A triangle is the simplest type of polygon, having three angles and three sides. The sides are formed by segments that are connected to each other by three points on the plane, thus forming a rigid shape. Equality 2 triangles can be confirmed by several methods.

Instructions

1. If triangles ABC and DEF are two sides equal, and the angle?, the one placed between the two sides of the triangle ABC, is equal to the angle?, the one placed between the corresponding sides of the triangle DEF, then these two triangles are equal to each other.

2. If triangles ABC and DEF side AB is equal to side DE, and the angles adjacent to side AB are equal to the angles adjacent to side DE, then these triangles are considered equal.

3. If triangles ABC sides AB, BC and CD are equal to their corresponding sides of triangle DEF, then these triangles are congruent.

Note!
If you need to confirm the equality of 2 right triangles, then this can be done using the following equal signs of right triangles: - one of the legs and the hypotenuse; - two famous legs; - one of the legs and the adjacent acute angle; - along the hypotenuse and one of the acute angles. Triangles are acute (if all its angles are less than 90 degrees), obtuse (if one of its angles is greater than 90 degrees), equilateral and isosceles (if its two sides are equal).

Helpful advice
In addition to the triangles being equal to each other, the same triangles are similar. Similar triangles are those whose angles are equal to each other, and the sides of one triangle are proportional to the sides of another. It is worth noting that if two triangles are similar to each other, this does not guarantee their equality. When similar sides of triangles are divided by each other, the so-called similarity index is calculated. This indicator can also be obtained by dividing the areas of similar triangles.

Geometry as a separate subject begins for schoolchildren in the 7th grade. Until this time, they concern geometric problems of a fairly light form and mainly what can be considered with visual examples: the area of ​​a room, a plot of land, the length and height of walls in rooms, flat objects, etc. At the beginning of studying geometry itself, the first difficulties appear, such as, for example, the concept of a straight line, since it is not possible to touch this straight line with your hands. As for triangles, this is the simplest type of polygon, containing only three angles and three sides.

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The theme of triangles is one of the main ones important and large topics of the school curriculum in geometry for grades 7–9. Having mastered it well, it is possible to solve very complex problems. In this case, you can initially consider a completely different geometric figure, and then divide it for convenience into suitable triangular parts.

To work on the proof of equality ∆ ABC And ∆A1B1C1 You need to thoroughly understand the signs of equality of figures and be able to use them. Before studying the signs, you need to learn determine equality sides and angles of the simplest polygons.

To prove that the angles of triangles are equal, the following options will help:

1. ∠ α = ∠ β based on the construction of the figures.
2. Given in the task conditions.
3. With two parallel lines and the presence of a secant, both internal cross-lying and corresponding ones can be formed ∠ α = ∠ β.
4. By adding (subtracting) to (from) ∠ α = ∠ β equal angles.
5. Vertical ∠ α and ∠ β are always similar
6. General ∠ α, simultaneously belonging to ∆MNK And ∆MNH .
7. The bisector divides ∠ α into two equal parts.
8. Adjacent to ∠ 90°- angle equal to the original one.
9. Adjacent equal angles are equal.
10. The height forms two adjacent ∠ 90° .
11. In isosceles ∆MNK at the base ∠ α = ∠ β.
12. Equal ∆MNK And ∆SDH corresponding ∠ α = ∠ β.
13. Previously proven equality ∆MNK And ∆SDH .

This is interesting: How to find the perimeter of a triangle.

3 signs that triangles are equal

Proof of equality ∆ ABC And ∆A1B1C1 very convenient to produce, based on basic signs the identity of these simplest polygons. There are three such signs. They are very important in solving many geometric problems. Each one is worth considering.

The characteristics listed above are theorems and are proven by the method of superimposing one figure on another, connecting the vertices of the corresponding angles and the beginning of the rays. Proofs for the equality of triangles in grade 7 are described in a very accessible form, but are difficult for schoolchildren to study in practice, since they contain a large number of elements indicated in capital Latin letters. This is not entirely familiar to many students when they start studying the subject. Teenagers get confused about the names of sides, rays, and angles.

A little later, another important topic “Similarity of triangles” appears. The very definition of “similarity” in geometry means similarity of shape with different sizes. For example, you can take two squares, the first with a side of 4 cm, and the second 10 cm. These types of quadrangles will be similar and, at the same time, have a difference, since the second will be larger, with each side increased by the same number of times.

In considering the topic of similarity, 3 signs are also given:

• The first is about the two correspondingly equal angles of the two triangular figures in question.
• The second is about the angle and the sides that form it ∆MNK, which are equal to the corresponding elements ∆SDH .
• The third one indicates the proportionality of all corresponding sides of the two desired figures.

How can you prove that the triangles are similar? It is enough to use one of the above signs and correctly describe the entire process of proving the task. Theme of similarity ∆MNK And ∆SDH is easier to perceive by schoolchildren based on the fact that by the time of studying it, students already freely use the designations of elements in geometric constructions, do not get confused in a huge number of names and know how to read drawings.

By completing the extensive topic of triangular geometric figures, students should already know perfectly how to prove the equality ∆MNK = ∆SDH on two sides, set the two triangles to be equal or not. Considering that a polygon with exactly three angles is one of the most important geometric figures, you should take the material seriously, paying special attention to even the smallest facts of the theory.

From ancient times to this day, the search for signs of equality of figures is considered a basic task, which is the basis of the foundations of geometry; hundreds of theorems are proven using equality tests. The ability to prove equality and similarity of figures is an important task in all areas of construction.

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Putting the skill into practice

Suppose we have a figure drawn on a piece of paper. At the same time, we have a ruler and a protractor with which we can measure the lengths of segments and the angles between them. How to transfer a figure of the same size to a second sheet of paper or double its scale.

