A triangle with equal angles is called. Triangle and its types

Perhaps the most basic, simple and interesting figure in geometry is a triangle. In a secondary school course, its basic properties are studied, but sometimes knowledge on this topic is formed incomplete. The types of triangles initially determine their properties. But this view remains mixed. So now let's take a closer look at this topic.

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-angled. 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 considered one is called obtuse-angled.

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

It is impossible not to say about the right 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 circles also lies in the same place. To solve problems, you need to know only one side, since the angles are initially set for you, and the other two sides are known. That is, the figure is given by only one parameter. There are Their main feature - the equality of the two sides and angles at the base.

Sometimes there is a question about whether there is a triangle with given sides. What you are really asking is whether this 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 to find the cosines of the angles of a triangle with sides 3,5,9, then the obvious can be explained here without complex mathematical tricks. 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 - 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 line AB, you will have to go an extra distance. Here a contradiction arises. This is, of course, a hypothetical explanation. Mathematics knows more than one way to prove that all kinds of triangles obey the basic identity. It says that the sum of two sides is greater than the length of the third.

Each 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 circumscribed around any triangle. It is also possible to inscribe a circle so that it has only three points of contact and does not go beyond the outer sides.

Now you have got acquainted 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.

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.

Of all the polygons triangles have the least number of angles and sides.

Triangles can be distinguished by the shape of their angles.

If all angles of a triangle are acute, then it is called an acute triangle.(Fig. 113, a).

If one of the angles of a triangle is right, then it is called a right triangle.(Fig. 113, b).

If one of the angles of a triangle is obtuse, then it is called an obtuse triangle.(Fig. 113, c).

They say that we classified triangles according to their angles.

Triangles can be classified not only by the type of angles, but also by the number of equal sides.

If two sides of a triangle are equal, then it is called an isosceles triangle.

Figure 114, a shows an isosceles triangle ABC, in which AB \u003d BC. In the figure, equal sides are marked with an equal number of dashes. Equal sides AB and BC are called sides, and the side AC − basis isosceles triangle ABC.

If the sides of a triangle are equal, then it is called an equilateral triangle.

The triangle shown in Figure 114b is equilateral, it has MN = NE = EM.

A triangle with three sides of different lengths is called a scalene triangle.

The triangles shown in Figure 113 are scalene. If the side of an equilateral triangle is a, then its perimeter is calculated by the formula:

P = 3a

Example 1 . Using a ruler and a protractor, construct a triangle whose two sides are 3 cm and 2 cm and the angle between them is 50°.

Using a protractor, we will construct an angle A, the degree measure of which is 50 ° (Fig. 115). On the sides of this angle from its top, using a ruler, set aside a segment AB 3 cm long and a segment AC 2 cm long ( fig. 116). Connecting points B and C with a segment, we get the desired triangle ABC ( fig. 117).

Example 2 . Using a ruler and a protractor, construct a triangle ABC whose side AB is 2 cm and whose angles CAB and CBA are respectively 40° and 110°.

Solution. Using a ruler, we build a segment AB 2 cm long ( fig. 118). From the beam AB with the help of a protractor we set aside an angle with a vertex at point A, the degree measure of which is 40 °. From the ray BA in the same direction from the straight line AB, in which the first angle was plotted, we lay off the angle with the vertex at point B, the degree measure of which is 110 ° (Fig. 119).

Having found the point C of the intersection of the sides of the angles A and B, we obtain the desired triangle ABC (Fig. 120).

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.

Tasks:

1. Introduce students to different types of triangles depending on the type of angles (rectangular, acute-angled, obtuse-angled). Learn to find triangles and their types in the drawings. To fix the basic geometric concepts and their properties: straight line, segment, ray, angle.

2. Development of thinking, imagination, mathematical speech.

3. Education of attention, activity.

During the classes

I. Organizational moment.

How much do we need guys?
For our skillful hands?
Draw two squares
And they have a big circle.
And then some more circles
Triangle cap.
So it came out very, very
Cheerful Weird.

