What is the second sign of equality of triangles. The third sign of equality of triangles

Two triangles are said to be congruent if they can be overlapped. Figure 1 shows equal triangles ABC and A 1 B 1 C 1. Each of these triangles can be superimposed on another so that they are completely compatible, that is, their vertices and sides are paired together. It is clear that in this case the angles of these triangles will be combined in pairs.

Thus, if two triangles are equal, then the elements (i.e., sides and angles) of one triangle are respectively equal to the elements of the other triangle. Note that in equal triangles against respectively equal sides(i.e. overlapping when superimposed) lie equal angles and back: opposite correspondingly equal angles lie equal sides.

So, for example, in equal triangles ABC and A 1 B 1 C 1, shown in Figure 1, equal angles C and C 1 lie against respectively equal sides AB and A 1 B 1. The equality of triangles ABC and A 1 B 1 C 1 will be denoted as follows: Δ ABC = Δ A 1 B 1 C 1. It turns out that the equality of two triangles can be established by comparing some of their elements.

Theorem 1. The first sign of equality of triangles. If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are equal (Fig. 2).

Proof. Consider triangles ABC and A 1 B 1 C 1, in which AB \u003d A 1 B 1, AC \u003d A 1 C 1 ∠ A \u003d ∠ A 1 (see Fig. 2). Let us prove that Δ ABC = Δ A 1 B 1 C 1 .

Since ∠ A \u003d ∠ A 1, then the triangle ABC can be superimposed on the triangle A 1 B 1 C 1 so that the vertex A is aligned with the vertex A 1, and the sides AB and AC overlap, respectively, on the rays A 1 B 1 and A 1 C one . Since AB \u003d A 1 B 1, AC \u003d A 1 C 1, then side AB will be combined with side A 1 B 1 and side AC - with side A 1 C 1; in particular, points B and B 1 , C and C 1 will coincide. Therefore, the sides BC and B 1 C 1 will be aligned. So, triangles ABC and A 1 B 1 C 1 are completely compatible, which means they are equal.

Theorem 2 is proved similarly by the superposition method.

Theorem 2. The second sign of the equality of triangles. If the side and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle, then such triangles are equal (Fig. 34).

Comment. Based on Theorem 2, Theorem 3 is established.

Theorem 3. The sum of any two interior angles of a triangle is less than 180°.

Theorem 4 follows from the last theorem.

Theorem 4. An external angle of a triangle is greater than any internal angle not adjacent to it.

Theorem 5. The third sign of the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal ().

Example 1 In triangles ABC and DEF (Fig. 4)

∠ A = ∠ E, AB = 20 cm, AC = 18 cm, DE = 18 cm, EF = 20 cm. Compare triangles ABC and DEF. What angle in triangle DEF is equal to angle B?

Solution. These triangles are equal in the first sign. Angle F of triangle DEF is equal to angle B of triangle ABC, since these angles lie opposite the corresponding equal sides DE and AC.

Example 2 Segments AB and CD (Fig. 5) intersect at point O, which is the midpoint of each of them. What is segment BD equal to if segment AC is 6 m?

Solution. Triangles AOC and BOD are equal (by the first criterion): ∠ AOC = ∠ BOD (vertical), AO = OB, CO = OD (by condition).
From the equality of these triangles follows the equality of their sides, i.e. AC = BD. But since, according to the condition, AC = 6 m, then BD = 6 m.

The video lesson "The third sign of the equality of triangles" contains the proof of the theorem, which is a sign of the equality of two triangles on three sides. This theorem is an important part of geometry. It is often used to solve practical problems. Its proof is based on the signs of the equality of triangles already known to students.

The proof of this theorem is complex, therefore, in order to improve the quality of education, to form the ability to prove geometric statements, it is advisable to use this visual aid, which will help to focus students' attention on the material being studied. Also, with the help of animation, visual demonstration of constructions and proofs, it makes it possible to improve the quality of education.

