1. Mastering the "angular" axiom, we can preliminarily apply the "angular" axiom to judge the congruence of two triangles; You should know that the diagonal lines of two sides correspond to two equal triangles and one of them is not necessarily identical.
2. Experience the process of exploring the congruence condition of triangle, and draw mathematical conclusions through practice and induction.
3. Will use the "edge" axiom to prove the congruence of two triangles and master the format of comprehensive proof.
4. By exploring the congruence condition of triangle, cultivate good thinking quality of bold guess and the ability to find problems.
Second, guide self-study.
Question: 1. Which two triangles are called congruent triangles?
A: Two that can completely overlap.
This triangle is called congruent triangles.
2. If △ABC and △A'B'C' satisfy that three sides are equal and three angles are equal, are △ABC and △A'B'C' congruent? Why?
A: △ABC and △A'B'C are the same.
Because two triangles that can completely coincide are congruent.
3. If △ABC and △ A ′ B ′ C ′ satisfy some of the above six conditions, are △ABC and △ A ′ B ′ C ′ identical?
Answer: △ABC and △ A ′ B ′ C ′ satisfy one or two of the above six conditions, and △ABC and △ A ′ B ′ C ′ are not necessarily the same.
△ABC and △ A ′ B ′ C ′ satisfy the trilateral correspondence equation, and △ABC and △ A ′ B ′ C ′ must be congruent.
3. If △ABC and △ A ′ B ′ C ′ satisfy some of the above six conditions, are △ABC and △ A ′ B ′ C ′ identical?
Answer: △ABC and △ A ′ B ′ C ′ satisfy one or two of the above six conditions, and △ABC and △ A ′ B ′ C ′ are not necessarily the same.
△ABC and △ A ′ B ′ C ′ satisfy the trilateral correspondence equation, and △ABC and △ A ′ B ′ C ′ must be congruent.
4. How many cases do 4.△ ABC and △ A ′ B ′ C ′ satisfy three of the above six conditions?
A: Apart from "trilateral equality", there are five other situations:
(2) Two sides and their included angles are equal;
(3) The diagonal lines of two sides and one side are equal;
(4) Two corners and their clamping edges are equivalent;
(5) The opposite sides of two angles and one angle are equal;
(6) The three angles are equal.
(1) Explore the conditions and draw a conclusion.
Question 5: Does it satisfy the congruence of △ABC and △ A ′ B ′ C ′ with equal angles on both sides?
(1) draw a △ABC at will, and then draw a △ a ′ b ′ c ′ so that AB = a ′ b ′, ∠ a = ∠ a ′, and AC = a ′ c ′.
(2) Cut the painted △ A ′ B ′ C ′ and put it on △ABC. Are all equal?
Painting: 1. Draw ∠ da 'e = ∠ a;
2. Intercepts A'B'=AB and a' c' = AC on rays A'D and A'E respectively.
3. connect the line segment B'C'. c '。
△ a ′ b ′ c ′ is a triangle.
(2) Cut the painted △A'B'C and put it on △ABC. They are the same.
Third, the teacher explained (1) the conditions and conclusions of the inquiry.
What law does the result of inquiry 5 reflect?
The method of determining the consistency of two triangles is obtained:
The angle between two sides and them is equivalent to the combination of two triangles.
Can be abbreviated as "corner edge" or "SAS".
Symbolic expression: in △ABC and △A'B'C,
∴△ABC?△a ' b ' c(SAS)。
As shown in the picture, there is a pond. To measure the distance between A and B at both ends of the pond, you can first take a point C on the flat ground that can directly reach A and B, connect it with AC, and extend it to D, so that CD = Ca. The connection BC extends to E, and CE = CB. Connect DE, then the measured DE length is the distance between A and B, why?
It is proved that in △ABO and △DEO,
∴△ABO?△deo(SAS)。
∴ AB=DE (the corresponding sides of congruent triangles are equal).
That is to say, the measured DE length is the distance between a and B.
Inquiry 6: We know that two triangles with equal included angles are congruent. Can △ABC and △ A ′ B ′ C ′ judge congruence with the condition that "both sides are equal to one of the diagonals"? Why?
We can answer by drawing pictures:
(1) draw a △ABC at will, and then draw a △ a ′ b ′ c ′ so that AB = a ′ b ′, ∠ b = ∠ b ′, AC = a ′ c ′, where AB > AC.
(2) Cut the painted △ A ′ B ′ C ′ and put it on △ABC. Are all equal?
We can answer by drawing pictures:
(1) draw a △ABC at will, and then draw a △ a ′ b ′ c ′ so that AB = a ′ b ′, ∠ b = ∠ b ′, AC = a ′ c ′, where AB > AC.
(2) Cut the painted △ A ′ B ′ C ′ and put it on △ABC. Are all equal?
Painting: 1. Draw ∠ db' e = ∠ b;
2. Intercept a' b' = ab on ray b' d.
3. Because the line segment A ′ c ′ is not on the ray B ′ e and A ′ c ′ = AC, there may be two C ′ points on the ray B ′ e, both of which make A ′ c ′ = AC.
Therefore, the △ a ′ b ′ c ′ that satisfies the condition may not be unique.
(2) Cut the painted △ A ′ B ′ C ′ and put it on △ABC. They are not necessarily identical.
We can also answer through experiments:
One end A of two thin sticks, one long and one short, is hinged together with screws, so that the other end of the long stick coincides with the end point B of the ray BC. After properly adjusting the angle between the long stick and the ray, fix the long stick and set up the short stick, so that the other end of the short stick falls at two different positions C and D of the ray respectively.
As shown in the figure, △ABC and △ABD satisfy the condition that the diagonals of two sides and one of them are equal, but △ABC and △ABD are not equal.
Thinking: What law does the result of Inquiry 6 reflect?
Answer: Two triangles with two sides and a diagonal are not necessarily the same.
1. As shown in the figure, two cars start from one end A of the north-south road section AB, travel the same distance to the east and west respectively, and arrive at C and D. Are the distances from C and D to B equal? Why?
Solution: At this time, the distances from C, D to B are equal.
* ba⊥dc
∴∠dab =∠cab = 90°
In △DAB and △CAB,
∴△dab?△cab(SAS)
∴ DB=CB (the corresponding sides of congruent triangles are equal).
In other words, the distances from C and D to B are equal.