What is the equality of triangles. How to establish and prove that triangles are congruent. Signs of equality of right triangles

>>Geometry: The third sign of equality of triangles. Complete lessons

LESSON TOPIC: The third sign of equality of triangles.

Lesson objectives:

• Educational – repetition, generalization and testing of knowledge on the topic: “Signs of equality of triangles”; development of basic skills.
• Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
• Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.

Lesson objectives:

• Develop skills in constructing triangles using a scale ruler, protractor and drawing triangle.
• Test students' problem-solving skills.

Lesson plan:

1. From the history of mathematics.
2. Signs of equality of triangles.
3. Updating basic knowledge.
4. Right triangles.

From the history of mathematics.
The right triangle occupies a place of honor in Babylonian geometry, and mention of it is often found in the Ahmes papyrus.

The term hypotenuse comes from the Greek hypoteinsa, meaning stretching under something, contracting. The word originates from the image of ancient Egyptian harps, on which the strings were stretched over the ends of two mutually perpendicular stands.

The term leg comes from the Greek word “kathetos”, which meant plumb line, perpendicular. In the Middle Ages, the word cathet meant height right triangle, while its other sides were called the hypotenuse, respectively the base. In the 17th century, the word cathet began to be used in the modern sense and became widespread starting from the 18th century.

Euclid uses the expressions:

“sides concluding a right angle” - for legs;

“the side subtending a right angle” - for the hypotenuse.

First, we need to refresh our memory of the previous signs of equality of triangles. And so let's start with the first one.

1st sign of equality of triangles.

There are three signs of equality for two triangles. In this article we will consider them in the form of theorems, and also provide their proofs. To do this, remember that the figures will be equal in the case when they completely overlap each other.

First sign

Theorem 1

Two triangles will be equal if two sides and the angle between them in one of the triangles are equal to two sides and the angle lying between them in the other.

Proof.

Consider two triangles $ABC$ and $A"B"C"$, in which $AB=A"B"$, $AC=A"C"$ and $∠A=∠A"$ (Fig. 1).

Let us combine the heights $A$ and $A"$ of these triangles. Since the angles at these vertices are equal to each other, the sides $AB$ and $AC$ will overlap, respectively, the rays $A"B"$ and $A"C" $. Since these sides are pairwise equal, the sides $AB$ and $AC$, respectively, coincide with the sides $A"B"$ and $A"C"$, and therefore the vertices $B$ and $B"$. , $C$ and $C"$ will be the same.

Therefore, side BC will completely coincide with side $B"C"$. This means that the triangles will completely overlap each other, which means they are equal.

The theorem is proven.

Second sign

Theorem 2

Two triangles will be equal if two angles and their common side of one of the triangles are equal to two angles and their common side in a different.

Proof.

Let's consider two triangles $ABC$ and $A"B"C"$, in which $AC=A"C"$ and $∠A=∠A"$, $∠C=∠C"$ (Fig. 2).

Let us combine the sides $AC$ and $A"C"$ of these triangles, so that the heights $B$ and $B"$ will lie on the same side of it. Since the angles at these sides are pairwise equal to each other, then the sides $AB$ and $BC$ will overlap, respectively, the rays $A"B"$ and $B"C"$. Consequently, both the point $B$ and the point $B"$ will be the intersection points of the combined rays (that is, for example, the rays $AB$ and $BC$). Since the rays can have only one intersection point, the point $B$ will coincide with the point $B"$. This means that the triangles will completely overlap each other, which means they are equal.

The theorem is proven.

Third sign

Theorem 3

Two triangles will be equal if three sides of one of the triangles are equal to three sides of the other.

Proof.

Consider two triangles $ABC$ and $A"B"C"$, in which $AC=A"C"$, $AB=A"B"$ and $BC=B"C"$ (Fig. 3).

Proof.

Let us combine the sides $AC$ and $A"C"$ of these triangles, so that the heights $B$ and $B"$ will lie on opposite sides of it. Next we will consider three different cases of the resulting arrangement of these vertices. We will consider them in the pictures.

First case:

Since $AB=A"B"$, the equality $∠ABB"=∠AB"B$ will be true. Likewise, $∠BB"C=∠B"BC$. Then, as a sum, we get $∠B=∠B"$

Second case:

Since $AB=A"B"$, the equality $∠ABB"=∠AB"B$ will be true. Likewise, $∠BB"C=∠B"BC$. Then, as a difference, we get $∠B=∠B"$

Therefore, by Theorem 1, these triangles are equal.

Third case:

Since $BC=B"C"$, the equality $∠ABC=∠AB"C$ will be true

Therefore, by Theorem 1, these triangles are equal.

The theorem is proven.

Sample tasks

Example 1

Prove the equality of the triangles in the figure below

Signs of equality of triangles

Triangles whose corresponding sides are equal are called congruent.

Theorem (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 congruent.

Theorem (second criterion for the 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.

Theorem (the third criterion for the equality of triangles).
If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Signs of similarity of triangles

Triangles whose angles are equal and whose similar sides are proportional are called similar: , where is the similarity coefficient.

I sign of similarity of triangles. If two angles of one triangle are respectively equal to two angles of another, then these triangles are similar.

