Among the huge number of polygons, which are essentially closed disjoint polyline, triangle - a figure with the fewest angles.In other words, it is a simple polygon.But, despite its simplicity, this figure hides a lot of mysteries and interesting discoveries, which highlights a special branch of mathematics - geometry.This discipline in the schools start teaching the seventh grade, and the theme "Triangle" are given special attention.Children not only learn about the rules of the figure, but also compare their learning 1, 2 and 3, a sign of equality of triangles.
Getting
One of the first rules that are familiar with the students, goes something like this: sum of all the angles of a triangle equals 180 degrees.To confirm this, it is enough using a protractor to measure each of the tops and put all the resulting values.For this reason, when the two known values easy to determine the third. example : In one corner of the triangle is 70 °, and the other - 85 °, what is the value of the third angle?
180 - 85 - 70 = 25.
answer to 25 °.
Tasks can be more complex, if you specify only one angle, and about a second value said only on how much or how many times it is more or less.
In the triangle to determine one or another of its features can be carried out special lines, each of which has its own name:
- height - perpendicular line drawn from the vertex to the opposite side;
- all three heights conducted simultaneously in the center of the figure intersect forming orthocenter, which depending on the kind of the triangle can be located both inside and outside;
- median - line connecting the top to the middle of the opposite side;
- median is the point of intersection of its severity, is inside the figure;
- bisector - the line that runs from the top to the point of intersection with the opposite side, the point of intersection of the three bisectors is the center of the inscribed circle.
Simple truths about triangles
Triangles, as, indeed, and all the figures have their own characteristics and properties.As mentioned above, this figure is a simple polygon but with its characteristic features:
- against the longest side is always a corner with a greater magnitude, and vice versa;
- equal sides lie opposite equal angles, example - an isosceles triangle;
- sum of the interior angles is always 180 °, which has already been demonstrated by the example;
- extension at one side of the triangle is formed beyond the outside corner will always be equal to the sum of the angles, not related to him;
- any of the parties is always less than the sum of the other two parties, but most of their differences.
Types of triangles
next stage of dating is to identify the group to which the triangle is shown.Belonging to a particular type depends on the angles of the triangle.
- Isosceles - with two equal sides are called lateral, the third in this case acts as a base figure.The angles at the base of the triangle are the same, and the median drawn from the top, is the bisector and height.
- correct, or an equilateral triangle - is one that has all its sides equal.
- Square: one of its angles is 90 °.In this case, the side opposite this angle is called the hypotenuse, and two others - two sides.
- acute triangle - all the angles less than 90 °.
- Obtuse - one of the corners more than 90 °.
Equality and similarity of triangles
The training is not only considered separately taken shape, but also to compare the two triangles.And this seemingly simple theme has a lot of rules and theorems, which can prove that the figures considered - equal triangles.Signs of equality of triangles have the following definition: the triangles are equal if their corresponding sides and angles are equal.In this equation, if we impose these two figures at each other, all their lines converge.Also, the figure may be similar, in particular, this applies to almost the same figures, differing only in magnitude.In order to make such a conclusion on the submitted triangles, observance of the following conditions:
- two angles of one figure equal to two different angles;
- two sides proportional to two sides of the second triangle and the angles formed by the sides are equal;
- three sides of the second figure is the same as in the first.
course, indisputable equality, which does not cause the slightest doubt, you must have the same values of all elements of both figures, however, using the theory of the problem is greatly simplified, and to prove the congruence of triangles exception of a few conditions.
first sign of equality of triangles
tasks on the subject are solved on the basis of the proof, which goes like this: "If the two sides of the triangle, and the angle which they form, are equal to two sides and angle of another triangle, then the figure is also equala. "
How sound proof of the theorem about the first sign of equality of triangles?Everyone knows that the two segments are equal if they are the same length or circumference are equal if they have the same radius.And in case of the triangles have several attributes with which it can be assumed that the figures are identical, which is very useful in solving various geometric problems.
How sounds theorem "The first sign of the equality of triangles", described above, but the proof:
- For example, triangles ABC and A1V1S1 have the same side of AB and A1B1 and, accordingly, BC and B1C1, and corners,these sides are formed to have the same value, i.e. equal.Then I put it on △ ABC △ A1V1S1 obtain concurrence of lines and vertices.This implies that these triangles are identical and, therefore, are equal.
theory of the "first sign of equality of triangles" is also called "On the two sides and the angle."Actually, this is the essence of it.
Theorem on the second sign
second sign of equality is proved similarly, the proof is based on the fact that the imposition of the figures at each other, they are identical in all the tops and sides.A theorem sounds like this: "If one side and two angles in the formation of which it is involved, to the parties and the two corners of the second triangle, then these figures are identical, ie equal."
third sign and proof of
If both 2 and 1 sign of equality applies to both sides of the triangles, angles and shapes, the third refers only to the parties.Thus, the theorem has the following wording: "If all sides of the triangle are equal to the three sides of the second triangle, the figures are identical."
To prove this theorem, it is necessary to delve in greater detail into the very definition of equality.In fact, what is meant by "equal triangles?"Identity says that if we place a piece to the other, all elements thereof are aligned, this may only be the case when their sides and angles are equal.At the same time, the angle subtended by a party which is the same as the other triangle is equal to the corresponding vertex of the second figure.It should be noted that at this point the proof easily translate into one sign of equality of triangles.If such a sequence is not observed, the equality of triangles is simply impossible except in those cases when the figure is the mirror image of the first.
Right Triangles
The structure of such triangles is always a top with the angle of 90 °.Therefore, the following assertions:
- triangles with right angles are equal, if one identical legs of the second leg of a triangle;
- figures are equal if they are equal to the hypotenuse and one of the legs;
- these triangles are equal if their legs and acute angle identical.
This feature refers to the right-angled triangle.To prove the theorem used the drawings to each other, resulting in the folded legs of the triangles so that the two lines came straight angle with the sides CA and CA1.
Practical application
In most cases, in practice, applied the first sign of equality of triangles.In fact, this seemingly simple theme 7th grade geometry and plane geometry is used to calculate the length, for example, the phone cable without a measurement area, in which it will take place.Using this theorem is easy to make the necessary calculations to determine the length of the island, located in the middle of the river, not swimming across it.Either strengthen the fence by placing the bar in the bay so that it is divided into two equal triangles, or calculate complex elements working in carpentry or in the calculation of the roof truss system during construction.
first sign of equality of triangles has wide application in a real "adult" life.Although the school years is the topic for many seems boring and totally unnecessary.
