The main types of triangles. Obtuse triangle: length of sides, sum of angles

The science of geometry tells us what a triangle, square, cube is. IN modern world it is studied in schools by all without exception. Also, a science that directly studies what a triangle is and what properties it has is trigonometry. She explores in detail all the phenomena associated with data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems related to them.

What is a triangle? Definition

This is a flat polygon. It has three corners, which is clear from its name. It also has three sides and three vertices, the first of which are segments, the second are points. Knowing what two angles are equal to, you can find the third one by subtracting the sum of the first two from the number 180.

What are triangles?

They can be classified according to various criteria.

First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first have acute angles, that is, those that are less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.

Right triangle

It is impossible not to talk about what a right triangle is.

Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The other two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. With its help, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can recall the isosceles. This is one in which two of the sides are equal, and two of the angles are also equal.

What is the leg and hypotenuse?

The leg is one of the sides of a triangle that form an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. From it, a perpendicular can be lowered onto the leg. The ratio of the adjacent leg to the hypotenuse is called the cosine, and the opposite is called the sine.

- what are its features?

It is rectangular. Its legs are three and four, and the hypotenuse is five. If you saw that the legs of this triangle are equal to three and four, you can be sure that the hypotenuse will be equal to five. Also, according to this principle, it can be easily determined that the leg will be equal to three if the second is equal to four, and the hypotenuse is five. To prove this statement, you can apply the Pythagorean theorem. If two legs are 3 and 4, then 9 + 16 \u003d 25, the root of 25 is 5, that is, the hypotenuse is 5. Also, the Egyptian triangle is called a right triangle, whose sides are 6, 8 and 10; 9, 12 and 15 and other numbers with a ratio of 3:4:5.

What else could be a triangle?

Triangles can also be inscribed and circumscribed. The figure around which the circle is described is called inscribed, all its vertices are points lying on the circle. A circumscribed triangle is one in which a circle is inscribed. All its sides are in contact with it at certain points.

How is

The area of ​​any figure is measured in square units (square meters, square millimeters, square centimeters, square decimeters, etc.). This value can be calculated in a variety of ways, depending on the type of triangle. The area of ​​any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite angle, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle between these sides, and divide this by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this, you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the four values ​​obtained. Next, find out the number that came out. The area of ​​an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by which is circumscribed around it times four.

The area of ​​the described triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: we square the side, multiply the resulting figure by the root of three, then divide this number by four. Similarly, you can calculate the height of a triangle in which all sides are equal, for this you need to multiply one of them by the root of three, and then divide this number by two.

Triangle theorems

The main theorems that are associated with this figure are the Pythagorean theorem, described above, and cosines. The second (sine) is that if you divide any side by the sine of the angle opposite to it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosine) is that if the sum of the squares of the two sides is subtracted from their product, multiplied by two and the cosine of the angle located between them, then the square of the third side will be obtained.

Dali triangle - what is it?

Many, faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. The Dali Triangle is the common name for three places that are closely associated with the life of the famous artist. Its “tops” are the house where Salvador Dali lived, the castle that he gave to his wife, and the museum of surrealistic paintings. During the tour of these places you can learn a lot. interesting facts about this peculiar creative artist known all over the world.

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

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

Rice. 1. Illustration for example

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

Rice. 2. Quadrangles

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 straight line, and three segments connecting these points in pairs.

The points are called triangle vertices, 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. Triangles are classified according to the angle 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 right-angled 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, i.e. greater than 90° (Fig. 6).

Rice. 6. Obtuse Triangle

According to the number of equal sides, triangles are equilateral, isosceles, scalene.

An isosceles triangle is a triangle 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 angles at the base are equal.

Isosceles triangles are acute and obtuse(Fig. 8) .

Rice. 8. Acute and obtuse isosceles triangles

An equilateral triangle is called, 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 triangle is called versatile, in which all three sides have different lengths (Fig. 10).

Rice. 10. Scalene triangle

Complete the task. Divide 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: #2, #6.

Obtuse triangles: #4, #5.

These triangles are divided 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.

Review the drawings.

Think about what piece of wire each triangle is made of (fig. 12).

Rice. 12. Illustration for the task

You can argue like this.

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

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

The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.

Today in the lesson we got acquainted with different types of triangles.

Bibliography

1. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
2. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
3. M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.

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

Homework

1. Finish the phrases.

a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.

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

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

d) According to the number of equal sides, triangles are ..., ..., ....

2. Draw

a) a right triangle

b) an acute triangle;

c) an obtuse triangle;

d) an equilateral triangle;

e) scalene triangle;

e) an isosceles triangle.

3. Make a task on the topic of the lesson for your comrades.

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

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

Rice. 1. Illustration for example

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

Rice. 2. Quadrangles

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 straight line, and three segments connecting these points in pairs.

The points are called triangle vertices, 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. Triangles are classified according to the angle 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 right-angled 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, i.e. greater than 90° (Fig. 6).

