A certain triangle in which all sides are not of the same length is usually called versatile.

A triangle with two equal sides is denoted as isosceles. Identical sides are usually called lateral, third party - basis. The following definition will be equally true triangle bases is the side of an isosceles triangle that is not equal to the other two sides.

IN isosceles triangle the angles at the base are equal. Height, median, bisector of an isosceles triangle, drawn to its base, are aligned.

Triangle, with all equal sides, is denoted as equilateral or correct. In an equilateral triangle, all angles are 60°, and the centers of the inscribed and circumscribed circles are aligned.

Types of triangles depending on angle parameters.

A triangle in which only angles less than 90 0 (acute) is called acute-angled.

A triangle containing an angle of 90 0 is called rectangular. The sides of a triangle forming a right angle are usually designated legs, and the side opposite the right angle is hypotenuse.

Triangle - definition and general concepts

A triangle is a simple polygon consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.

All vertices of any triangle, regardless of its type, are designated by capital Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, only not in capital letters, but in small ones. So, for example, a triangle with vertices labeled A, B and C has sides a, b, c.

If we consider a triangle in Euclidean space, then it is a geometric figure that is formed using three segments connecting three points that do not lie on the same straight line.

Look carefully at the picture shown above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms angles inside it.

Types of triangles

According to the size of the angles of triangles, they are divided into such varieties as: Rectangular;
Acute angular;
Obtuse.

Rectangular triangles include those that have one right angle and the other two have acute angles.

Acute triangles are those in which all its angles are acute.

And if a triangle has one obtuse angle and the other two acute angles, then such a triangle is classified as obtuse.

Each of you understands perfectly well that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:

Isosceles;
Equilateral;
Versatile.

Assignment: Draw different types of triangles. Define them. What difference do you see between them?

Basic properties of triangles

Although these simple polygons may differ from each other in the size of their angles or sides, each triangle has the basic properties that are characteristic of this figure.

In any triangle:

The total sum of all its angles is 180º.
If it belongs to equilaterals, then each of its angles is 60º.
An equilateral triangle has equal and equal angles.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are equal angles, and vice versa.
If we take a triangle and extend its side, we end up with an external angle. It is equal to the sum of the internal angles.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:

1. a< b + c, a >b–c;
2.b< a + c, b >a–c;
3. c< a + b, c >a–b.

Exercise

The table shows the already known two angles of the triangle. Knowing the total sum of all angles, find what the third angle of the triangle is equal to and enter it into the table:

1. How many degrees does the third angle have?
2. What type of triangle does it belong to?

Tests for equivalence of triangles

I sign

II sign

III sign

Height, bisector and median of a triangle

The altitude of a triangle - the perpendicular drawn from the vertex of the figure to its opposite side is called the altitude of the triangle. All altitudes of a triangle intersect at one point. The point of intersection of all 3 altitudes of a triangle is its orthocenter.

A segment drawn from a given vertex and connecting it at the middle of the opposite side is the median. Medians, as well as altitudes of a triangle, have one common point of intersection, the so-called center of gravity of the triangle or centroid.

The bisector of a triangle is a segment connecting the vertex of an angle and a point on the opposite side, and also dividing this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.

The segment that connects the midpoints of 2 sides of a triangle is called the midline.

Historical reference

A figure such as a triangle was known back in Ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron’s formula, the study of the properties of the triangle moved to a higher level, but still, this happened more than two thousand years ago.

In the 15th – 16th centuries, a lot of research began to be carried out on the properties of a triangle, and as a result, a science such as planimetry arose, which was called “New Triangle Geometry”.

Russian scientist N.I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application in mathematics, physics and cybernetics.

Thanks to knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when drawing up maps, measuring areas, and even when designing various mechanisms.

What is the most famous triangle you know? This is of course the Bermuda Triangle! It received this name in the 50s because of the geographical location of the points (vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The vertices of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.

Assignment: What theories about the Bermuda Triangle have you heard?

Did you know that in Lobachevsky’s theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemann's geometry, the sum of all the angles of a triangle is greater than 180º, and in the works of Euclid it is equal to 180 degrees.

