Lesson 7: Determining equilateral triangles.
Decision theorem 1: A triangle with three equal angles is an equilateral triangle, paraphrase it as if all three angles are equal in a triangle, then the triangle is an equilateral triangle, write it in mathematical language. As shown in the figure, the triangle ABC has the middle angle A=Angle B=Angle C, and the verification AB=AC=BC.
How do you prove that three sides are equal? Starting from the known conditions, the known conditions have told us that the angle a = angle b, then according to the equiangular equal side then bc=ac, and because the angle a is also equal to ac, the angle c then ab=bc, at this time the three sides are equal, write out the process of solving the problem.
Decision theorem 2: There is an isosceles triangle with an angle of 60° is an equilateral triangle, paraphrase it as if there is an angle of 60° in an isosceles triangle, then this isosceles triangle is an equilateral triangle, write it in mathematical language. In the triangle abc ab=ac, the apex angle is 60°, then the triangle is an equilateral triangle, that is, the three sides are equal.
How do you prove that three sides are equal? Only from the known conditions, according to ab = ac, then the angle b = angle c, which is the property of an isosceles triangle. And because the apex angle is 60°, so according to the inner angle of the triangle and the angle b and angle c are equal to 180° minus 60°, 120° and divided by 2 is equal to 60°, at this time the three angles are all 60°, all equal, so the sides they are opposite are also equal, at this time the triangle ABC is an equilateral triangle.
Write out the process of solving the problem, in theorem 2, there is an isosceles triangle with an angle of 60°, just now the angle of 60° is used as the top angle, when the base angle is 60°, can you prove that it is an equilateral triangle? Next, it is proved that in the second case, in the triangle ABC ab=ac, the angle b=60°, the triangle abc is an equilateral triangle, that is, the three sides are equal.
If we want to verify that the three sides are equal, we need to find out that the three angles they are facing are equal, then we must start from the known conditions, because ab=ac, then the angle b = angle c is also equal to 60°, and according to the sum of the inner angles of the triangle is equal to 180°, then the angle a is also 60°, so the three sides are also equal. Write out the process of solving the problem, so that you can derive the determination theorem of equilateral triangles.
Let's write its decision theorem and property theorem for it.
To sum up, let's look at isosceles triangles with equilateral triangles.
First, what is its nature? Say equilateral to equilateral.
There is also a property called three-in-one.
The third property is an equilateral triangle, in which the three inner angles are all equal, and each angle is sixty degrees.
Its decision theorem is that equiangular to equilateral is equilateral, an isosceles triangle with an angle of sixty degrees is equilateral and equilateral, and a triangle with three equal angles is an equilateral triangle.
The decision theorem about equilateral triangles has been learned.