Determination of the type of triangle. Triangle

Today we go to the country of geometry, where we will get acquainted with various types of triangles.

Consider geometric shapes and find "extra" among them (Fig. 1).

Fig. 1. Illustration for example

We see that Figures number 1, 2, 3, 5 are quadrangles. Each of them has its name (Fig. 2).

Fig. 2. Quadrangles

So, the "superfluous" figure is a triangle (Fig. 3).

Fig. 3. Illustration for example

The triangle is called the figure, which consists of three points that are not lying on one straight line, and three segments, pairwise connect these points.

Coints are called vertices of a triangle, segments - it parties. The side of the triangle form in the vertices of the triangle three corners.

The main signs of the triangle are three sides and three corners. The magnitude of the angle of triangles are acreditated, rectangular and stupid.

The triangle is called acutely, if all three angles are sharp, that is, less than 90 ° (Fig. 4).

Fig. 4. Acute triangle

The triangle is called rectangular, if one of its corners is 90 ° (Fig. 5).

Fig. 5. Rectangular triangle

The triangle is called stupolized, if one of its corners are stupid, that is, more than 90 ° (Fig. 6).

Fig. 6. Stupid triangle

According to the number of equal parties, triangles are equilateral, equilibried, versatile.

It is an equally called triangle, in which two sides are equal (Fig. 7).

Fig. 7. Equal triangle

These parties are called side, the third side - base. In an equilibried triangle, the angles at the base are equal.

Equal triangles are acredit and stupid and stupid(Fig. 8) .

Fig. 8. Acredit and stupid iscessed triangles

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

Fig. 9. Equipical triangle

In equilateral triangle all corners are equal. Equally triangles always outlifted.

A versatile is called a triangle, in which all three sides have a different length (Fig. 10).

Fig. 10. Diversified triangle

Perform a task. Distribute these triangles into three groups (Fig. 11).

Fig. 11. Illustration for task

First we distribute the magnitude of the corners.

Acreditated triangles: No. 1, No. 3.

Rectangular triangles: No. 2, No. 6.

Stupid triangles: No. 4, No. 5.

These same triangles distribute into groups by the number of equal parties.

Versatile triangles: No. 4, No. 6.

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

Equipical triangle: No. 1.

Consider drawings.

Think from which pieces of the wire made each triangle (Fig. 12).

Fig. 12. Illustration for task

You can talk like that.

The first piece of wire is divided into three equal parts, so an equilateral triangle can be made of it. In the figure, it is depicted third.

The second piece of wire is divided into three different parts, so you can make a versatile triangle from it. In the figure, it is depicted first.

The third piece of wire is divided into three parts, where the two parts have the same length, it means that it is possible to make an equifiable triangle. In the picture, it is depicted second.

Today, we met various types of triangles in the classroom.

Bibliography

1. M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: In 2 parts, part 1. - M.: Enlightenment, 2012.
2. M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: in 2 parts, part 2. - M.: "Education", 2012.
3. M.I. Moro. Mathematics lessons: Methodical recommendations for the teacher. Grade 3. - M.: Enlightenment, 2012.
4. Regulatory document. Control and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. School of Russia: Primary School Programs. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: test work. Grade 3. - M.: Enlightenment, 2012.
7. V.N. Rudnitskaya. Tests. - M.: Exam, 2012.

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

Homework

1. Finish phrases.

a) The triangle is called a figure, which consists of ... who are not lying on one straight line, and ..., in pairwise connect these points.

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

c) the magnitude of the angle of triangles are ..., ..., ....

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

2. History

a) rectangular triangle;

b) acute triangle;

c) stupid triangle;

d) equilateral triangle;

e) a versatile triangle;

e) an equifiable triangle.

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

Triangle - definition and general concepts

The triangle is such a simple polygon consisting of three sides and has as many corners. Its planes are limited to 3 points and 3 segments, in pairwise connecting dial points.