At the beginning of the lesson, the title of the topic is demonstrated and the theorem is formulated that triangles are equal if all sides of one triangle are pairwise equal to all sides of the second triangle. The text of the theorem is shown on the screen and can be written by students in a notebook. Next, we consider the proof of this theorem.

To prove the theorem, triangles ΔABC and ΔA 1 B 1 C 1 are constructed. From the conditions of the theorem it follows that the sides are pairwise equal, that is, AB \u003d A 1 B 1, BC \u003d B 1 C 1 and AC \u003d A 1 C 1. At the beginning of the proof, the imposition of the triangle ΔАВС on ΔА 1 В 1 С 1 is demonstrated so that the vertices A and A 1 , as well as B and B 1 of these triangles are aligned. In this case, the peaks C and C 1 should be located on opposite sides of the superimposed sides AB and A 1 B 1. With this construction, several options for the arrangement of triangle elements are possible:

1. Beam C 1 C lies inside the angle ∠A 1 C 1 B 1 .
2. Beam C 1 C coincides with one of the sides of the angle ∠A 1 C 1 B 1.
3. Ray C 1 C lies outside the angle ∠A 1 C 1 B 1.

Each case must be considered separately, since the proof cannot be the same for all given cases. In the first case, two triangles formed as a result of construction are considered. Since, according to the condition, in these triangles the sides are AC \u003d A 1 C 1, and BC \u003d B 1 C 1, then the resulting triangles ΔB 1 C 1 C and ΔA 1 C 1 are equilateral. Using the studied property of isosceles triangles, we can assert that the angles ∠1 and ∠2 are equal to each other, and also ∠3 and ∠4 are equal. Since these angles are equal, the sum of ∠1 and ∠3, as well as ∠2 and ∠4 will also give equal angles. Therefore, the angles ∠С and ∠С 1 are equal. Having proved this fact, we can reconsider the triangles ΔABC and ΔA 1 B 1 C 1, in which the sides BC \u003d B 1 C 1 and AC \u003d A 1 C 1 according to the condition of the theorem, and it is proved that the angles between them ∠C and ∠C 1 are also equal. Accordingly, these triangles will be equal according to the first criterion for the equality of triangles, which is already known to the students.

In the second case, when the triangles are superimposed, the points C and C 1 lie on one straight line passing through the point B (B 1). In the sum of two triangles ΔABC and ΔA 1 B 1 C 1, a triangle ΔCAC 1 is obtained, in which the two sides AC \u003d A 1 C 1, according to the condition of the theorem, are equal. Accordingly, this triangle is isosceles. In an isosceles triangle with equal sides, there are equal angles, so it can be argued that the angles ∠С=∠С 1. It also follows from the conditions of the theorem that the sides BC and B 1 C 1 are equal to each other, therefore, ΔABC and ΔA 1 B 1 C 1, taking into account the stated facts, are equal to each other according to the first sign of equality of triangles.

The proof in the third case, similarly to the first two, uses the first criterion for the equality of triangles. A geometric figure constructed by imposing triangles, when connected by a segment of vertices C and C 1, is transformed into a triangle ΔB 1 C 1 C. This triangle is isosceles, since its sides B 1 C 1 and B 1 C are equal by condition. And with equal sides in an isosceles triangle, the angles ∠С and ∠С 1 are also equal. Since, according to the condition of the theorem, the sides AC \u003d A 1 C 1 are equal, then the angles at them in the isosceles triangle ΔACS 1 are also equal. Taking into account the fact that the angles ∠С and ∠С 1 are equal, and the angles ∠DCAand ∠DC 1 A are equal to each other, then the angles ∠ACB and ∠AC 1 B are also equal. Considering this fact, to prove the equality of triangles ΔАВС and ΔА 1 В 1 С 1, you can use the first sign of equality of triangles, since the two sides of these triangles are equal in terms of conditions, and the equality of the angles between them is proved in the course of reasoning.