Rice. 6. Obtuse Triangle

According to the number of equal sides, triangles are equilateral, isosceles, scalene.

An isosceles triangle is a triangle 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 angles at the base are equal.

Isosceles triangles are acute and obtuse(Fig. 8) .

Rice. 8. Acute and obtuse isosceles triangles

An equilateral triangle is called, 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 triangle is called versatile, in which all three sides have different lengths (Fig. 10).

Rice. 10. Scalene triangle

Complete the task. Divide 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: #2, #6.

Obtuse triangles: #4, #5.

These triangles are divided 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.

Review the drawings.

Think about what piece of wire each triangle is made of (fig. 12).

Rice. 12. Illustration for the task

You can argue like this.

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

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

The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.

Today in the lesson we got acquainted with different types of triangles.

Bibliography

1. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
2. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
3. M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.

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

Homework

1. Finish the phrases.

a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.

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

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

d) According to the number of equal sides, triangles are ..., ..., ....

2. Draw

a) a right triangle

b) an acute triangle;

c) an obtuse triangle;

d) an equilateral triangle;

e) scalene triangle;

e) an isosceles triangle.

3. Make a task on the topic of the lesson for your comrades.

The simplest polygon that is studied at school is a triangle. It is more understandable for students and encounters fewer difficulties. Despite the fact that there are different kinds triangles that have special properties.

What shape is called a triangle?

Formed by three points and line segments. The former are called vertices, the latter are called sides. Moreover, all three segments must be connected so that corners form between them. Hence the name of the figure "triangle".

Differences in the names in the corners

Since they can be sharp, obtuse and straight, the types of triangles are determined by these names. Accordingly, there are three groups of such figures.

• First. If all the angles of a triangle are acute, then it will be called an acute triangle. Everything is logical.

• Second. One of the angles is obtuse, so the triangle is obtuse. Easier nowhere.

• Third. There is an angle equal to 90 degrees, which is called a right angle. The triangle becomes rectangular.

Differences in names on the sides

Depending on the features of the sides, the following types of triangles are distinguished:

the general case is versatile, in which all sides have an arbitrary length;

isosceles, two sides of which have the same numerical values;

equilateral, the lengths of all its sides are the same.

If the task does not specify a specific type of triangle, then you need to draw an arbitrary one. In which all angles are acute, and the sides have different lengths.

Properties common to all triangles

1. If you add up all the angles of a triangle, you get a number equal to 180º. And it doesn't matter what kind it is. This rule always applies.
2. The numerical value of any side of the triangle is less than the other two added together. Moreover, it is greater than their difference.
3. Each outer corner has a value that is obtained by adding two inner corners that are not adjacent to it. Moreover, it is always larger than the adjacent internal one.
4. The smallest side of a triangle is always opposite the smallest angle. Conversely, if the side is large, then the angle will be the largest.

These properties are always valid, no matter what types of triangles are considered in problems. All the rest follow from specific features.

Properties of an isosceles triangle

• The angles adjacent to the base are equal.
• The height that is drawn to the base is also the median and the bisector.
• The heights, medians and bisectors, which are built to the sides of the triangle, are respectively equal to each other.

Properties of an equilateral triangle

If there is such a figure, then all the properties described a little above will be true. Because an equilateral will always be an isosceles one. But not vice versa, an isosceles triangle will not necessarily be equilateral.

• All its angles are equal to each other and have a value of 60º.
• Any median of an equilateral triangle is its height and bisector. And they are all equal to each other. To determine their values, there is a formula that consists of the product of the side and the square root of 3 divided by 2.

Properties of a right triangle

• Two acute angles add up to 90º.
• The length of the hypotenuse is always greater than that of any of the legs.
• The numerical value of the median drawn to the hypotenuse is equal to half of it.
• The leg is equal to the same value if it lies opposite an angle of 30º.
• The height, which is drawn from the top with a value of 90º, has a certain mathematical dependence on the legs: 1 / n 2 \u003d 1 / a 2 + 1 / in 2. Here: a, c - legs, n - height.

Problems with different types of triangles

No. 1. Given an isosceles triangle. Its perimeter is known and is equal to 90 cm. It is required to know its sides. As additional condition: the lateral side is 1.2 times less than the base.

The value of the perimeter directly depends on the quantities that need to be found. The sum of all three sides will give 90 cm. Now you need to remember the sign of a triangle, according to which it is isosceles. That is, the two sides are equal. You can make an equation with two unknowns: 2a + b \u003d 90. Here a is the side, b is the base.

It's time for an additional condition. Following it, the second equation is obtained: b \u003d 1.2a. You can substitute this expression into the first one. It turns out: 2a + 1.2a \u003d 90. After transformations: 3.2a \u003d 90. Hence a \u003d 28.125 (cm). Now it's easy to find out the reason. It is best to do this from the second condition: v \u003d 1.2 * 28.125 \u003d 33.75 (cm).

