What should we pay attention to when using the judgment and properties of parallel lines?

What is parallelism? That is, in the same plane, two lines that never intersect are parallel to each other. Although parallel lines are defined in a plane, they are also applicable to solid geometry. The judgment and properties of parallel lines are the basic knowledge of geometry and the key content of junior high school geometry. Because students are unfamiliar with "judgment" and "nature", they are not clear about their relationship, and they are not familiar with the introduction of reasoning proof, so some students have difficulties in learning. This article talks about some views, hoping to help them.

Properties of parallel lines

1. Two parallel lines are cut by a third straight line and have the same angle.

2. Two parallel lines are cut by a third straight line, and the internal dislocation angles are equal.

3. Two parallel lines intersect with the third straight line and complement each other.

The above characteristics can be simply described as:

1. Two straight lines are parallel and the included angle is equal.

2. The two straight lines are parallel and the internal dislocation angles are equal.

3. Two straight lines are parallel and complementary.

Determination of parallel lines

1. Definition of parallel lines (in the same plane, two lines that do not intersect are called parallel lines. )

2. Parallel axiom inference: Two lines parallel to the same line are parallel to each other.

3. In the same plane, two straight lines perpendicular to the same straight line are parallel to each other.

4. The same angle is equal, and two straight lines are parallel.

5. The internal dislocation angles are equal and the two straight lines are parallel.

6. The internal angles on the same side are complementary and the two straight lines are parallel.

The judgment and nature of parallel lines are all about the figure where two straight lines are cut by a third straight line. First of all, we know what parallel lines are by drawing.

The drawing method of parallel lines can be summarized as follows: one "falling", two "leaning", three "pushing" and four "drawing", that is, one "falling": one side of a triangle falls on a known straight line; The second "lean": the ruler leans on the other side of the triangle; Three "push": push the triangle along the ruler, so that the edge that begins to fall on the known straight line passes through the known point; Draw a known point and draw a straight line along the edge of the triangle. The concept of three-line octagon. When we study the determination and properties of parallel lines, we should involve isosceles angle, internal angle and internal angle of the same side. The key to determine these angular positions is to find the intersection of two straight lines and the third straight line. It can be said that this number is a necessary prerequisite for their similarity. The difference between them is that the nature of parallel lines and the conditions and conclusions for judging parallel lines are just the opposite. To judge whether two straight lines are parallel, we must first study the quantitative relationship among congruent angle, internal angle and internal angle on the same side. When we know that congruent angle or internal angle is equal to or complementary to the internal angle on the same side, we can judge that these two lines are parallel. They are judging from "number" to "shape". The "nature" of parallel lines is that when two lines are known to be parallel, we can deduce the quantitative relationship of the same angle, the same internal angle and the complementary internal angle of the same side, that is, the nature of the graphic "parallel line". They are reasoning from "shape" to "number".

The "judgment" and "nature" of parallel lines are closely related and fundamentally different, and are often confused. In the proof of parallel lines, beginners are often confused when to use the property theorem of parallel lines and when to use the judgment theorem. To understand this problem, we must first understand the structure of these two theorems (as shown in the following table). It is not difficult to see from the table that the conditions and conclusions of the two theorems are just the opposite. So, which theorem should be used when solving problems?

How to use judgment and nature to solve problems? Let me give a few examples to illustrate.

Example 1 known: As shown in the figure, BD divides ∠ABC, ∠ 1=∠2, ∠C=70, and finds the degree of ∠ADE.

Analysis: This question is to find the angle. First, determine to use the judgment of parallel lines to solve the problem. To explain the relationship between the sizes of angles, we need to prove the positional relationship of straight lines and use the property theorem of parallel lines, which can just use the condition that two angles are known to be equal. In addition, through the analysis and reasoning of problems, students can gradually form the idea of proof.

Solution: ∠ 1=∠2 (known) ED∨BC (internal dislocation angles are equal and two straight lines are parallel).

