The first sign of the equality of triangles. The second and third signs of the equality of triangles

Among the huge number of polygons, which in fact are a closed non-intersecting broken line, the triangle is the figure with the least number of angles. In other words, this is the simplest polygon. But, despite all its simplicity, this figure contains many mysteries and interesting discoveries, which are covered by a special section of mathematics - geometry. This discipline is being taught in schools since the seventh grade, and the topic "Triangle" is given special attention here. Children not only learn the rules about the figure itself, but also compare them, studying 1, 2 and 3 signs of equality of triangles.

First meeting

One of the first rules with which students are acquainted, sounds like this: the sum of the magnitudes of all angles of the triangle is 180 degrees. To confirm this, it is enough to measure each vertex with the help of a protractor and add all the resulting values. Proceeding from this, for two known quantities it is easy to determine the third. For example : In a triangle, one of the angles is 70 °, and the other - 85 °, what is the value of the third angle?

180 - 85 - 70 = 25.

Answer: 25 °.

Problems can be more complicated if only one value of the angle is specified, and the second value says only how much or how many times it is greater or less.

In the triangle, to determine any of its features, special lines can be drawn, each of which has its own name:

• Height - a perpendicular 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 both inside and outside;
• Median - the line connecting the vertex with the middle of the opposite side;
• The intersection of the medians is the point of gravity, is inside the figure;
• Bisectrix is a line passing from the vertex to the point of intersection with the opposite side, the intersection point of the three bisectors is the center of the inscribed circle.

Simple truths about triangles

Triangles, as, indeed, 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 there is always an angle with a larger value, and vice versa;
• Equal angles lie on equal sides, an isosceles triangle is an example;
• The sum of the internal angles is always 180 °, which has already been demonstrated by the example;
• When one side of the triangle is extended 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;
• Any of the parties is always less than the sum of the other two parties, but more than their difference.

Types of triangles

The next stage of acquaintance is to determine the group to which the represented triangle belongs. Belonging to one form or another depends on the angles of the triangle.

• Equal - 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 bisectrix and the height.
• A regular or equilateral triangle is one with all its sides equal.
• Rectangular: one of its angles is 90 °. In this case, the side opposite this corner is called the hypotenuse, and the other two are called the legs.
• Sharply triangle - all angles are less than 90 °.
• Obtuse-angled - one of the angles is greater than 90 °.

Equality and the similarity of triangles

In the process of learning, not only consider a single figure, but also compare two triangles. And this seemingly simple topic has a lot of rules and theorems on which one can prove that the figures under consideration are equal triangles. The signs of equality of triangles have the following definition: triangles are equal if their respective sides and angles are the same. With this equality, if you superimpose these two figures on each other, all their lines will converge. Also, the figures can be similar, in particular, it concerns almost identical figures, differing only in size. In order to make such a conclusion about the represented triangles, one of the following conditions must be observed:

• Two corners of one figure are equal to two angles of the other;
• The two sides of one are proportional to the 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 will not cause the slightest doubt, it is necessary to have the same values for all elements of both figures, but using the theorems the problem is greatly simplified, and only a few conditions are allowed to prove the equality of the triangles.

The first sign of the equality of triangles

The problems on this topic are solved on the basis of the proof of the theorem, which reads: "If the two sides of the triangle and the angle they form are equal to two sides and the corner of the other triangle, then the figures are also equal."

How does the proof of the theorem for the first sign of the 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 for solving various geometric problems.

How does the theorem "The first sign of the equality of triangles" sound, is described above, but its proof:

• Suppose triangles ABC and A ₁ B ₁ C ₁ have the same sides AB and A ₁ B ₁ and, respectively, 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 △ ABC to △ A ₁ B ₁ C _1, we get the coincidence of all lines and vertices. Hence it follows that these triangles are absolutely identical, and therefore 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 characterization theorem

The second sign of equality is proved similarly, the proof is based on the fact that when the figures are superimposed on each other they completely coincide on all the 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 angles of the second triangle, then these figures are identical, that is, equal."

The third sign and proof

If both the 2's and the 1's of the equality of the triangles touched both the sides and the 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 go into more detail in the very definition of equality. In effect, what does the expression "triangles equal" mean? Identity means that if you superimpose one figure on another, all of their elements will coincide, it 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 at this point 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 when the figure is a mirror image of the first.

Rectangular triangles

In the structure of such triangles, there are always vertices with an angle of 90 °. Therefore, the following assertions are true:

• Triangles with a right angle are equal if the legs of one are identical to the legs of the second;
• Figures are equal if their hypotenuse and one of the legs are equal;
• Such triangles are equal if their legs and acute angle are identical.

This attribute refers to rectangular triangles. To prove the theorem apply the application of the figures to each other, as a result of which the triangles are folded by the legs so that from the two straight lines there is an unfolded angle with the sides CA and CA ₁ .

Practical use

In most cases, in practice, the first sign of equality of triangles is applied. In fact, such a seemingly simple theme of the 7th class 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. With the help of this theorem, it is easy to make the necessary calculations to determine the length of the island in the middle of the river, not crossing it. Either reinforce the fence by placing the bar in the span so that it divides it into two equal triangles, or calculate complex elements of the carpentry work, or when calculating the roof truss system during construction.

The first sign of equality of triangles has wide application in real "adult" life. Although in school years it is this topic for many seems boring and completely unnecessary.