Parallel lines and intersecting lines

? Speaking of intersecting lines and parallel lines, let's first discuss the relationship between points, lines and surfaces. First of all, what is the positional relationship between two points? There are two answers: two-point coincidence and two-point non-coincidence, and there is nothing else except these two situations. What about three points? There are also two relationships: the three-point line and the three-point line. How about four o'clock? There can be three situations at four points, four-point line, three-point line and arbitrary three-point line.

So, what is the positional relationship between the two straight lines? Obviously, one is intersecting and the other is parallel, but there is a special type of intersection, which is vertical. So, how to define intersection? Intersection means that two straight lines have a common point. What about parallel? Two straight lines that do not intersect in the same plane are called parallel straight lines. Why are they on the same plane? Because if you don't stipulate that you are on a plane, those two lines may be in the air, that is, they don't intersect or parallel.

Then, let's discuss what interesting angles are formed by the intersection of straight lines. Draw a picture first,

When we cross the straight line AB at point O, we will find ∠ AOC+∠ AOC = 180. This is because of the definition of straight angle, we can know that ∠ COD = 180. We call such an angle an adjacent complementary angle. Secondly, you will find that two groups of antipodal angles are formed on the way, and the adjacent complementary angles just obtained can prove that the antipodal angles are equal. Therefore, whenever you meet two mutually diagonal angles, their degrees must be equal.

? We will also find that the sum of two angles on a straight line is equal to 180 degrees, so such angles are called complementary angles. If the sum of two angles is equal to 90 degrees, then the two angles are complementary.

If (assuming) ∠ 2+∠ 3 = 180, we can know from the figure that ∠ 2+∠ 4 = 180, then if these two formulas are simplified, we will get ∠3 =∞. So we call this conclusion: the complementary angles of the same angle are equal. In the same way, we can also get that the complementary angles of the same angle are equal.

So what conclusion can we draw from the vertical line? If you draw a straight line more than one point, then if you think this distance is the shortest, you must draw a vertical line. So we can conclude that the vertical segment is the shortest.

After crossing the vertical line, we should say parallel lines. There is a very important figure in parallel lines, that is, three lines and octagon (two lines are cut by the third line). So, what's the mystery of this three-line octagon?

If ∠ 1=∠2, then A∨b has no basis, so we call this method "self-evident". We call this angle the isosceles angle, so its literal language is that the isosceles angle is equal and the two lines are parallel. So we call this "parallel line judgment theorem 1".

Besides the isosceles angle, what other special angles are there? Can you judge the parallelism? Is there a parallel judgment theorem 2 and a parallel judgment theorem 3?

There must be some, as well as the inner angle and the inner angle on the same side.

Let's talk about the inner angle first, as shown in the figure, we can know that ∠ 1=∠8, they form the inner angle, so how to prove it? By making the vertex angles equal, we can know that ∠4=∠8, then ∠ 1 and ∠4 are the same angles, then we can use the parallel line judgment theorem 1 to get a ∨ B. Therefore, this is the parallel line judgment theorem 2: the inner angles are equal and the two lines are parallel.

Next, the ipsilateral internal angle.

As shown in the figure, it is known that ∠ 1+∠ 2 = 180, and find a ∨ b. We know that ∠ 1+∠ 2 = 180, and we can get ∠ 2+∠ 8 = 180 by using the definition of right angle, so we can get ∠1by using the principle that the complementary angles of the same angle are equal. So we use the judgment theorem 2 of parallel lines to find a ∑ b. Therefore, this is the judgment theorem 3 of parallel lines: the inner angles of the same side are complementary and the two straight lines are parallel.

These three lines are the judgment theorems of parallel lines. Then we can deduce the property theorem of parallel lines reciprocal to him in the same way.

There are three more:

Let's prove it together draw

Let's talk about congruence angle first, which is the first theorem of parallel lines. Given a∨b, then ∠ 1 = ∠ 2. Why is this happening? Why didn't it prove the process? We know that the judgment theorem of parallel lines 1 is self-evident, and so is the property theorem of parallel lines 1. like this

So now that we know that the parallel congruence angles of two straight lines are equal (that is, the property theorem of parallel lines 1), we can try to find the property theorem of parallel lines 2 and parallel lines 3. Let's start with the property theorem 2 of parallel lines.

As shown in the figure, a∨b is known, and ∠3=∠4 is verified. From a∨b, it can be concluded that ∠4 is equal to ∠8, and based on the property theorem of parallel lines, 1, it can be concluded that ∠3 is equal to ∠8 through vertex angle equality, so ∠3=∠4 can be obtained through equivalent substitution. This is the property theorem 2 of parallel lines. The two lines are parallel and the internal angles are equal.

Next, let's explore the inner corner of the same side.

As shown in the figure, a∨b is known, and ∠ 2+∠ 3 = 180 is verified. We will find that it has two solutions, one is to use the property theorem of parallel lines, and the other is to use the property theorem of parallel lines. The above are the methods and steps.

So, what problems can be solved with parallel lines? For example, the sum of the inner and outer angles of triangles (including quadrangles and polygons); There is also the external angle theorem of triangle: an external angle is equal to the sum of the degrees of two internal angles that are not adjacent to this angle, and so on.

What about the future? What shall we explore in the future? In the future, we may explore the outer corner of the triangle and congruent triangles ... So are you interested in it? !

This is my exploration of parallel lines and intersecting lines.