As can be seen from the figure, ED and BC are cut by AC, and ∠C=∠ADE (two straight lines are parallel and have the same angle).

And ∠C=70 (known), ∠ADE=70.

Example 2 is as follows: BE bisects ∠ABC, EC bisects ∠BCD, ∠ E = 90, then AB∨CD? Why? Analysis: This shows that the positional relationship between two straight lines should use the property theorem. Before solving the problem, students can talk about the idea of solving the problem and what is the basis of each step, so that students can gradually feel that all the steps proved are reasonable. You can't say what you want without a general idea.

Solution: ∠ e = 90 (known), ∠1+∠ 2 = 90 (internal angle and properties of triangle).

Divided ∠ABC (known) and EC are divided into BCD (known).

∠ Abe+∠ dec = 90 (definition of angular bisector).

∠ ABC +∠ BCD = 180 (equivalent substitution)

AB∨CD (complementary to the inner angle on the same side, two straight lines are parallel).

For beginners, it is best to let students talk about the problem-solving ideas first, because language is the embodiment of thinking, and you can write if you can speak.

Example 3. As shown in the figure, DE∥BC, ∠ADE=∠EFC.

Fill in the reasons for the establishment of ∠ 1=∠2 completely.

Solution: ∫DE∨BC (known)

∴∠ADE=∠ABC (Two straight lines are parallel and have the same angle. )

∠∠ADE =∠EFC (known)

∴∠∠EFC =∠ABC

∴DB∥EF (same angle, two parallel straight lines)

∴∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠873

Now I have learned how to use judgment and nature to solve problems, but often because the seventh-grade students are just beginning to learn proof, the writing process is also disorganized. By supplementing the proof process, they can gradually become familiar with the writing format of the proof questions.

Example 4, as shown in the figure, BD⊥AC, EF⊥AC, D and F are vertical feet, ∠ 1=∠2, try to explain ∠ adg = ∠ C.

Solution: ∵∠ adg+∠1+∠ fdb =180 (definition of right angle).

∠ 2+∠ C+∠ CFE = 180 (triangle interior angle and its definition)

∴∠ADG+∠ 1+∠FDB=∠2+∠C+∠CFE

∫≈ 1 =∠2 (known)

∠ FDB =∠ CFE = 90 (definition of vertical line)

∴∠ADG =∠C (shift term and sign)

This is also a comprehensive problem, because the relationship between the size of the angle is proved by the relationship between the size of the angle, so both judgment and essence should be used. When students solve problems, they can find solutions through backward deduction, which can also help students make rational use of known conditions.

Example 5. As shown in the figure, A, F, C, D are in a straight line, AF= CD, AB//DE, AB = DE, judge whether EF and BC are parallel, and explain the reasons.

∫AC-FC = DF-FC

∴AC=DF

∫ED and AB are intercepted by AD.

∫AB//DE (known)

∴∠EDF =∞∠cab (two straight lines are parallel and the internal dislocation angles are equal)

AB = DE (known)

∠EDF=∠CAB (certification)

AC=DF (known)

∴ Triangle ABC Triangle Definition (SAS)

∴∠BCF=∠EFD (the corresponding sides of congruent triangles are equal).

∴EF//BC (the inner angles are equal and the two straight lines are parallel) is more difficult. We should not only be familiar with the usage of judgment and nature, but also be clear about the nature and judgment of all triangles. Knowledge points are interrelated. When solving a problem, you must carefully examine the problem and don't rush for success.

Example 6 shows that BE is the extension line of AB, DF is the extension line of AD, ∠ CBF = ∠ A = ∠ C.

1. From ∠CBF=∠A, which two straight lines are parallel can be determined? What is the basis?

2. From ∠CBE=∠C, which two straight lines are parallel can be determined? What is the basis?

3. What angles are needed to prove that AF∨BC is equal?

4. Which angles are equal to proving AE∨DC?