it is different. Last time, all eight angles were equal, which proved to be parallel lines. This time it will be different. All eight angles are not equal. Then how do you prove it? For the angles marked in the figure, the relationship between angle 1 and angle 2 is the same as that between angle 3 and angle 4, and they all belong to internal dislocation angles. As the name implies, it is the internal dislocation angle. What happens if a pair of internal dislocation angles are equal? Suppose these two lines are parallel. After measuring two angles, they are equal (angles 4 and 3 are both 58 degrees). Judging from the indirect vertical verification method in conjecture 1;
? Parallel!
? So if I don't have to guess one, can I prove it? Imagine, when the straight line C rotates clockwise or counterclockwise, will the equal relationship between angles 3 and 4 change?
? After measurement, angles 3 and 4 are both 158 degrees. I found that no matter how the straight line C rotates, it changes the size of angle 3 and angle 4, but the only constant is the relationship-equality! So, click here to prove parallel lines!
? So if we use the four corners outside to guess, can we also prove parallel lines? If we only say four outer corners, there will be the same outer corners as the inner corners. Angle 5 and Angle 7 are good examples, as are Angle 6 and Angle 8. Truth equals the inner angle. So what happens if you put the outer corner and the inner corner together? These are two straight lines and the same straight line. After the straight line A is vertically translated for a certain distance, B is obtained, both of which form an angle with C. Look at the relationship between angle 5 and angle 2, and then look at 6 4, 6543 8+0 7, 3 8. Do they belong to the same position? And if two straight lines are parallel, the degree of a diagonal line will be equal, and this diagonal line is called congruence angle. Also, on the left and right sides of the straight line C, there are two groups of relationship angles, which are called "ipsilateral inner angles" (also ipsilateral outer angles). Unlike the wrong angle, they are on the same side. Their laws are not equal, but complementary equals 180 degrees! Understand?
? If it is proved according to the proof method of "Cause Seven" and Euclid, it is as follows:
? Known: all B.
? Verification: Angle 5= Angle 2
? Verification: Because allb
? So the angle 1= angle 2 (internal dislocation angles are equal).
? Because angle 4? Angle 2= 180 degrees, angle 1? Angle 3= 180 degrees, angle 3? Angle 5= 180 degrees.
? So 180- angle 4= 180- angle 3= angle 5= angle 2.
? Known: all B.
? Verification: Angle 1? Angle 4= Angle 2? Angle 3= 180 (complementary)
? Verification: Because allb
? So angle 1= angle 2, angle 3= angle 4 (internal dislocation angles are equal).
? Because angle 4? Angle 2= 180 degrees, angle 1? Angle 3= 180 degrees.
? What about angle 2? Angle 4= Angle 2? Angle 3= 180 degrees, angle 1? Angle 3= Angle 4= 180 degrees.
We reached the final conclusion from one condition to another. When you know how to judge a parallel line, you have mastered every point, figured out every method and understood the essence of parallel lines. Just like learning fractions, the basic properties of fractions will help you calculate, and so will parallel lines.
? External chapter:
? After the school started, we followed the pace of teacher Zhao Junjie and relearned the judgment and nature of parallel lines. It is found that these three methods (or theorems) for determining parallel lines are all obtained by different methods. Today, we will review the past and learn the new, and re-understand these three theorems:
? Theorem 1: equipotential angle
? When you think about it, there seems to be no clear reasoning process to reach this conclusion. Because this is a very intuitive phenomenon, it has not been processed and thought by your brain, and it is the original face. Therefore, we have to regard it as a kind of existence in nature, which is beyond doubt. If you want to know each other, you can rotate the third line between two parallel lines and the line that intersects them constantly. Through many experiments, it is found that the same angle is equal in any case.
? Language description: When a straight line intersects with two straight lines, if the congruence angle is equal, the last two straight lines are parallel to each other.
? Theorem 2: Internal Angle Equivalence
? This is not the original "axiom", because this theorem can be deduced through logical thinking. Of course, under known conditions, it includes the equality of isosceles angles, because this theorem is derived on the basis of the equality of isosceles angles. According to my drawings, you can see the following:
? The known angle 1 is equal to angle 2.
? Verification: A and B are parallel.
? Prove: Because the angle 1= angle 2, the vertex angles are equal (known conditions).
? Therefore, the angle 1= angle 5 (equal to the vertex angle).
? So ... Angle 2= Angle 5 (equivalent replacement)
? So ... Allb (same angle, two parallel straight lines)
Language description: When a straight line intersects with two straight lines, if the internal angles are equal, allb.
? Theorem 3: Complementarity of internal angles on the same side
? This, like Theorem 2, is derived by logical reasoning. Complementarity means that the sum of two angles is equal to 180 degrees, which is exactly a flat angle. But he has two ways to prove it. The first is to use the theorem 1:
? It is known that angle 2 and angle 3 are complementary.
? Verification: allb
? Proof: Because. Angle 2? Angle 3= 180, angle 5? Angle 3= 180 degrees (known condition and definition of right angle)
? So ... A quarter? Angle 3= Angle 2? Angle 3= 180 degrees (equivalent replacement)
So ... Angle 5= Angle 2 (complementary angles of the same angle are equal).
So ... Allb (same angle, two parallel straight lines)
? At the same time, we can also use Theorem 2:
? It is known that angle 2 and angle 3 are complementary.
? Verification: allb
? Proof: Because. Angle 2? Angle 3= 180 degrees, angle 1? Angle 3= 180 degrees (known condition and definition of right angle)
? So ... Angle 1? Angle 3= Angle 2? Angle 3= 180 degrees (equivalent replacement)
? So ... Angle 1= angle 2.
? So ... Allb (internal dislocation angles are equal and two straight lines are parallel)
? Are you clear?
? After we get the judgment theorem of parallel lines, we can use this theorem to summarize the properties of parallel lines. In the past, we used theorems to judge parallel lines. This time we will use theorems to explore the properties of parallel lines. In fact, nature is a theorem, just the other way around. In fact, to put it bluntly, it is through "two straight lines are parallel" that the next one gets its own properties. Before it's too late, come on!
? Nature 1:
? Because the idea of equal apposition angle can't be logically reasoned, it is a natural phenomenon, so we can directly draw the conclusion that "two straight lines are parallel and the apposition angle is equal" through judgment.
? Nature 2:
? Known: all B.
? Verification: Angle 2= Angle 1 (internal dislocation angles are equal)
Proof: Because. known conditions
? Therefore, angle 2= angle 5, and angle 5= angle 1 (equivalent replacement).
? Therefore, angle 2= angle 1 (two straight lines are parallel and the internal dislocation angles are equal).
? Nature 3:
? Known: all B.
? Proof: Angle 2? Angle 3= 180 degrees (complementary to the lateral inner angle)
? Proof: Because. Allb, angle 3? Angle 5= 180 degrees (known condition)
? So ... Angle 2= Angle 5 (I don't know)
? So ... Angle 2? Angle 3= 180 degrees (two straight lines are parallel and complementary to the side inner angle)
? Nature 4:
? Known: all B, bllc
? Proof: allc (equal internal angles)
? Proof: Because. Allb, bllc (known condition)
? So ... Allc (two lines parallel to the same line are parallel)
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