Teaching plan 1: isosceles triangle (1) The purpose of eighth grade mathematics teaching in People's Education Press

1. The concept of isosceles triangle.

2. The nature of isosceles triangle.

3. Application of the concept and properties of isosceles triangle.

Teaching focus

The concept and properties of 1. isosceles triangle.

2. Application of the nature of isosceles triangle.

Teaching difficulties

Understanding and application of the nature of isosceles triangle with three lines in one

teaching process

First, ask questions and create situations

In the previous research, we have known the axisymmetric figure, explored its properties, and made a simple plane figure about a straight line. We can also design some beautiful patterns through axisymmetric transformation. In this lesson, we only know some familiar geometric figures from the angle of axial symmetry. Let's study: ① Is a triangle an axisymmetric figure? ② What kind of triangle is an axisymmetric figure?

Some triangles are axisymmetric figures, while others are not.

Question: What kind of triangle is an axisymmetric figure?

A triangle that satisfies the axisymmetric condition is an axisymmetric figure, that is, an axisymmetric figure is a figure in which the two parts can completely overlap after the triangle is folded in half along a straight line.

In this lesson, we will know a triangle with axisymmetric graphics-isosceles triangle.

Two. Introduction of new lesson: Let students make an isosceles triangle through their own thinking.

Make a straight line L, take a point A on L, take a point B outside L, make a symmetrical point C of point B about the straight line L, and connect AB, BC and CA to get an isosceles triangle.

Definition of isosceles triangle: A triangle with two equal sides is called an isosceles triangle. Two equal sides are called waist and the other side is called bottom. The angle between the two waists is called the top angle, and the angle between the buttocks and the waist is called the bottom angle. Students indicate their waist, bottom, top angle and bottom angle in their isosceles triangle.

Thinking:

1. Is the isosceles triangle an axisymmetric figure? Please find its symmetry axis.

2. What is the relationship between the two base angles of an isosceles triangle?

3. Is the straight line where the vertex bisector is located the symmetry axis of the isosceles triangle?

4. Is it a straight line with the center line of the bottom as the symmetry axis of the isosceles triangle? What about the straight line with the height on the bottom edge?

Conclusion: isosceles triangle is an axisymmetric figure, and its symmetry axis is the straight line where the vertex bisector lies. Because the two waists of the isosceles triangle are equal, we can know that the isosceles triangle is an axisymmetric figure, and its symmetry axis is the straight line where the bisector of the vertex lies.

Ask the students to fold their isosceles triangle, find out its symmetry axis and see what the relationship between its two base angles is.

Folding along the bisector of the vertex of the isosceles triangle, it is found that the two sides of the isosceles triangle overlap each other, which shows that the two bottom angles of the isosceles triangle are equal, and it is also known that the bisector of the vertex is both the middle line and the height of the bottom.

From this, we can get the properties of isosceles triangle:

1. The two base angles of an isosceles triangle are equal (abbreviated as "equilateral equilateral angle").

2. The bisector of the top angle of an isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide (usually called "three lines in one").

Inspired by the above folding process, two congruent triangles are obtained by making the symmetry axis of isosceles triangle, so these properties are proved by the congruence of triangles. Now let's write these proof processes.

As shown in the figure on the right, in △ABC, AB=AC, which is the middle line AD of the bottom BC, because

So △ bad△ CAD (SSS).

So ∠ b = ∠ c.

] As shown on the right, in △ABC, AB=AC, which is the bisector AD of the vertex angle ∠BAC, because

So delta is not good △ CAD.

So BD=CD, ∠ BDA = ∠ CDA = ∠ BDC = 90.

[Example 1] As shown in the figure, in △ABC, AB=AC, D is on AC, BD=BC=AD,

Find: the degree of each angle of △ABC.

Analysis: According to the properties of equilateral corners, we can get

∠A=∠ABD，∠ABC=∠C=∠BDC

From ∠BDC=∠A+∠ABD, we can get ∠ ABC = ∠ C = ∠ BDC = 2 ∠ A.

From the sum of the interior angles of the triangle to 180, three interior angles of △ABC can be obtained.

If ∠A is set to X, then ∠ABC and ∠C can both be represented by X, which makes the process simpler.

Solution: Because AB=AC, BD=BC=AD,

So ∠ ABC = ∠ C = ∠ BDC.

∠A=∠ABD (equilateral and equiangular).

Let ∠A=x, then ∠BDC=∠A+∠ABD=2x,

So ∠ ABC = ∠ C = ∠ BDC = 2x.

