Just today, when we were having math class, you sent a very interesting question. You are what we call European geometry. European geometry is something we only learn in junior high school. It sounds tall, but European geometry is actually very simple, but it is very troublesome. Why do you say that? Because Euclidean geometry is no longer divided like our primary school, for example, it should be clear that proving that the sum of the internal angles of a triangle is 180 degrees is no longer a hands-on operation, but a strict reasoning. Because of this, one of the biggest characteristics of Euclid's geometry is that when you write a question, you must first have a mathematical axiom, which is self-evident, does not need reasoning proof, and the number of mathematical axioms is small. When you solve another problem, it is best to use only one mathematical axiom, and then use this mathematical axiom to deduce other mathematical theorems, with a very rigorous mathematical thinking and reasoning in the middle. This is the case with European geometry, and in European geometry, if you want to forget what you have learned before, you can only use our existing mathematical axioms. Just today, we found a very amazing thing, that is, the three-line octagon in European geometry.

What is a three-line octagon? As can be seen from the name, there are three lines, two of which are parallel and the other intersects with these two lines, and this line passes through any point in these two lines, as shown in the figure.

So this is a picture, and now we need a mathematical axiom. We determined that the kilometer is angle 3 equals angle 4, angle 7 equals angle 8, and angle 5 equals angle 6. We can understand it this way, because line A and line B are parallel, so line A and line B can completely coincide. Then when line A and line B coincide, angle 4 and angle 3 will coincide, and angle 7 and angle 8 will also coincide, but this method is not consistent. So we have to think of other ways to prove it, but we find that there is no better way, but in Euclidean geometry, we can directly say that angle 1 and angle 2 are the same, they are the same angle, which is what you may ask, and I have no proof! There is no need to prove that this is Euclidean geometry. As I just said, it is a mathematical axiom, which goes without saying. So now we have our first mathematical axiom, which is equal to the complementary angle.

So now we need some other theorems. I have a guess that foot seven plus angle two may be equal to 180 degrees, but it cannot be said out of thin air, and strict reasoning is needed.

So now the purpose of our speculation is

∫a∨b

∴<； 7 = & lt8，& lt3 = & lt2

∵& lt； 7+& lt； 3= 180 degrees

∴<； 7+& lt； 2= 180 degrees

This is a theorem that we have calculated with a mathematical axiom. These are the internal angles on the same side, two angles next to the same line, so now we have a conclusion that the sum of the internal angles on the same side is equal to 180 degrees. There is a proper term in European geometry, that is, complementarity, which is 180 degrees.

Now we have another guess, that is, foot seven and angle four are equal, which needs to be proved by very strict mathematical reasoning.

Now we can know that hexagon and pentagon are equal because A is parallel to B, that is, six plus four, that is, 180 degrees, and pentagon plus heptagon is also 180 degrees, so 180 degrees minus pentagon degrees, that is, heptagon degrees, 180 degrees minus hexagon degrees. It is such a process.

∫A∨b

∴<； 5 = & lt6,

∵& lt； 5+& lt； 7= 180 degrees,

∴<； 7 =< Four

This is the second theorem we derived, and this is the inner angle, so the inner angles are equal.

While exploring here, we found some interesting examples. For example, we used to talk about rectangles in primary schools. When we are learning a rectangle, some teachers may tell you directly that the diagonal corners in the rectangle are all the same, and the teacher may give each of us a rectangle and then let us always fold it in half to prove it. So in European geometry, you have no such concept at all. We must have strict reasoning to prove it.

First of all, we definitely need a rectangle, as shown in the following figure:

For example, the saint now has a rectangle, and then I made some changes to make it easier for us to understand.

As shown in the figure, what we know now is that A is parallel to B, and just because A is parallel to B, then angle plus angle 2 equals 180 degrees. If we get to this point, we have already applied it. At this time, we don't need to recalculate a theorem we just got. This theorem is the tool we use. C is also parallel to D, so we can also get an angle plus. Now it is simple and clear that there is an angle 1 in both formulas, so now it can be drawn directly that angle 3 is equal to angle 2, so the two diagonals in the rectangle are equal. The whole process is as follows.

∫a∨b

∴<； 1+& lt； 2= 180 degrees

∫c∨d

∴<； 1+& lt； 3= 180 degrees

∴<； 2 =< Three

In addition, we also found triangles. In the past, when we were studying triangles, the teacher would let us cut these three angles and put them together. After observation, it is proved that the sum of the internal angles of the triangle is 180 degrees, so now we will use reasoning to prove whether the sum of the internal angles of the triangle is 180 degrees.

You need to have a triangle first, as shown in the figure.

I also added something to this triangle. In order to facilitate our understanding, I also marked angles, one angle, two angles and three angles, and angles A and B. Now we know that A and B are parallel, we can use this to get that angle 2 is equal to angle A, and use the inscribed angle obtained above. In addition, we can also get that angle 3 is equal to angle b, and angle 1 plus angle 2 plus angle 3 is 65438. They form a right angle, so the angle plus angle A plus angle B is 180 degrees. Because angle A replaces angle 2 and angle B replaces angle 3, we can also find that the sum of the internal angles of the triangle is 180 degrees. The process is as follows.

∫a∨b

∴<； 2 =< 1,<3 =<b

∵& lt； 1+& lt； 2+& lt； 3= 180 degrees

∴<； 1+& lt； a+& lt； B= 180 degrees

We can find the sum of the interior angles of a triangle, which we deduce by reasoning. Besides exploring these, we can also use what we know now to explore the secrets of other rectangles, triangles or other figures. How amazing is this?

This is the first time I feel Euclidean geometry. When you start studying Euclidean geometry, it means that you have jumped into a rigorous thinking pit. You have drawn some theorems through rigorous reasoning, and when you derive another theorem, you will feel a great sense of accomplishment. How wonderful! I think this is the charm of European geometry.