To check, you can add three values: 28.125 * 2 + 33.75 = 90 (cm). All right.

Answer: the sides of the triangle are 28.125 cm, 28.125 cm, 33.75 cm.

No. 2. The side of an equilateral triangle is 12 cm. You need to calculate its height.

Solution. To search for an answer, it is enough to return to the moment where the properties of the triangle were described. This is the formula for finding the height, median and bisector of an equilateral triangle.

n \u003d a * √3 / 2, where n is the height, a is the side.

Substitution and calculation give the following result: n = 6 √3 (cm).

This formula does not need to be memorized. Suffice it to recall that the height divides the triangle into two rectangular ones. Moreover, it turns out to be a leg, and the hypotenuse in it is the side of the original one, the second leg is half of the known side. Now you need to write down the Pythagorean theorem and derive a formula for the height.

Answer: the height is 6 √3 cm.

No. 3. MKR is given - a triangle, 90 degrees in which makes an angle K. The sides MP and KR are known, they are equal to 30 and 15 cm, respectively. You need to find out the value of the angle P.

Solution. If you make a drawing, it becomes clear that MP is the hypotenuse. Moreover, it is twice as large as the leg of the CD. Again, you need to turn to the properties. One of them is just related to the corners. From it it is clear that the angle of the KMR is 30º. So the desired angle P will be equal to 60º. This follows from another property which states that the sum of two acute angles must equal 90º.

Answer: angle R is 60º.

No. 4. You need to find all the angles of an isosceles triangle. It is known about him that the external angle from the angle at the base is 110º.

Solution. Since only the outer corner is given, this should be used. It forms with an internal angle developed. So they add up to 180º. That is, the angle at the base of the triangle will be equal to 70º. Since it is isosceles, the second angle has the same value. It remains to calculate the third angle. By a property common to all triangles, the sum of the angles is 180º. So the third is defined as 180º - 70º - 70º = 40º.

Answer: the angles are 70º, 70º, 40º.

No. 5. It is known that in an isosceles triangle the angle opposite the base is 90º. A dot is marked on the base. The segment connecting it with a right angle divides it in a ratio of 1 to 4. You need to know all the angles of the smaller triangle.

Solution. One of the corners can be determined immediately. Since the triangle is right-angled and isosceles, those that lie at its base will be 45º, that is, 90º / 2.

The second of them will help to find the relation known in the condition. Since it is equal to 1 to 4, then the parts into which it is divided are only 5. So, to find out the smaller angle of the triangle, you need 90º / 5 = 18º. It remains to find out the third. To do this, from 180º (the sum of all the angles of a triangle), you need to subtract 45º and 18º. The calculations are simple, and it turns out: 117º.

A triangle (from the point of view of Euclid's space) is such geometric figure, which is formed by three segments connecting three points that do not lie on one straight line. The three points that form a triangle are called its vertices, and the line segments connecting the vertices are called sides of the triangle. What are triangles?

Equal Triangles

There are three signs of the equality of triangles. What triangles are called equal? These are the ones who:

• two sides and the angle between these sides are equal;
• one side and two angles adjacent to it are equal;
• all three sides are equal.

Right triangles have the following signs of equality:

• along an acute angle and hypotenuse;
• along an acute angle and leg;
• on two legs;
• along the hypotenuse and cathetus.

What are triangles

According to the number of equal sides, a triangle can be:

• Equilateral. It is a triangle with three equal sides. All angles in an equilateral triangle are 60 degrees. In addition, the centers of the circumscribed and inscribed circles coincide.
• Unequilateral. A triangle with no equal sides.
• Isosceles. It is a triangle with two equal sides. Two identical sides are the sides, and the third side is the base. In such a triangle, the bisector, median and height coincide if they are lowered to the base.

According to the size of the angles, a triangle can be:

1. Obtuse - when one of the angles has a value of more than 90 degrees, that is, when it is obtuse.
2. Acute-angled - if all three angles in the triangle are acute, that is, they have a value of less than 90 degrees.
3. Which triangle is called a right triangle? This is one that has one right angle equal to 90 degrees. The legs in it will be called the two sides that form this angle, and the hypotenuse is the side opposite the right angle.

Basic properties of triangles

1. A smaller angle always lies opposite the smaller side, and a larger angle always lies opposite the larger side.
2. Equal angles always lie opposite equal sides, and opposite sides always lie different angles. In particular, in an equilateral triangle, all angles have the same value.
3. In any triangle, the sum of the angles is 180 degrees.
4. An external angle can be obtained by extending one of its sides to a triangle. The value of the outer angle will be equal to the sum of the inner angles not adjacent to it.
5. The side of a triangle is greater than the difference of its other two sides, but less than their sum.

In the spatial geometry of Lobachevsky, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere, this value is greater than 180 degrees. The difference between 180 degrees and the sum of the angles of a triangle is called a defect.