Homework

Solve a crossword puzzle on a given topic

Questions for the crossword:

1. What is the name of the perpendicular that is drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the largest side of the triangle?
6. What is the name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90°?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, the capital letter A is...?
11. What is the name of the segment dividing the angle of a triangle in half?

Questions on the topic of triangles:

1. Define it.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called remarkable.
7. What device can you use to measure the angle?
8. If the clock hands show 21 o'clock. What angle do the hour hands make?
9. At what angle does a person turn if he is given the command “left”, “circle”?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?

Subjects > Mathematics > Mathematics 7th grade

Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.

Consider the geometric shapes and find the “extra” one among them (Fig. 1).

Rice. 1. Illustration for example

We see that figures No. 1, 2, 3, 5 are quadrilaterals. Each of them has its own name (Fig. 2).

Rice. 2. Quadrilaterals

This means that the “extra” figure is a triangle (Fig. 3).

Rice. 3. Illustration for example

A triangle is a figure that consists of three points that do not lie on the same line and three segments connecting these points in pairs.

The points are called the vertices of the triangle, segments - his parties. The sides of the triangle form There are three angles at the vertices of a triangle.

The main features of a triangle are three sides and three corners. According to the size of the angle, triangles are acute, rectangular and obtuse.

A triangle is called acute-angled if all three of its angles are acute, that is, less than 90° (Fig. 4).

Rice. 4. Acute triangle

A triangle is called rectangular if one of its angles is 90° (Fig. 5).

Rice. 5. Right Triangle

A triangle is called obtuse if one of its angles is obtuse, that is, more than 90° (Fig. 6).

Rice. 6. Obtuse triangle

Based on the number of equal sides, triangles are equilateral, isosceles, scalene.

An isosceles triangle is one in which two sides are equal (Fig. 7).

Rice. 7. Isosceles triangle

These sides are called lateral, Third side - basis. In an isosceles triangle, the base angles are equal.

There are isosceles triangles acute and obtuse(Fig. 8) .

Rice. 8. Acute and obtuse isosceles triangles

An equilateral triangle is one in which all three sides are equal (Fig. 9).

Rice. 9. Equilateral triangle

In an equilateral triangle all angles are equal. Equilateral triangles Always acute-angled.

A scalene is a triangle in which all three sides have different lengths (Fig. 10).

Rice. 10. Scalene triangle

Complete the task. Distribute these triangles into three groups (Fig. 11).

Rice. 11. Illustration for the task

First, let's distribute according to the size of the angles.

Acute triangles: No. 1, No. 3.

Right triangles: No. 2, No. 6.

Obtuse triangles: No. 4, No. 5.

We will distribute the same triangles into groups according to the number of equal sides.

Scalene triangles: No. 4, No. 6.

Isosceles triangles: No. 2, No. 3, No. 5.

Equilateral triangle: No. 1.

Look at the pictures.

Think about what piece of wire each triangle was made from (Fig. 12).

Rice. 12. Illustration for the task

You can think like this.

The first piece of wire is divided into three equal parts, so you can make an equilateral triangle from it. He is shown third in the picture.

The second piece of wire is divided into three different parts, so it can be used to make a scalene triangle. It is shown first in the picture.

The third piece of wire is divided into three parts, where two parts have the same length, which means that an isosceles triangle can be made from it. In the picture he is shown second.

Today in class we learned about different types of triangles.

Bibliography

1. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
2. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
3. M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
5. “School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
6. S.I. Volkova. Mathematics: Test papers. 3rd grade. - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

1. Nsportal.ru ().
2. Prosv.ru ().
3. Do.gendocs.ru ().

Homework

1. Complete the phrases.

a) A triangle is a figure that consists of ... that do not lie on the same line, and ... that connect these points in pairs.

b) The points are called … , segments - his … . The sides of the triangle form at the vertices of the triangle ….

c) According to the size of the angle, triangles are ... , ... , ... .

d) Based on the number of equal sides, triangles are ... , ... , ... .