At the end of the video tutorial, an important application of the third criterion for the equality of triangles is demonstrated - the rigidity of a given geometric figure. An example explains what this statement means. As an example of a flexible design, two battens connected with a nail are given. These slats can be moved apart and shifted at any angle. If we attach another one to the rails, connected by ends to the existing rails, then we will get a rigid structure in which it is impossible to change the angle between the rails. Getting a triangle with given sides and other angles is not possible. This corollary of the theorem is of great practical importance. The screen displays engineering structures in which this property of triangles is used.

The video lesson "The third sign of the equality of triangles" makes it easier for the teacher to present new material in a geometry lesson on this topic. Also, the video lesson can be successfully used for distance learning in mathematics, it will help students to understand the complexities of the proof on their own.

The second sign of equality of triangles

If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

MN=PR N=R M=P

As in the proof of the first sign, you need to make sure that this is enough for the triangles to be equal, can they be completely combined?

1. Since MN = PR, then these segments are combined if their end points are combined.

2. Since N = R and M = P , then the rays \(MK\) and \(NK\) overlap the rays \(PT\) and \(RT\), respectively.

3. If the rays coincide, then their points of intersection \(K\) and \(T\) coincide.

4. All the vertices of the triangles are combined, that is, Δ MNK and Δ PRT are completely compatible, which means they are equal.

The third sign of equality of triangles

If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

MN = PR KN = TR MK = PT

Again, let's try to combine the triangles Δ MNK and Δ PRT by overlapping and make sure that the correspondingly equal sides guarantee the equality of the corresponding angles of these triangles and they completely coincide.

Let's combine, for example, identical segments \(MK\) and\(PT\). Let us assume that the points \(N\) and \(R\) do not coincide in this case.

Let \(O\) be the midpoint of the segment \(NR\). According to this information MN = PR , KN = TR . The triangles \(MNR\) and \(KNR\) are isosceles with a common base \(NR\).

Therefore, their medians \(MO\) and \(KO\) are heights, so they are perpendicular to \(NR\). The lines \(MO\) and \(KO\) do not coincide, because the points \(M\), \(K\), \(O\) do not lie on the same line. But through the point \(O\) of the line \(NR\) it is possible to draw only one line perpendicular to it. We have come to a contradiction.

It is proved that the vertices \(N\) and \(R\) must also coincide.

The third sign allows us to call the triangle a very strong, stable figure, sometimes they say that triangle - rigid figure . If the lengths of the sides do not change, then the angles do not change either. For example, a quadrilateral does not have this property. Therefore, various supports and fortifications are made triangular.

But a kind of stability, stability and perfection of the number \ (3 \) people have been evaluating and highlighting for a long time.

Fairy tales talk about it.

There we meet "Three Bears", "Three Winds", "Three Little Pigs", "Three Comrades", "Three Brothers", "Three Lucky Men", "Three Craftsmen", "Three Princes", "Three Friends", "Three hero", etc.

There are given “three attempts”, “three advice”, “three instructions”, “three meetings”, “three wishes” are fulfilled, you need to endure “three days”, “three nights”, “three years”, go through “three states ”,“ three underground kingdoms ”, endure“ three trials ”, swim through the“ three seas ”.

Among the huge number of polygons, which are essentially a closed non-intersecting 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 covered by a special section of mathematics - geometry. This discipline in schools begins to be taught from 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 sign of equality of triangles.

First meeting

One of the first rules that students learn 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 measure each of the vertices with the help of a protractor 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 value of the third angle?

180 - 85 - 70 = 25.

Answer: 25°.

Tasks can be even more complex 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 larger or smaller.

In a triangle, to determine one or another 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 drawn simultaneously intersect in the center of the figure, forming the orthocenter, which, depending on the type of triangle, can be both inside and outside;
• median - a line connecting the top with the middle of the opposite side;
• the intersection of the medians is the point of its gravity, located inside the figure;
• bisector - a line passing from the vertex to the point of intersection with the opposite side, the point of intersection of three bisectors is the center of the inscribed circle.