All the vertices of any triangle, regardless of its varieties, are denoted by capital latin letters, and its parties are depicted by the corresponding designations of opposite vertices, only in large letters, but small. For example, a triangle with vertices indicated by letters A, B and C has the parties a, b, c.

If we consider the triangle in the Euclidean space, then this is such a geometric shape that was formed using three segments connecting three points that do not lie on one straight line.

Look carefully on the drawing, which is depicted at the top. On it, points A, B and C are the vertices of this triangle, and its segments are the names of the sides of the triangle. Each vertex of this polygon forms inside its corners.

Types of triangles

According to the magnitude, the corners of the triangles, they are divided into such varieties as: rectangular;
Acute-angular;
Tomb.

These triangles belong to rectangular, who have one straight angle in the presence, and the other two have sharp corners.

The acute triangles are those that all its corners are sharp.

And if the triangle has one stupid angle, and the two remaining angle is sharp, then such a triangle refers to the stupid.

Each of you perfectly understands that not all triangles have equal parties. And accordingly, what length there are his parties, triangles can be divided into:

Anoseced;
Equilateral;
Versatile.

Task: Draw different types of triangles. Give them a definition. What difference between them you see?

The main properties of triangles

Although these simple polygons may differ from each other of the values \u200b\u200bof the corners or sides, but in each triangle there are basic properties, characteristic of this figure.

In any triangle:

The total amount of all its corners is 180º.
If it belongs to the equilateral, then each of its angle is 60º.
The equilateral triangle has the same and smooth angles.
The less the side of the polygon, the smaller the angle is located opposite it and on the contrary, on the contrary, there is a larger angle.
If the parties are equal, then equal angles are located opposite them, and vice versa.
If you take a triangle and extend its side, then in the end we are formed an outer angle. It is equal to the sum of the inner corners.
In any triangle, his side, regardless of which you have not chosen, will still be less than the sum of 2 other 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.

The task

The table shows already known two corners of the triangle. Knowing the total amount of all angles, find what is equal to the third corner of the triangle and bring to the table:

1. How many degrees has the third angle?
2. What kind of triangles does he belong to?

Signs of the equality of triangles

I sign

II sign

III Sign

Height, bisector and median triangle

The height of the triangle is perpendicular, conducted from the top of the figure to its opposite side, is called the height of the triangle. All heights of the triangle intersect at one point. The intersection point of all 3 heights of the triangle is its orthoctor.

The segment conducted from this vertex and connecting it to the middle of the opposite side is median. Medians, as well as the height of the triangle, have one common intersection point, the so-called center of gravity of the triangle or centroid.

The bisector of the triangle is a segment connecting the peak of the angle and the point of the opposite side, as well as the dividing the corner in half. All triangle bisectors intersect at one point, which is called the center of the circle inscribed in the triangle.

A segment that connects the middle of the 2 sides of the triangle is called the middle line.

Historical reference

Such a figure, like a triangle, was known in ancient times. This figure and its properties mentioned on Egyptian papyrus four thousand years ago. A little later, thanks to the Pythagoreo theorem and the Geron formula, the study of the properties of the triangle, passed to a higher level, but still, it happened more than two thousand years ago.

In the XV - XVI centuries began to carry out a lot of research on the properties of the triangle and in the end there was such a science as the planimetry that was called the "new triangle geometry".

A scientist from Russia N. I.lobachevsky made a huge contribution to the knowledge of the properties of triangles. Its works later found the use of both in mathematics and physics and cybernetics.

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

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

Task: What are theories about the Bermuda triangle you heard?

And whether you know that in the theory of Lobachevsky, when the corners of the triangle are addition, their sum always has the result less than 180º. In the geometry of Riemann, the sum of all the corners of the triangle are greater than 180º, and in the writings of Euclide, it is 180 degrees.

