We all say that points move into lines, lines move into planes, and planes move into bodies. So what is the definition of parallel lines? Roger that. Parallel lines must be the positional relationship between lines. Although they all have different positional relationships, such as point and line, line and surface, point and point, line and surface, body and body, surface and point and body, body and surface, body and line, we mainly focus on the relationship between lines, but we need to have a special restriction on parallel lines, that is, they must be in the same plane, if not in the same plane. This discussion doesn't make much sense. The definition of parallel lines in our primary school may be two straight lines that never intersect in the same plane, namely parallel lines and intersecting lines. These two lines have one thing in common, that is, they are on the same plane. We can also draw pictures according to this definition. Now, we have learned something about parallelism and intersection, and other things. Please read the following.
In junior high school, how should we define two straight lines as intersecting straight lines? Let's express it in written language first L 1 and l2 intersect at point A, but the symbolic language of intersection is not specified, so we can also read it like this. The picture below shows the graphic language.
So how do we define two lines as parallel lines? We can define parallel straight lines and parallel straight lines according to the definition of intersecting lines, that is, two straight lines with no common point in the same plane. Symbolic language, that is, graphic language, is as follows.
So what special positional relationships between intersecting lines need to be named? In that case, two straight lines intersect on the same plane to form four angles, one of which is 90 degrees. In fact, if one angle is 90 degrees, then all four angles are 90 degrees. In this case, there is a special point, which is the point that intersects in a straight line. We call this point the vertical foot. If you want to express it in words, L 1 is perpendicular to l2 at point O, and there is a special symbol.
Of course, there are some special angles, such as the angle 1 and the angle 2 in the picture I drew below. These two angles seem to be equal, but they need to be proved by strict logical reasoning to reach this conclusion. So how can we prove it?
(The following is the proof process)
As shown in the figure, the straight line L 1 intersects with L2 at point A. ..
Verification: < 1 = < 2
Proof: ∫∠2+∠4 = 180 degrees (flat angle definition)
∠ 1+∠ 4 = 180 degrees (angle definition)
∴ 180 degrees MINUS angle 4 equals angle 3 (the basic property of the equation)
180 degrees minus angle four equals angle one (the basic property of the equation)
∴∠ 1 = ∠ 2 (equivalent substitution)
So now we come to a conclusion that the vertex angles of two straight lines intersecting the same plane are equal. We can also judge the intersection of two straight lines according to this property. Later, we also found two other different relationships. When two angles add up to 90 degrees, they are complementary angles. When the two angles add up to 180 degrees, the two angles complement each other, as shown in the following figure.
Now angle 1 and angle 2 are complementary angles, which are the same as him, but add up to 180 degrees.
So how many straight lines can be made perpendicular to the known straight lines after a little bit? Only one straight line perpendicular to the known straight line can be made in the same plane, as shown below.
Given a point outside the straight line L and the straight line A, we now make a straight line ao passing through the point A, which is perpendicular to L and has a vertical foot of O, so how do we define the distance from the point to the straight line? In fact, the length of hammer line segment ao is the distance from point A to straight line L. We may often say that the line segment between two points is the shortest, so point A to line A is the shortest vertical line segment.
Next, we began to explore parallel lines.
Which of the following three ways to draw parallel lines do you prefer? You will find that if we operate physically, the first method will definitely not work, but the second and third methods can draw parallel lines. Next, we will study a very interesting phenomenon, that is, two parallel lines are cut by another straight line, so what are the special angles? As shown in the figure below.
This is actually what we often call a three-line octagon, so can we judge whether two lines are parallel lines according to the three-line octagon?
Through observation, we found that several groups of these eight angles are the same. Angle 1 is the same as angle 2, angle 3 is the same as angle 4, angle 5 is equal to angle 7, and angle 6 is equal to angle 8. We also call this angle conformal angle, which is symbolic language.
∵∠ 1=∠2
∴a∥b
So let's sum up, when line AB is cut by line C, if the congruence angle is equal, then these two lines are parallel, which is our parallel line judgment theorem 1.
Then we found several groups of angles with equal angles, that is, angle six equals angle seven and angle two equals angle three, but it needs to be proved by strict logical reasoning, and the proof process is as follows.
We call such an angle an inner angle, so now we get the parallel line judgment theorem 2. Let's describe it in written language, that is, the straight line AB is cut by the straight line C. If the inner angles are equal, then the two lines are parallel, and the symbolic language is:
∵∠2=∠3
∴a∥b
There is one last discovery, such as angle 2 plus angle 7= 180 degrees, which we can also infer. The reasoning process is as follows.
This is what we call the ipsilateral internal angle. Describe it in written language, that is, the straight line AB is cut by the straight line C. If the inner angles on the same side are complementary, then the two straight lines are parallel. In symbolic language, it is:
∵∠2+∠7= 180
∴a∥b
These are three ways to judge parallel lines through our reasoning and then proved, but the judgment of parallel lines is only a way for us to judge whether two straight lines are parallel, but do they have some properties? In other words, we now know that two straight lines are parallel, so what conclusion can we draw?
We can draw two parallel lines with a ruler and a triangular ruler, and then draw a straight line C to intersect the two parallel lines. Then we can measure the degrees of eight angles formed by three straight lines with a protractor. What patterns can you find?
Let's take a look at the picture below. Finally, my measurement results are: angle 1, angle 4, angle 5 and angle 7 are all 120 degrees, angle 2, angle 3, angle 6 and angle 8 are all 80 degrees. So what guesses can we make from this picture? My guess is that two straight lines are parallel and have equal juxtaposition angles, so in this picture, that is, angle one equals angle five, angle two equals angle six, angle four equals angle seven, and angle three equals angle eight. We can call this discovery parallel line property theorem one. Although we usually call it that, it is actually not a complete theorem, because it is actually an axiom and self-evident. Just like the judgment theorem of parallel lines, let's first
Now we can use this theorem to deduce other theorems. Now I have a guess, that is, two parallel lines have the same internal angle, for example, angle three equals angle six, so how can we prove it? The proof process is as follows.
So we also get the second theorem of parallel lines. Such an angle is called an inner angle, so the two lines are parallel and the inner angles are equal.
Of course, I have other guesses, that is, angle three plus angle five equals 180 degrees. How can I prove it? The proof process is as follows.
So now we also get the property theorem 3 of parallel lines. We call such angles ipsilateral internal angles, so two straight lines are parallel, and the sum of ipsilateral internal angles is equal to 180 degrees, which means that ipsilateral internal angles are complementary.
So this is the essence of parallelism, so now we have made clear the judgment and essence of parallelism. In the future study, we will encounter various situations and situations, and then we will use various relationships to prove that you are ready for brainstorming. Let's look at a problem!
I think this geometry problem is very interesting. The fun of it is that you get a lot of information, but the results can't be combined. But at a certain moment, you suddenly find that two lines or several angles can take a very special relationship. Then you can use your known conditions to finally find out that an angle or two lines are parallel, which is based on the nature of parallel lines and the judgment of parallel lines, although this is very important for solving problems. And solving a difficult problem is very fulfilling, but I enjoy the process of solving this problem more, especially the problem that has no clue at first, which requires you to keep reasoning and your brain is running at full speed. I think this is the charm of mathematics.