Proofs of theorems on angles associated with a circle. Problems of Proving Geometric Facts from Hia Proof of Equal Angles

The triangle is the simplest of the polygon types, with three corners and three sides. The sides are formed by segments that are joined together by three points on the plane, thus forming a rigid shape. Equality 2 triangles it is allowed to confirm by several methods.

Instructions

1. If triangles ABC and DEF are two sides equal, and the angle?, Which is located between the two sides of triangle ABC, is equal to the angle ?, which is located between the corresponding sides of 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 angles adjacent to side AB are equal to angles adjacent to side DE, then these triangles are considered equal.

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

Note!
If it is required to confirm the equality of 2 right-angled triangles among themselves, then this can be done using the following equal signs of right-angled triangles: - one of the legs and the hypotenuse; - along two famous legs; - one of the legs and an acute angle adjacent to it; - along the hypotenuse and one of the acute angles. Triangles are acute-angled (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).

Useful advice
In addition to the equality of triangles with each other, these same triangles are similar. Similar triangles are those in which the angles are equal to each other, and the sides of one triangle are proportional to the sides of the other. It should be noted that if two triangles are similar to each other, this does not guarantee their equality. When dividing the similar sides of the triangles by each other, the so-called similarity index is calculated. Also, this indicator can be obtained by dividing the areas of similar triangles.

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 proved using equality tests. The ability to prove the equality and similarity of figures is an important task in all areas of construction.

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Putting the skill into practice

Let's say we have a shape 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 the segments and the angles between them. How to transfer a shape of the same size to a second sheet of paper or double its scale.

We know that a triangle is a shape made up of three line segments called sides that make up corners. Thus, there are six parameters — three sides and three corners — that define this shape.

However, having measured the size of all three sides and angles, it will be difficult to transfer this shape to another surface. In addition, it makes sense to ask the question: is it not enough to know the parameters of two sides and one corner, or just three sides.

Having measured the length of the two sides and between them, then we 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 determine what the minimum number of parameters is enough to know in order to get confidence that the triangles are the same.

Important! Shapes are said to be the same if the line segments that form their sides and the angles are equal to each other. Similar are those figures whose sides and angles are proportional. Thus, equality is similarity with a proportional factor 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 the same if their two 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 the same when all of their sides are of equal length.

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

Proof of 1 feature

For a long time, among the first mathematicians, this criterion was considered an axiom, however, as it turned out, it can be geometrically proven 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 of the same point, then we can say that between these pairs of rays are the same angles (this is given by the condition of the theorem). Let's make 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 put on the ray K 1 M 1 a segment of length KM, which originates at point K. Since, according to the condition, the obtained segment will be equal to the segment K 1 M 1, then points M and M 1 coincide. Similarly, with the segments KN and K 1 N 1. Thus, transferring K 1 M 1 N 1 so that the points K 1 and K coincide, and the two sides overlap, we get a complete coincidence of the figures themselves.

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

Proof of 2 signs

Let us prove the second equality criterion for two corners and a side, based on the first.

Proof of 2 signs

Consider KMN and PRS. K is equal to P, N is equal to S. The side of KN has the same length as PS. It is necessary to prove that KMN and PRS are the same.

Reflect point M relative to the ray KN. The resulting point will be called 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 180 degrees, KLN is equal to PRS, which means that PRS and KLN are the same (similar) on both sides and angle, according to the first attribute.

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

Proof of 3 signs

How to establish that triangles are equal. This follows directly from the proof of the second feature.

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

This means that both figures are similar to each other. But since their sides are the same, then 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, you need to know their properties, whether they are the same:

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

Using the sign of equality of triangles to solve problems

Consequences of the first sign

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

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

Let us prove this simple theorem.

There are two right-angled triangles. One side has a, b, c, where c is the hypotenuse; a, b - legs. The second side has 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, respectively, will be equal on the third basis (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 tutorial on geometry grade 7

Geometry 7 First sign of equality of triangles

Conclusion

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

Geometry as a separate subject begins with students in grade 7. Until that time, they relate to geometric problems of a fairly light form and mainly what can be considered with illustrative examples: the area of a room, a plot of land, the length and height of walls in rooms, flat objects, and so on. At the beginning of studying geometry directly, the first difficulties appear, such as, for example, the concept of a straight line, since there is no way to touch this straight line with your hands. As for triangles, this is the simplest kind of polygons, containing only three corners and three sides.

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The theme of triangles is one of the main important and big topics of the school curriculum in the geometry of 7-9 grades. 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 learn well the signs of equality of figures and be able to use them. Before studying signs, you need to learn determine equality sides and corners of the simplest polygons.

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

∠ α = ∠ β based on the construction of figures.
Given in the statement of the task.
With two parallel lines and the presence of a secant, both internal criss-crossing lines and the corresponding ∠ α = ∠ β can be formed.
Adding (subtracting) equal angles to (from) ∠ α = ∠ β.
Vertical ∠ α and ∠ β are always similar
General ∠ α, simultaneously belonging to ∆ MNK and ∆ MNH .
The bisector divides ∠ α into two equivalent ones.
Adjacent to ∠ 90 °- angle equal to the original.
Adjacent equal angles are equal.
The height forms two adjacent ∠ 90 ° .
In isosceles ∆ MNK at the base ∠ α = ∠ β.
Equal ∆ MNK and ∆ SDH corresponding to ∠ α = ∠ β.
The previously proved equality ∆ MNK and ∆ SDH .

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

3 signs of equality of triangles

Proof of Equality ∆ ABC and ∆A1B1C1 it is very convenient to produce, relying on the main signs the identity of these simplest polygons. There are three such signs. They are very important in solving many geometric problems. Each is worth considering.

The signs listed above are theorems and are proved by the method of superimposing one figure on another, connecting the vertices of the corresponding angles and the beginning of the rays. The proofs of 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 customary for many students at the time of the beginning of the study of the subject. Teenagers get confused about the names of sides, rays, angles.

A little later, another important topic, "Similarity of triangles", appears. The very definition of "similarity" in geometry means similarity of shape with a difference in size. 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, differ, since the second will be larger, and each side is enlarged by the same number of times.

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

The first is about two correspondingly equal angles of the two considered triangular figures.
The second is about the angle and the sides forming it. ∆ MNK that are equal to the corresponding elements ∆ SDH .
The third - indicates the proportionality of all the corresponding sides of the two desired figures.

How can one prove that triangles are similar? It is enough to use one of the above signs and correctly describe the entire process of proving the assignment. Similarity theme ∆ MNK and ∆ SDH it is more easily perceived by schoolchildren on the basis that by the time of its study, students already freely use the designations of elements in geometric constructions, do not get confused in a huge number of names and are able to read drawings.

By completing the extensive topic of triangular geometric shapes, students should already perfectly know how to prove equality ∆ MNK = ∆ SDH on both sides, whether two triangles are 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.