Homework

Decide the crossword on a given topic

Questions to the crossword

1. What is the name of the perpendicular, which spent from the top of the triangle to the straight line, located on the opposite side?
2. How, in short, you can call the sum of the lengths of the side of the triangle?
3. What are the triangle, whose two parties are equal?
4. Name the triangle, which has an angle equal to 90 °?
5. What is the name bearing big, from the side of the triangle?
6. The title of the side of an equifiable triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle, which has one of the corners exceeds 90 °?
9. The title of the segment connecting the vertex of our figure from the mid-opposite side?
10. In a simple polygon ABC, the capital letter A is ...?
11. What kind of name is a segment dividing the corner of the triangle in half.

Questions to the topic of triangles:

1. Give the definition.
2. How many heights it has?
3. How many biscomers have a triangle?
4. What is its sum of the corners?
5. What types of this simple polygon are you known?
6. Name the points of triangles that are called wonderful.
7. What device can be measured the angle value?
8. If the clock arrows show 21 hours. What angle form watch arrows?
9. What kind of an angle is the man turns if the team "left", "circle" is given?
10. What other definitions are known to you who are associated with a figure having three angle and three sides?

Today we go to the country of geometry, where we will get acquainted with various types of triangles.

Consider geometric shapes and find "extra" among them (Fig. 1).

Fig. 1. Illustration for example

We see that Figures number 1, 2, 3, 5 are quadrangles. Each of them has its name (Fig. 2).

Fig. 2. Quadrangles

So, the "superfluous" figure is a triangle (Fig. 3).

Fig. 3. Illustration for example

The triangle is called the figure, which consists of three points that are not lying on one straight line, and three segments, pairwise connect these points.

Coints are called vertices of a triangle, segments - it parties. The side of the triangle form in the vertices of the triangle three corners.

The main signs of the triangle are three sides and three corners. The magnitude of the angle of triangles are acreditated, rectangular and stupid.

The triangle is called acutely, if all three angles are sharp, that is, less than 90 ° (Fig. 4).

Fig. 4. Acute triangle

The triangle is called rectangular, if one of its corners is 90 ° (Fig. 5).

Fig. 5. Rectangular triangle

The triangle is called stupolized, if one of its corners are stupid, that is, more than 90 ° (Fig. 6).

Fig. 6. Stupid triangle

According to the number of equal parties, triangles are equilateral, equilibried, versatile.

It is an equally called triangle, in which two sides are equal (Fig. 7).

Fig. 7. Equal triangle

These parties are called side, the third side - base. In an equilibried triangle, the angles at the base are equal.

Equal triangles are acredit and stupid and stupid(Fig. 8) .

Fig. 8. Acredit and stupid iscessed triangles

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

Fig. 9. Equipical triangle

In equilateral triangle all corners are equal. Equally triangles always outlifted.

A versatile is called a triangle, in which all three sides have a different length (Fig. 10).

Fig. 10. Diversified triangle

Perform a task. Distribute these triangles into three groups (Fig. 11).

Fig. 11. Illustration for task

First we distribute the magnitude of the corners.

Acreditated triangles: No. 1, No. 3.

Rectangular triangles: No. 2, No. 6.

Stupid triangles: No. 4, No. 5.

These same triangles distribute into groups by the number of equal parties.

Versatile triangles: No. 4, No. 6.

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

Equipical triangle: No. 1.

Consider drawings.

Think from which pieces of the wire made each triangle (Fig. 12).

Fig. 12. Illustration for task

You can talk like that.

The first piece of wire is divided into three equal parts, so an equilateral triangle can be made of it. In the figure, it is depicted third.

The second piece of wire is divided into three different parts, so you can make a versatile triangle from it. In the figure, it is depicted first.

The third piece of wire is divided into three parts, where the two parts have the same length, it means that it is possible to make an equifiable triangle. In the picture, it is depicted second.

Today, we met various types of triangles in the classroom.

