Teaching objectives:
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: Exercise 12.3, Question 1, 2, 3, 4 of textbook P56.
blackboard-writing design
12.3. 1. 1 isosceles triangle
First, the design scheme is to make an isosceles triangle.
Second, the nature of isosceles triangle: 1. Equilateral equilateral corner 2. Three lines in one