We know that a triangle is a figure made up of three segments called sides that form the angles. Thus, there are six parameters - three sides and three angles - that define this figure.

However, having measured the size of all three sides and angles, transferring this figure to another surface will be a difficult task. In addition, it makes sense to ask the question: wouldn’t it be enough to know the parameters of two sides and one angle, or just three sides?

Having measured the length of the two sides and between them, we will then put this angle on a new piece of paper, so we can completely recreate the triangle. Let's figure out how to do this, learn how to prove the signs by which they can be considered the same, and decide what minimum number of parameters is enough to know in order to be confident that the triangles are the same.

Important! Figures are called identical if the segments forming their sides and angles are equal to each other. Similar figures are those whose sides and angles are proportional. Thus, equality is similarity with a proportionality coefficient of 1.

What are the signs of equality of triangles? Let’s give their definition:

• the first sign of equality: two triangles can be considered identical if two of their sides are equal, as well as the angle between them.
• the second sign of equality of triangles: two triangles will be the same if two angles are the same, as well as the corresponding side between them.
• third sign of equality of triangles : Triangles can be considered identical when all their sides are of equal length.

How to prove that triangles are congruent. Let us give a proof of the equality of triangles.

Evidence of 1 sign

For a long time, among the first mathematicians this sign was considered an axiom, however, as it turned out, it can be proven geometrically based on more basic axioms.

Consider two triangles - KMN and K 1 M 1 N 1 . The KM side has the same length as K 1 M 1, and KN = K 1 N 1. And the angle MKN is equal to the angles KMN and M 1 K 1 N 1.

If we consider KM and K 1 M 1, KN and K 1 N 1 as two rays that come out from the same point, then we can say that the angles between these pairs of rays are the same (this is specified by the condition of the theorem). Let us carry out a parallel transfer of rays K 1 M 1 and K 1 N 1 from point K 1 to point K. As a result of this transfer, rays K 1 M 1 and K 1 N 1 will completely coincide. Let us plot on the ray K 1 M 1 a segment of length KM, originating at point K. Since, by condition, the resulting segment will be equal to the segment K 1 M 1, then the points M and M 1 coincide. Similarly with the segments KN and K 1 N 1. Thus, by transferring K 1 M 1 N 1 so that the points K 1 and K coincide, and the two sides overlap, we obtain a complete coincidence of the figures themselves.

Important! On the Internet there are proofs of the equality of triangles by two sides and an angle using algebraic and trigonometric identities with numerical values ​​of the sides and angles. However, historically and mathematically, this theorem was formulated long before algebra and earlier than trigonometry. To prove this feature of the theorem, it is incorrect to use anything other than the basic axioms.

Evidence 2 signs

Let us prove the second sign of equality in two angles and a side, based on the first.

Evidence 2 signs

Let's consider KMN and PRS. K is equal to P, N is equal to S. Side KN has the same length as PS. It is necessary to prove that KMN and PRS are the same.

Let us reflect the point M relative to the ray KN. Let's call the resulting point L. In this case, the length of the side KM = KL. NKL is equal to PRS. KNL is equal to RSP.

Since the sum of the angles is equal to 180 degrees, then KLN is equal to PRS, which means PRS and KLN are the same (similar) on both sides and the angle, according to the first sign.

But, since KNL is equal to KMN, then KMN and PRS are two identical figures.

Evidence 3 signs

How to determine that triangles are congruent. This follows directly from the proof of the second feature.

Length KN = PS. Since K = P, N = S, KL=KM, and KN = KS, MN=ML, then:

This means that both figures are similar to each other. But since their sides are the same, they are also equal.

Many consequences follow from the signs of equality and similarity. One of them is that in order to determine whether two triangles are equal or not, it is necessary to know their properties, whether they are the same:

• all three sides;
• both sides and the angle between them;
• both angles and the side between them.

Using the triangle equality test to solve problems

Consequences of the first sign

In the course of the proof, one can come to a number of interesting and useful consequences.

1. . The fact that the point of intersection of the diagonals of a parallelogram divides them into two identical parts is a consequence of the signs of equality and is quite amenable to proof. The sides of the additional triangle (with a mirror construction, as in the proofs that we performed) are the sides of the main one (the sides of the parallelogram).
2. If there are two right triangles that have the same acute angles, then they are similar. If the leg of the first is equal to the leg of the second, then they are equal. This is quite easy to understand - all right triangles have a right angle. Therefore, the signs of equality are simpler for them.
3. Two triangles with right angles, in which two legs have the same length, can be considered identical. This is due to the fact that the angle between the two legs is always 90 degrees. Therefore, according to the first criterion (by two sides and the angle between them), all triangles with right angles and identical legs are equal.
4. If there are two right triangles, and their one leg and hypotenuse are equal, then the triangles are the same.

Let's prove this simple theorem.

There are two right triangles. One has sides a, b, c, where c is the hypotenuse; a, b - legs. The second has sides n, m, l, where l is the hypotenuse; m, n - legs.

According to the Pythagorean theorem, one of the legs is equal to:

;

.

Thus, if n = a, l = c (equality of legs and hypotenuses), respectively, the second legs will be equal. The figures, accordingly, will be equal according to the third characteristic (on three sides).

Let us note one more important consequence. If there are two equal triangles, and they are similar with a similarity coefficient k, that is, the pairwise ratios of all their sides are equal to k, then the ratio of their areas is equal to k2.

The first sign of equality of triangles. Video lesson on geometry 7th grade

Geometry 7 The first sign of equality of triangles

Conclusion

The topic we have discussed will help any student better understand basic geometric concepts and improve their skills in the interesting world of mathematics.

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