Signs of similarity and equality of triangles. Properties of similar triangles

A triangle is the simplest closed figure on a plane. When studying a school course in geometry, special attention is paid to the consideration of its properties. In this article, we will reveal the question of signs of similarity and equality of triangles.

Which triangles are called similar and which are equal?

It is logical to assume that the two figures under consideration will be equal to each other if they have all the same angles and lengths of the sides. As for the similarity, the situation here is a bit more complicated. Two triangles will be similar when each corner of one is equal to the corresponding angle of the other, and the sides opposite the equal angles of both figures are proportional. The figure below shows two similar triangles.

Using this figure, we write in the form of mathematical equations the above definition: B = G, A = E, C = F, BA / GE = AC / EF = BC / GF = r, here one Latin letter means an angle, and two letters - side length. The value of r is called the similarity coefficient. It is clear that if r = 1, then not only similar, but equal triangles take place.

Signs of similarity

Speaking about the properties and signs of the similarity and equality of triangles, one should list three main criteria by which it can be determined whether the figures in question are similar or not.

So, two figures will be similar to each other if one of the following conditions is true:

1. Their two angles are equal. Since the sum of the angles of the triangle is equivalent to 180 ^o , the equality of the first two of them automatically means that the third will be the same. Using the figure above, this characteristic can be written as follows: if B = G and A = E, then ABC and GEF are similar. If in this case they are equal on at least one side of both figures, then we can talk about the complete equivalence of the triangles.
2. The two sides are proportional and the angles between them are the same. For example, BA / GE = AC / EF and A = E, then GEF and ABC will be similar. Note that angles A and E lie between the respective proportional sides.
3. All three sides are mutually proportional. In mathematical terms, we get: BA / GE = AC / EF = BC / GF = r, then the figures in question are also similar.

We note again that to prove the similarity it is enough to cite any one of the presented features. It is logical that all others will be executed as well.

Rectangular triangles: when are they similar and when are they equal?

Speaking about the signs of equality and similarity of right-angled triangles, it should be noted immediately that each of them has one corner equal to (90 ^o ).

The latter fact leads to the following wording of the above similarity criteria:

1. If in two right-angled triangles there is only one angle that is not right, then such figures are similar to each other.
2. If the legs are proportional to each other, then the figures will also be similar, since the angle between the legs is straight.
3. Finally, the proportionality of just any two sides for both right triangles is enough to prove their similarity. The reason for this is that the sides of these figures are interconnected by the Pythagorean theorem, so the proportionality of 2 of them leads to proportionality with a similar similarity coefficient for third parties.

As for the equality of triangles with right angles, here it’s easy to remember: if two any elements (right angle is not considered) of both figures are equal, then the figures themselves are equal. For example, these two elements can be an acute angle and a leg, a leg and a hypotenuse, or a hypotenuse and an acute angle.

Properties of triangles like

From the considered signs of similarity and equality of triangles of property, we can distinguish the following:

1. The perimeters of these figures relate to each other as a similarity coefficient, that is, P ₁ / P ₂ = r, where P ₁ and P ₂ are the perimeters of the 1st and 2nd triangles, respectively.
2. The areas of such figures are referred to as the square of the similarity coefficient, that is: S ₁ / S ₂ = r ² , where S ₁ and S ₂ are the areas of the 1st and 2nd triangles, respectively.

Both of these properties can be proved independently. The essence of the proof is to apply a mathematical notation of similarity between the sides of the figures. Here we give only the proof of the first property.

Let a, b, c be the lengths of the sides of one triangle and a ', b', c 'the sides of the second. Since the figures are similar, we can write: a = r * a ', b = r * b', c = r * c '. Now we substitute these expressions with respect to their perimeters, we obtain: P ₁ / P ₂ = (a + b + c) / (a ​​'+ b' + c ') = (r * a' + r * b '+ r * c ') / (a' + b '+ c') = r (a '+ b' + c ') / (a' + b '+ c') = r.

Problem solving example

Signs of similarity and equality of triangles can be used to solve various geometric problems. The following is one example.

There are two triangles. The sides of one of them are 7.6 cm, 4.18 cm and 6.65 cm, and the other 3.5 cm, 2.2 cm and 4 cm. It is necessary to determine whether these figures are similar.

Since the values ​​of three sides are given, we can immediately check the third criterion of similarity. The difficulty here is that you need to understand between which parties to take the relationship. Here we should use simple logical reasoning: the similarity coefficients can be equal if we divide the smallest side of one triangle into the same for the other and so on. Therefore, we have: 4.18 / 2.2 = 1.9; 6.65 / 3.5 = 1.9; 7.6 / 4 = 1.9. After checking the ratio of all sides, we can confidently say that the triangles are similar, since the 3rd criterion is fulfilled.