Simple Truths About Triangles

Triangles, like, in fact, all shapes, have their own characteristics and properties. As already mentioned, this figure is the simplest polygon, but with its own characteristic features:

• opposite the longest side there is always an angle with a larger value, and vice versa;
• equal angles lie against equal sides, an example of this is an isosceles triangle;
• the sum of the interior angles is always 180°, which has already been demonstrated with an example;
• when one side of a 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;
• either side is always less than the sum of the other two sides, but greater than their difference.

Types of triangles

The next stage of acquaintance is to determine the group to which the presented triangle belongs. Belonging to a particular species depends on the magnitude 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 vertex is the bisector and height.
• A regular or equilateral triangle is one in which all of its sides are equal.
• Rectangular: one of its angles is 90°. In this case, the side opposite this angle is called the hypotenuse, and the other two are the legs.
• Acute triangle - all angles are less than 90°.
• Obtuse - one of the angles is greater than 90°.

Equality and similarity of triangles

In the learning process, they not only consider a single figure, but also compare two triangles. And this seemingly simple topic has a lot of rules and theorems by which you can prove that the figures in question are equal triangles. Triangles are equal if their corresponding sides and angles are the same. With this equality, if you put these two figures on top of each other, all their lines will converge. Also, figures can be similar, in particular, this applies to almost identical figures that differ only in size. In order to make such a conclusion about the triangles presented, one of the following conditions must be met:

• two angles of one figure are equal to two angles 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 those of the first.

Of course, for indisputable equality, which will not cause the slightest doubt, it is necessary to have the same values ​​\u200b\u200bof all the elements of both figures, however, using theorems, the task is greatly simplified, and only a few conditions are allowed to prove the equality of triangles.

The first sign of equality of triangles

Problems on this topic are solved on the basis of the proof of the theorem, which sounds like this: "If two sides of a triangle and the angle that they form are equal to two sides and an angle of another triangle, then the figures are also equal to each other."

How does the proof of the theorem about the first criterion for the equality of triangles sound? Everyone knows that two segments are equal if they have 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, we can assume that the figures are identical, which is very convenient to use when solving various geometric problems.

How the theorem “The first sign of equality of triangles” sounds is described above, but here is its proof:

• Suppose 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, by superimposing △ ABC on △ A 1 B 1 C 1, we get the coincidence of all lines and vertices. It follows from this that these triangles are absolutely identical, which means they are equal to each other.

The theorem "The first criterion for the equality of triangles" is also called "By two sides and an angle." Actually, this is its essence.

Second feature 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 in 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 angles of the second triangle, then these figures are identical, that is, equal."

Third Sign and Proof

If both 2 and 1 sign of equality of triangles concerned both the sides and corners of the figure, then the 3rd applies 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 into the very definition of equality in more detail. Essentially, what does the expression "triangles are equal" mean? Identity says that if you superimpose one figure on another, all their elements will coincide, this can only be the case when 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 be easily translated to 1 criterion for the equality of triangles. If such a sequence is not observed, the equality of triangles is simply impossible, except in cases where 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:

• 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 hypotenuses and one of the legs are equal;
• such triangles are congruent if their legs and acute angle are identical.

This sign refers to To prove the theorem, the figures are applied to each other, as a result of which the triangles are folded with legs so that two straight lines come out with sides CA and CA 1.

Practical use

In most cases, in practice, the first sign of the equality of triangles is used. In fact, such a seemingly simple topic of grade 7 in geometry and planimetry is also used to calculate the length, for example, of a telephone cable without measuring the terrain along 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 swimming across it. Either strengthen the fence by placing a plank in the span so that it divides it into two equal triangles, or calculate the complex elements of work in 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, this particular topic seems boring and completely unnecessary for many.