So in △ABC, there is

∠A+∠ABC+∠C=x+2x+2x= 180，

X = 36。 In △ABC, ∠ A = 35, ∠ ABC = ∠ C = 72.

Let's consolidate what we have learned in this lesson through practice.

Three. Exercise in class: 1. Textbook P5 1 exercise 1, 2, 3.2. Look at the textbook P49 ~ P5 1 and make a summary.

Ⅳ. Class summary

This lesson mainly discusses the properties of isosceles triangle and makes a simple application of the properties. An isosceles triangle is an axisymmetric figure, and its two base angles are equal (equal corners). The symmetry axis of an isosceles triangle is the bisector of its vertex, and the bisector of its vertex is both the middle line and the height of its base.

Through this lesson, we must first understand and master these properties and use them flexibly.

ⅴ. Homework: Problems in textbook P56+0,2,3,4 12.3.

blackboard-writing design

isosceles triangle

First, the design scheme is to make an isosceles triangle.

Second, the nature of the isosceles triangle:

1. equilateral and equilateral

2. Three lines in one

Teaching plan 2: isosceles triangle (2) the eighth grade mathematics teaching goal of People's Education Press

1, understand and master the judgment theorem and inference of isosceles triangle.

2. We can use its nature and judgment to prove that line segments or angles are equal.

Teaching focus

Judgement theorem of isosceles triangle and application of inference

Teaching difficulties

Correctly distinguish the judgment and nature of isosceles triangle, we can use the judgment theorem of isosceles triangle to prove that the line segments are equal.

teaching process

First, review the nature of isosceles triangle.

Second, new funding.

I ask questions and create situations.

Show me the slides. In order to estimate the width of an east-west river, a geological expert chose a tree (point B) on the north bank of the river as the B mark, and then walked 60 degrees from the southeast to point C (point A on the south bank as the mark), and the measured ∠ACB was 30. At this time, geologists know the width of the river by measuring the length of AC.

Students want to know, what is the basis for estimating river width in this way? Use this question to guide students to learn the judgment of isosceles triangle.

II introduction of new curriculum

1. The research content comes from the hypothesis of property theorem and the change of conclusion-in △ABC, if bitter ∠B=∠C, then AB= AC?

What does it matter to make a triangle with two equal angles and then observe the two equal angles?

2. Guide students to write the known content and verify it according to the figure.

2. Summary: Through argumentation, this proposition is true, that is, the judgment theorem of isosceles triangle.

It is emphasized that this theorem is an important basis for transforming the equal relationship of angles in a triangle into the equal relationship of sides, which is similar to the property theorem and can be called "equal angles and equal sides" for short.

4. Guide students to tell the basis of geological expert investigation method in the cited examples.

Examples and exercises

1. as shown in figure 2.

Where △ABC is an isosceles triangle and []

2.① As shown in Figure 3, it is known that in △ABC, AB = AC. ∠ A = 36, then∠ C _ _ _ _ (according to what? ).

(2) As shown in Figure 4, it is known that in △ABC, ∠ A = 36, ∠ C = 72, and △ABC is a _ _ _ _ triangle (according to what? ).

③ If ∠ A = 36, ∠ C = 72, BD bisects ∠ABC and crosses AC to D, it is determined that the isosceles triangle in Figure 5 has _ _ _ _ _.

(4) if known AD = 4 cm, BC _ _ _ _ _ cm.

3. Deduct L _ _ _ _ in the form of questions.

4. In the form of question 2 _ _ _ _ _, draw inferences from one another.

Example: If the bisector of the outer corner of a triangle is parallel to one side of the triangle, it is proved that the triangle is an isosceles triangle.

Analysis: instruct students to make pictures according to the meaning of the question, write what is known, verify and analyze the proof.

Exercise: 5. (l) As shown in Figure 6, in △ABC, the bisectors of AB=AC, ∠ABC and ∠ACB intersect at point F, which triangles are isosceles triangles when passing through F, which ones are DE//BC, AB at point D and AC at point E?

(2) In the above question, if the condition AB=AC is removed and other conditions remain unchanged, is there an isosceles triangle in Figure 6?

Exercise: P53 exercise 1, 2, 3.

IV class summary

1. How many methods are there to judge whether a triangle is an isosceles triangle?

2. How many methods are there to judge whether a triangle is an equilateral triangle?

3. What is the relationship between the property theorem of isosceles triangle and the judgment theorem?

4. Now it is proved that the problem of line segment equality should be considered from several aspects.

Homework: P56 Exercise 12.3 Questions 5 and 6