Among the huge number of polygons, which in essence are a closed disjoint broken line, a triangle is a figure with the least number of angles. In other words, this is the simplest polygon. But, despite all its simplicity, this figure is fraught with many mysteries and interesting discoveries, which are illuminated by a special section of mathematics - geometry. This discipline in schools is beginning to be taught from the seventh grade, and the topic "Triangle" is given special attention. Children not only learn the rules about the figure itself, but also compare them, studying the 1, 2 and 3 signs of equality of triangles.
First meeting
One of the first rules that students learn about is something like this: the sum of the values ​​of all the angles of a triangle is 180 degrees. To confirm this, it is enough to use a protractor to measure each of the vertices and add up all the resulting values. Based on this, with two known values, it is easy to determine the third. For example : In a triangle, one of the angles is 70 ° and the other is 85 °, what is the size of the third corner?
180 - 85 - 70 = 25.
Answer: 25 °.
Tasks can be more complicated if only one value of the angle is indicated, and the second value is said only by how much or how many times it is more or less.
In the triangle, to determine one or another of its features, special lines can be drawn, each of which has its own name:
- height - a perpendicular straight line drawn from the top to the opposite side;
- all three heights held simultaneously in the center of the figure intersect, forming an orthocenter, which, depending on the type of triangle, can be located both inside and outside;
- median - a line connecting the top with the middle of the opposite side;
- the intersection of medians is the point of its gravity, located inside the figure;
- bisector - a line from the top to the point of intersection with the opposite side, the intersection point of three bisectors is the center of the inscribed circle.
Simple Truths About Triangles
Triangles, like, in fact, all figures, have their own characteristics and properties. As already mentioned, this figure is the simplest polygon, but with its own characteristic features:
- against the longest side is always an angle with a larger magnitude, and vice versa;
- equal angles lie against equal sides, an example is an isosceles triangle;
- the sum of the internal angles is always equal to 180 °, which has already been demonstrated by example;
- when extending one side of the triangle beyond its limits, an external angle is formed, which will always be equal to the sum of the angles that are not adjacent to it;
- either side is always less than the sum of the other two parties, but greater than their difference.
Types of Triangles
The next stage of acquaintance is to determine the group to which the represented triangle belongs. Belonging to a particular species depends on the values ​​of the angles of the triangle.
- Isosceles - with two equal sides, which are called lateral, the third in this case acts as the base of the figure. The angles at the base of such a triangle are the same, and the median drawn from the top is the bisector and height.
- A regular or equilateral triangle is one in which all its sides are equal.
- Rectangular: one of its angles is 90 °. In this case, the side opposite this corner is called hypotenuse, and the other two are called legs.
- An acute-angled triangle - all angles are less than 90 °.
- Obtuse - one of the angles is greater than 90 °.
Equality and likeness of triangles
In the learning process, not only a single figure is considered, but also two triangles are compared. And this seemingly simple topic has a lot of rules and theorems on which it can be proved that the figures in question are equal triangles. Signs of equality of triangles have the following definition: triangles are equal if their respective sides and angles are the same. With such equality, if these two figures are superimposed on each other, all their lines will converge. Also, the figures can be similar, in particular, this applies to almost identical figures, differing only in size. In order to make such a conclusion about the presented triangles, one of the following conditions must be met:
- two corners of one figure are equal to two corners of another;
- two sides of one are proportional to two sides of the second triangle, and the angles formed by the sides are equal;
- the three sides of the second figure are the same as the first.
Of course, for indisputable equality, which does not cause the slightest doubt, it is necessary to have the same values ​​for all elements of both figures, however, using theorems, the problem is greatly simplified, and only a few conditions are allowed to prove the equality of triangles.
The first sign of equality of triangles
The problems on this topic are solved on the basis of the proof of the theorem, which reads as follows: "If the two sides of the triangle and the angle that they form are equal to two sides and the corner of another triangle, then the figures are also equal to each other."
How does the proof of the theorem about the first sign of equality of triangles sound? Everyone knows that two segments are equal if they are of the same length, or circles are equal if they have the same radius. And in the case of triangles, there are several signs, having which, it can be assumed that the figures are identical, which is very convenient to use when solving different geometric problems.
How does the theorem “The first sign of the equality of triangles”, described above, but its proof:
- Suppose the triangles ABC and A 1 B 1 C 1 have the same sides AB and A 1 B 1 and, accordingly, BC and B 1 C 1 , and the angles that are formed by these sides have the same value, that is, they are equal. Then, applying â–ł ABC to â–ł A 1 B 1 C 1, we obtain the coincidence of all lines and vertices. It follows that these triangles are absolutely identical, which means they are equal to each other.
The theorem “The first sign of the equality of triangles” is also called “On two sides and a corner”. Actually, this is its essence.
The second criterion theorem
The second sign of equality is proved in the same way, the proof is based on the fact that when the figures are superimposed on each other, they completely coincide on all vertices and sides. And the theorem sounds like this: "If one side and two angles in the formation of which it participates correspond to the side and two corners of the second triangle, then these figures are identical, that is, equal."
Third sign and proof
If both 2 and 1, the sign of equality of triangles touched both sides and corners of the figure, then the third refers only to the sides. So, the theorem has the following formulation: "If all sides of one triangle are equal to three sides of the second triangle, then the figures are identical."
To prove this theorem, we need to delve deeper into the very definition of equality. In fact, what does the expression “triangles are equal” mean? Identity suggests that if you superimpose one figure on another, all of their elements will coincide, this can only be if their sides and angles are equal. At the same time, the angle opposite one of the sides, which is the same as that of the other triangle, will be equal to the corresponding vertex of the second figure. It should be noted that in this place the proof can easily be translated into 1 sign of equality of triangles. If such a sequence is not observed, the equality of the triangles is simply impossible, except in those cases when the figure is a mirror image of the first.
Right triangles
In the structure of such triangles there are always vertices with an angle of 90 °. Therefore, the following statements are true:
- right angle triangles are equal if the legs of one are identical to the legs of the second;
- figures are equal if their hypotenuses and one of the legs are equal;
- such triangles are equal if their legs and acute angle are identical.
This feature refers to right-angled triangles. To prove the theorem, we apply the application of the figures to each other, as a result of which the triangles are folded with legs so that the developed angle with the sides CA and CA 1 comes out of two straight lines.
Practical use
In most cases, the first sign of equality of triangles is applied in practice. In fact, this seemingly simple topic of class 7 in geometry and planimetry is also used to calculate the length, for example, of a telephone cable without measuring the terrain over which it will pass. Using this theorem, it is easy to make the necessary calculations to determine the length of an island in the middle of a river without crossing it. Either reinforce the fence by placing the bar in the span so that it divides it into two equal triangles, or calculate the complex elements of work in the carpentry, or when calculating the roof truss system during construction.
The first sign of equality of triangles is widely used in real "adult" life. Although in school years, for many, this topic seems boring and completely unnecessary.