II. Announcement of the topic of the lesson.

Today in the lesson we will make a trip around the city of Geometry and visit the Triangles microdistrict (that is, we will get acquainted with different types of triangles depending on their angles, we will learn to find these triangles in the drawings.) We will conduct a lesson in the form of a “competition game” by commands.

1 team - “Segment”.

2 team - "Ray".

Team 3 - "Corner".

And the guests will represent the jury.

The jury will guide us along the way

And will not leave without attention. (Evaluate by points 5,4,3,...).

And on what will we travel around the city of Geometry? Remember what types of passenger transport are in the city? There are so many of us, which one shall we choose? (Bus).

Bus. Clearly, briefly. Boarding begins.

Let's get comfortable and start our journey. Team captains get tickets.

But these tickets are not easy, and the tickets are “tasks”.

III. Repetition of the material covered.

First stop"Repeat."

Question for all teams.

Find a straight line in the drawing and name its properties.

Without end and edge, the line is straight!
At least a hundred years go along it,
You won't find the end of the road!

• The straight line has neither beginning nor end - it is infinite, so it cannot be measured.

Let's start our competition.

Protecting your team names.

(All teams read the first questions and discuss. In turn, the team captains read out the questions, 1 team reads 1 question).

1. Show a segment in the drawing. What is called a cut. Name its properties.

• The part of a straight line bounded by two points is called a line segment. A line segment has a beginning and an end, so it can be measured with a ruler.

(Team 2 reads 1 question).

1. Show the beam in the drawing. What is called a beam. Name its properties.

• If you mark a point and draw a part of a straight line from it, you get an image of a beam. The point from which a part of the line is drawn is called the beginning of the ray.

The beam has no end, so it cannot be measured.

(Team 3 reads 1 question).

1. Show the angle on the drawing. What is called an angle. Name its properties.

• Drawing two rays from one point, a geometric figure is obtained, which is called an angle. An angle has a vertex, and the rays themselves are called sides of the angle. Angles are measured in degrees using a protractor.

Fizkultminutka (to the music).

IV. Preparing to study new material.

Second stop"Fabulous".

On a walk, the Pencil met different angles. I wanted to say hello to them, but I forgot the name of each of them. Pencil will have to help.

(The angles of the study are checked using the model of a right angle).

Assignment to teams. Read questions #2 and discuss.

Team 1 reads question 2.

2. Find a right angle, give a definition.

• An angle of 90° is called a right angle.

Team 2 reads question 2.

2. Find an acute angle, give a definition.

• An angle less than a right angle is called an acute angle.

Team 3 reads question 2.

2. Find an obtuse angle, give a definition.

An angle greater than a right angle is called obtuse.

In the microdistrict where Pencil liked to walk, all the corners differed from other residents in that the three of us always walked, drank tea together, went to the cinema together. And the Pencil could not understand what kind of geometric figure three angles together make up?

A poem will give you a clue.

You on me, you on him
Look at all of us.
We have everything, we have everything
We only have three!

Which shape is being referred to?

• About the triangle.

What shape is called a triangle?

• A triangle is a geometric figure that has three vertices, three angles, and three sides.

(Learners show a triangle in the drawing, name the vertices, angles and sides).

Vertices: A, B, C (points)

Angles: BAC, ABC, BCA.

Sides: AB, BC, CA (segments).

V. Physical education:

stomp your foot 8 times,
Clap your hands 9 times
we will squat 10 times,
and bend over 6 times
we'll jump straight
so many (triangle display)
Hey, yes, count! Game and more!

VI. Learning new material.

Soon the corners became friends and became inseparable.

And now we will call the microdistrict: the Triangles microdistrict.

The third stop is “Znayka”.

What are the names of these triangles?

Let's give them names. And let's try to formulate the definition ourselves.

2. Find triangles of different types

1 team will find and show obtuse triangles.

2 command will find and show right triangles.

3 command will find and show acute triangles.

VIII. The next stop is Thinking.

Assignment to all teams.

After shifting 6 sticks, make 4 equal triangles from the lantern.

What kind of angles are triangles? (Acute-angled).

IX. Summary of the lesson.

What neighborhood did we visit?

What types of triangles are you familiar with?