II sign of similarity of triangles. If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

III sign of similarity of triangles. If two sides of one triangle are proportional to two sides of another triangle, and the angles between these sides are equal, then the triangles are similar.

Among the huge number of polygons, which are essentially a closed, non-intersecting broken line, the triangle is the figure with the fewest 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 special section mathematics - geometry. This discipline begins to be taught in schools from the seventh grade, and the topic “Triangle” is given here Special attention. Children not only learn the rules about the figure itself, but also compare them by studying the 1st, 2nd and 3rd sign of equality of triangles.

First meeting

One of the first rules that schoolchildren learn goes something like this: the sum of the values ​​of all the angles of a triangle equals 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 quantities 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 angle?

180 - 85 - 70 = 25.

Answer: 25°.

Problems can be even more complex if only one angle value is specified, and the second value is only told how much or how many times it is larger or smaller.

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

• height - a perpendicular straight line drawn from the vertex to the opposite side;
• all three heights, drawn simultaneously, intersect in the center of the figure, forming an orthocenter, which, depending on the type of triangle, can be located both inside and outside;
• median - a line connecting the vertex to the middle of the opposite side;
• the intersection of the medians is the point of its gravity, located inside the figure;
• bisector - a line running from a 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 all shapes, have their own characteristics and properties. As already mentioned, this figure is the simplest polygon, but with its own characteristic features:

• the angle with the larger value always lies opposite the longest side, and vice versa;
• Equal angles lie opposite equal sides, an example of this is an isosceles triangle;
• the sum of internal angles is always equal to 180°, which has already been demonstrated by 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 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 one type or another depends on the size 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 the height.
• Correct, 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 angle is called the hypotenuse, and the other two are called the legs.
• Acute triangle - all angles are less than 90°.
• Obtuse - one of the angles greater than 90°.

Equality and similarity of triangles

During the learning process, they not only consider a single figure, but also compare two triangles. And this one, it would seem, simple theme has a lot of rules and theorems by which it can be proven that the figures in question are equal triangles. The criteria for the equality of triangles have the following definition: triangles are equal if their corresponding sides and angles are the same. With such equality, if you superimpose these two figures on top of each other, all their lines will converge. Also, the 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 presented triangles, one of the following conditions must be met:

• two angles of one figure are equal to two angles of another;
• the two sides of one are proportional to the two sides of the second triangle, and the magnitudes of the angles formed by the sides are equal;
• three sides of the second figure are the same as the first.

Of course, for indisputable equality, which will not raise the slightest doubt, it is necessary to have same values all elements of both figures, however, with the use of 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 based on the proof of the theorem, which goes like this: “If two sides of a triangle and the angle they form are equal to two sides and the angle of another triangle, then the figures are also equal to each other.”

What does the proof of the theorem about the first sign of equality of triangles sound like? Everyone knows that two segments are equal if they are 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 different geometric problems.

What the theorem “The first sign of equality of triangles” sounds like 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 formed by these sides have the same size, that is, they are equal. Then, by superimposing △ ABC on △ A 1 B 1 C 1, we obtain the coincidence of all lines and vertices. It follows that these triangles are absolutely identical, and therefore equal to each other.

The theorem “The first sign of equality of triangles” is also called “On two sides and an angle.” Actually, this is its essence.

Theorem about the second sign

The second sign of equality is proved in a similar 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 angles of the second triangle, then these figures are identical, that is, equal.”

Third sign and proof

If both 2 and 1 signs of equality of triangles concerned both the sides and corners of the figure, then the 3rd one 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 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 to 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 to 1 criterion for the equality of triangles. If such a sequence is not observed, equality of triangles is simply impossible, except in cases where the figure is mirror image first.

Right Triangles

The structure of such triangles always has vertices with an angle of 90°. Therefore, the following statements are true:

• triangles with right angles are equal if the legs of one are identical to the legs of the second;
• figures are equal if their hypotenuses and one of their legs are equal;
• such triangles are equal if their legs and sharp corner identical.

This sign refers to To prove the theorem, they apply the application of figures to each other, as a result of which the triangles are folded by 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 equality of triangles is used. In fact, such a seemingly simple 7th grade topic on geometry and planimetry is also used to calculate the length, for example, of a telephone cable without measuring the area through which it will pass. Using this theorem, it is easy to make the necessary calculations to determine the length of an island located in the middle of the river without swimming across to it. Or strengthen the fence by placing the bar in the span so that it divides it into two equal triangle, or to calculate complex elements of work in carpentry, or when calculating a roof truss system during construction.

The first sign of equality of triangles is widely used in real “adult” life. Although during school years this particular topic seems boring and completely unnecessary for many.

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 on illustrative examples: area of ​​the room, land, 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.

In contact with

Classmates

The theme of triangles is one of the main ones important and big topics school curriculum in geometry 7-9 grades. Having mastered it well, it is possible to decide very complex tasks. 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 the 7th grade 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 by capital letters with 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 one appears important topic"Similarity of triangles." 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 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 are already fluent in using the notation of elements in geometric constructions, do not get confused by a huge number of names and know how to read drawings.

Concluding the passage of the extensive topic of triangular geometric shapes, students should already know perfectly how to prove 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.