Bibliography

1. M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: In 2 parts, part 1. - M.: Enlightenment, 2012.
2. M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: in 2 parts, part 2. - M.: "Education", 2012.
3. M.I. Moro. Mathematics lessons: Methodical recommendations for the teacher. Grade 3. - M.: Enlightenment, 2012.
4. Regulatory document. Control and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. School of Russia: Primary School Programs. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: test work. Grade 3. - M.: Enlightenment, 2012.
7. V.N. Rudnitskaya. Tests. - M.: Exam, 2012.

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

Homework

1. Finish phrases.

a) The triangle is called a figure, which consists of ... who are not lying on one straight line, and ..., in pairwise connect these points.

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

c) the magnitude of the angle of triangles are ..., ..., ....

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

2. History

a) rectangular triangle;

b) acute triangle;

c) stupid triangle;

d) equilateral triangle;

e) a versatile triangle;

e) an equifiable triangle.

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

The triangle (from the point of view of the Euclide space) is such a geometric shape that is formed by three segments connecting three points that are not lying on one straight line. Three points that formed a triangle are called its vertices, and the segments connecting vertices are called the sides of the triangle. What are the triangles?

Equal triangles

There are three signs of equality of triangles. What triangles are called equal? These are those whose:

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

Rectangular triangles exist the following signs of equality:

• in acute corner and hypotenuse;
• in acute corner and cathetu;
• in two categories;
• on hypotenuse and cathetu.

What are the triangles

By number of equal sides, a triangle can be:

• Equilateral. This is a triangle with three equal parties. All angles in the equilateral triangle are 60 degrees. In addition, the centers of the described and inscribed circles are coincided.
• Inequilage. A triangle that does not have equal parties.
• Equalued. This is a triangle with two equal parties. Two identical sides are lateral, and the third party is the basis. In such a triangle, the bisector, median and height, if they are omitted for the base.

The magnitude of the corners of the triangle can be:

1. Stupid - when one of the corners has a magnitude of more than 90 degrees, that is, when he is stupid.
2. Outlifted - if all three angle in the triangle are sharp, that is, they have a value of less than 90 degrees.
3. What triangle is called rectangular? This is such that has one straight angle equal to 90 degrees. Cates in it will be called two sides, which are formed by this angle, and the hypotenuse is opposed to the right corner side.

The main properties of triangles

1. Ahead of the smaller side always lies a smaller corner, and a larger angle always lies against most.
2. Equal angles always lie against equal parties, and different angles always lie against different sides. 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. External angle can be obtained if the triangle has one of its sides. The magnitude of the outer angle will be equal to the amount of non-adjacent internal angles.
5. The side of the triangle is greater than the difference of its two other sides, but less than their amount.

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

Triangle - This is a convex polygon with the smallest number of corners and sides. The triangle is formed by a closed broken, consisting of three stages, and the part of the plane that is inside the broken.

In the text, triangles are denoted by the symbol Δ and three capital latin letters standing at the vertices - Δ ABC:

In a triangle ABC Points A., B. and C. - this is vertices of trianglesSegments AB, BC. and CA. - triangle sides. The corners formed by the sides of the triangle are called triangle angles.

The lower side of the triangle is usually called base. In a triangle ABC side AC - Base.

Types of triangles

Triangles differ in each other, firstly, by the nature of the corners, secondly, by the nature of the parties.

By the nature of the corners, the triangle is called:

• OtterugalIf all its corners are sharp.
• Rectangularif one corner is straight. In a rectangular triangle, the sides forming the straight angle are called catetie, and the side lying opposite the direct angle - hypotenuse.
• Stupidif one of his corners are stupid.

By the nature of the parties, the triangle is called:

• VersatileIf all of his parties have a different length.
• EqualuedIf two sides are equal to each other. Equal parties are called sideways, and the third party - base. In an equilibried triangles, the angles at the base are equal.
• EquilateralIf all three sides are equal to each other. In equilateral triangles, all three corners are equal.

An equal side of the parties in the drawings are marked with the same amount of cerebrals.