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.
Classmates
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:
This is interesting: How to find the perimeter of a triangle.
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, and each side will be increased by the same number of times.
In considering the topic of similarity, 3 signs are also given:
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, mastering the material should be taken seriously, paying special attention to even minor facts of the theory.
Instructions
If triangles ABC and DEF have side AB 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 congruent.
If triangles ABC have sides AB, BC and CD equal to their corresponding sides of triangle DEF, then these triangles are congruent.
Please note
If you need to prove the equality of two right triangles, this can be done using the following signs of equality of right triangles:
One of the legs and the hypotenuse;
- on two known sides;
- along one of the legs and the acute angle adjacent to it;
- 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 more than 90 degrees), equilateral and isosceles (if two of its sides are equal).
Useful 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 the other. It is worth noting that if two triangles are similar to each other, this does not guarantee their equality. When dividing similar sides of triangles by each other, the so-called similarity coefficient is calculated. This coefficient can also be obtained by dividing the areas of similar triangles.
Sources:
Two triangles are equal if all the elements of one are equal to the elements of the other. But it is not necessary to know all the sizes of the triangles to draw a conclusion about their equality. It is enough to have certain sets of parameters for given figures.
Instructions
If it is known that two sides of one triangle are equal to another and the angles between these sides are equal, then the triangles in question are congruent. To prove it, align the vertices of equal angles of two figures. Continue layering. From the resulting point common to the two triangles, direct one side of the corner of the overlapping triangle along the corresponding side of the lower figure. By condition, these two sides are equal. This means that the ends of the segments will coincide. Consequently, another pair of vertices in the given triangles has coincided. The directions of the second sides of the angle from which it began will coincide due to the equality of these angles. And since these sides are equal, the last vertex will overlap. A single straight line can be drawn between two points. Therefore, the third sides of the two triangles will coincide. You have obtained two completely matching figures and the proven first sign that triangles are equal.
If a side and two adjacent angles in one triangle are equal to the corresponding angles in another triangle, then these two triangles are congruent. To prove the correctness of this statement, superimpose two figures, aligning the vertices of equal angles with equal sides. Due to the equality of the angles, the directions of the second and third sides will coincide and the place of their intersection will be unambiguously determined, that is, the third vertex of the first of the triangles will necessarily coincide with a similar point of the second. The second criterion for the equality of triangles has been proven.
1) on two sides and the angle between them
Proof:
Let triangles ABC and A 1 B 1 C 1 have angle A equal to angle A 1, AB equal to A 1 B 1, AC equal to A 1 C 1. Let's prove that the triangles are congruent.
Let's impose triangle ABC (or symmetrical to it) onto triangle A 1 B 1 C 1 so that angle A is aligned with angle A 1 . Since AB=A 1 B 1, and AC=A 1 C 1, then B will coincide with B 1, and C will coincide with C 1. This means that triangle A 1 B 1 C 1 coincides with triangle ABC, and therefore is equal to triangle ABC.
The theorem has been proven.
2) along the side and adjacent corners
Proof:
Let ABC and A 1 B 1 C 1 be two triangles in which AB is equal to A 1 B 1, angle A is equal to angle A 1, and angle B is equal to angle B 1. Let's prove that they are equal.
Let's impose triangle ABC (or symmetrical to it) onto the triangle A 1 B 1 C 1 so that AB coincides with A 1 B 1. Since ∠BAC =∠B 1 A 1 C 1 and ∠ABC=∠A 1 B 1 C 1, then ray AC will coincide with A 1 C 1, and BC will coincide with B 1 C 1. It follows that vertex C coincides with C 1. This means that triangle A 1 B 1 C 1 coincides with triangle ABC, and therefore is equal to triangle ABC.
The theorem has been proven.
3) on three sides
Proof :
Consider triangles ABC and A l B l C 1, in which AB = A 1 B 1, BC = B l C 1 CA = C 1 A 1. Let us prove that ΔАВС =ΔA 1 B 1 C 1.
Let's apply triangle ABC (or symmetrical to it) to the triangle A 1 B 1 C 1 so that vertex A is aligned with vertex A 1 , vertex B is aligned with vertex B 1 , and vertices C and C 1 are on opposite sides of straight line A 1 B 1 . Let's consider 3 cases:
1) Ray C 1 C passes inside the angle A 1 C 1 B 1. Since, according to the conditions of the theorem, the sides AC and A 1 C 1, BC and B 1 C 1 are equal, then the triangles A 1 C 1 C and B 1 C 1 C are isosceles. By the theorem on the property of the angles of an isosceles triangle, ∠1 = ∠2, ∠3 = ∠4, therefore ∠ACB=∠A 1 C 1 B 1 .
2) Ray C 1 C coincides with one of the sides of this angle. A lies on CC 1. AC=A 1 C 1, BC=B 1 C 1, C 1 BC - isosceles, ∠ACB=∠A 1 C 1 B 1.
3) Ray C 1 C passes outside the angle A 1 C 1 B 1. AC=A 1 C 1, BC=B 1 C 1, which means ∠1 = ∠2, ∠1+∠3 = ∠2+∠4, ∠ACB=∠A 1 C 1 B 1.
So, AC=A 1 C 1, BC=B 1 C 1, ∠C=∠C 1. Therefore, triangles ABC and A 1 B 1 C 1 are equal in
the first criterion for the equality of triangles.
The theorem has been proven.
2.
Dividing a segment into n equal parts.
Draw a ray through A, lay out n equal segments on it. Draw a straight line through B and A n and parallel lines to it through points A 1 - A n -1. Let us mark their points of intersection with AB. We obtain n segments that are equal according to Thales’ theorem.
Thales's theorem. If several equal segments are laid out in succession on one of two lines and parallel lines are drawn through their ends that intersect the second line, then they will cut off equal segments on the second line.
Proof. AB=CD
1. Draw straight lines through points A and C parallel to the other side of the angle. We get two parallelograms AB 2 B 1 A 1 and CD 2 D 1 C 1. According to the property of a parallelogram: AB 2 = A 1 B 1 and CD 2 = C 1 D 1.
2. ΔABB 2 =ΔCDD 2 ABB 2 CDD 2 BAB 2 DCD 2 and are equal based on the second criterion for the equality of triangles:
AB = CD according to the theorem,
as corresponding ones, formed at the intersection of parallel BB 1 and DD 1 straight line BD.
3. Similarly, each of the angles turns out to be equal to the angle with the vertex at the point of intersection of the secants. AB 2 = CD 2 as corresponding elements in congruent triangles.
4. A 1 B 1 = AB 2 = CD 2 = C 1 D 1
A geometric figure formed by three segments that connect three points that do not belong to the same straight line.
The sides of the triangle form three angles at the vertices of the triangle. To paraphrase, triangle is a polygon that has three angles .
Practical significance signs of equality of triangles boils down to the following: according to the wording triangles are equal, in the case when it is possible to superimpose them on each other so that they coincide; however, implementing triangle overlap can sometimes be difficult and sometimes impossible.
Tests for the equality of triangles make it possible to replace the overlap of triangles by finding and comparing individual fundamental components (sides and angles) and thus justify the equality of triangles.
3. All three sides:
They also highlight fourth sign, which is not as widely covered in the school mathematics course as the previous three. It is formulated as follows:
If two sides of the first triangle are respectively equal to two sides of the second triangle and the angle opposite the larger of these sides in the first triangle is equal to the angle opposite the corresponding side in the second triangle, then these triangles are equal.