2. Draw

a) right triangle;

b) acute triangle;

c) obtuse triangle;

d) equilateral triangle;

e) scalene triangle;

e) isosceles triangle.

3. Create an assignment on the topic of the lesson for your friends.

Perhaps the most basic, simple and interesting figure in geometry is the triangle. In a high school course, its basic properties are studied, but sometimes the knowledge on this topic is incomplete. The types of triangles initially determine their properties. But this view remains mixed. Therefore, now let’s look at this topic in a little more detail.

The types of triangles depend on the degree measure of the angles. These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute. If at least one angle of the triangle is 90 degrees, then you are dealing with a rectangular subspecies. Accordingly, in all other cases the one under consideration is called obtuse-angled.

There are many problems for acute-angled subtypes. A distinctive feature is the internal location of the intersection points of bisectors, medians and heights. In other cases, this condition may not be met. It is not difficult to determine the type of triangle figure. It is enough to know, for example, the cosine of each angle. If any values ​​are less than zero, then the triangle is in any case obtuse. In the case of a zero indicator, the figure has a right angle. All positive values ​​are guaranteed to tell you that you are looking at an angular view.

One cannot help but mention the regular triangle. This is the most ideal view, where all the intersection points of medians, bisectors and heights coincide. The center of the inscribed and circumscribed circle also lies in the same place. To solve problems, you need to know only one side, since the angles are initially given to you, and the other two sides are known. That is, the figure is specified by only one parameter. There are Their main feature is the equality of two sides and angles at the base.

Sometimes the question arises as to whether a triangle with given sides exists. What you are really asking is whether the given description fits the main species. For example, if the sum of two sides is less than the third, then in reality such a figure does not exist at all. If the task asks you to find the cosines of the angles of a triangle with sides of 3,5,9, then the obvious can be explained without complex mathematical techniques. Suppose you want to get from point A to point B. The distance in a straight line is 9 kilometers. However, you remembered that you need to go to point C in the store. The distance from A to C is 3 kilometers, and from C to B is 5. Thus, it turns out that when moving through the store, you will walk one kilometer less. But since point C is not located on straight AB, you will have to walk an extra distance. There is a contradiction here. This is, of course, a conditional explanation. Mathematics knows more than one way to prove that all types of triangles obey the basic identity. It states that the sum of two sides is greater than the length of the third.

Any type has the following properties:

1) The sum of all angles is 180 degrees.

2) There is always an orthocenter - the point of intersection of all three heights.

3) All three medians drawn from the vertices of the interior angles intersect in one place.

4) A circle can be drawn around any triangle. You can also inscribe a circle so that it has only three points of contact and does not extend beyond the outer sides.

Now you are familiar with the basic properties that different types of triangles have. In the future, it is important to understand what you are dealing with when solving a problem.

Triangle is a convex polygon with the smallest number of angles and sides. A triangle is formed by a closed broken line, consisting of three links, and that part of the plane that is inside the broken line.

In the text, triangles are designated by the symbol Δ and three capital Latin letters at the vertices - Δ ABC:

In a triangle ABC points A, B And C- This vertices of the triangle, segments AB, B.C. And C.A. - sides of the triangle. The angles formed by the sides of a triangle are called angles of the triangle.

The bottom side of the triangle is usually called basis. In a triangle ABC side A.C.- base.

Types of triangles

Triangles differ from each other, firstly, by the nature of the angles, and secondly, by the nature of the sides.

Based on the nature of the angles, a triangle is called:

• Acute-angled, if all its angles are acute.
• Rectangular if one angle is right. In a right triangle, the sides forming a right angle are called legs, and the side lying opposite the right angle is hypotenuse.
• Obtuse, if one of its angles is obtuse.

Based on the nature of the sides, a triangle is called:

• Versatile, if all its sides have different lengths.
• Isosceles, if its two sides are equal to each other. Equal sides are called sides, and the third party - basis. In isosceles triangles, the base angles are equal.
• Equilateral, if all three of its sides are equal to each other. In equilateral triangles, all three angles are equal.

Equal sides of the side in the drawings are marked with the same number of lines.