Triangle definition and types. Properties of a triangle

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-angled 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 triangle is one 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 you can use it 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

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

Nsportal.ru ().
Prosv.ru ().
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.

Subject: mathematics

Class: 3rd grade

Textbook: “Mathematics” part 2.

Subject: Types of triangles

Lesson type: discovery of new knowledge

Target: Learn to identify the types of triangles by measuring the lengths of their sides.

Tasks :

1) Update knowledge about geometric shapes - rectangle, square, triangle.

2) Update the addition and subtraction of three-digit numbers, dividing a two-digit number into single-digit, two-digit and round; multiplying a two-digit number by a single-digit number.

3) Introduce the terms: isosceles, equilateral, scalene triangle.

During the classes

1.Motivation for learning activities

Look, tell me, what is it?

(pyramid)

Tell me, what does it consist of? (from parts, levels...)

Can this pyramid be compared with our knowledge? (Yes)

Every day you build more and more new pyramids, each level of the pyramid is new knowledge that you gain in class. What will happen to the pyramid if we remove the blue level? (It will collapse and become smaller.)

What could cause our pyramid of knowledge to collapse? (Due to not completing homework, missing lessons, not listening carefully to the teacher.)

What needs to be done to make our pyramid stronger and grow? (Study homework, work well in class, do homework, don’t skip school.)

Guys, you said everything right. Now let's imagine that our pyramid cast a shadow. Tell me, what geometric figure does the shadow look like?

(On the triangle.)

Today we will continue to work with such a geometric figure as a triangle.

2.Updating knowledge and recording difficulties in a problem situation

What geometric shapes are you familiar with? (square, rectangle, triangle).

There is a table on the board, fill it out based on your knowledge (each student has a card with such a table):

What are the first two geometric figures called? (a rectangle and a square, in a word, these are quadrilaterals.)

Tell me, what types of quadrilaterals do you know? Their image on the slide will help you answer this question.

The names of the quadrilaterals appear after the children's answers.

(rhombus, square, rectangle, trapezoid, parallelogram - they are called by the images on the slide or board.)

Can you tell what is a rectangle and what is a square?

(A rectangle is a quadrilateral with all right angles.

A square is a rectangle with all sides equal)

Find the extra geometric figure based on the results of the table. (Triangle).

Okay, quadrilaterals are all very different, but what do you know about a triangle? (Triangles are: acute, obtuse, rectangular.)

What else do you know about the triangle? (Definition)

A triangle is a geometric figure that has 3 angles, 3 vertices, and 3 sides.

Fill out the following table based on your knowledge:

(The teacher fills out the table according to the children’s answers. Different opinions arise in the “title” columns, and some children leave them blank.)

3.Identification of the location and cause of the difficulty.

What task were you doing? (Fill out the table.)

Where did the problem arise? (When writing the names of triangles)

Why did the problem arise? (We don’t know what they are called)

What is the goal of the lesson? (Find out what other types of triangles there are besides those studied (obtuse, acute, rectangular), learn to identify these types of triangles.)

What is the topic of our lesson? (Types of triangles)

4.Discovery of new knowledge.

Let's go back to the table.

Let's enter the dimensions of the sides of the triangles. (Write in.)

Okay, now look and tell me what you noticed? (The first triangle has all equal sides, the second has 2 equal sides, and the third has all different sides.)

Right, can you come up with names for these triangles based on the explanation you just gave? (Yes)

What do you call a triangle with all sides equal? Come up with an adjective consisting of 2 words: equal sides. (Equilateral)

What do you call a triangle whose sides are all different? (Versatile)

What is the name of a triangle with 2 equal sides? (Children have doubts; to answer this question they use the textbook p. 73) (Isosceles) What other triangle can we call isosceles? (Equilateral)

Fill out the table yourself, based on your new knowledge.

Can we now define the types of triangles? (Yes)

Equilateral - a triangle in which all three sides are equal.

Isosceles - a triangle with at least two equal sides. An equilateral triangle is also an isosceles triangle.

Versatile - a triangle with all sides different.

Check your definitions p. 73 - textbook. (They check.)

Have you made the definitions correctly? (Yes.)

5.Primary consolidation with pronunciation in external speech

Complete the task from the textbook p.74 (under?)

1) Versatile: 2,3,5

2)Isosceles: 1,4 , 6, 7

(Students write in notebooks. Take turns saying the answers, giving reasons. The sample is recorded on the board).

6.Independent work with self-test according to the standard.

Completing the task yourself. At the end of the work - self-test according to the sample (on the board or on individual cards).

№1.Fill out the table , draw triangles schematically.

№2. Write down the numbers:

1) Scalene triangles.

2) Isosceles, from the numbers written out, underline the numbers of equilateral triangles.

Reference:

Task No. 1:

Task No. 2:

1) Scalene triangles: 2,3,4

2)Isosceles triangles (the number of the equilateral triangle is underlined): 1,5

7. Inclusion in the knowledge system and repetition

The boy drew triangles on the sand and encrypted the words; find the meanings of the expressions written in the triangles. First solve those written in scalene triangles, and then in isosceles triangles. And you will guess the encrypted words.

Hint: Write the numbers in ascending order and you will get the words.

Card:

Solution:

Answer: Types of triangles

8. Reflection of educational activities.

Draw accordingly a pyramid of knowledge consisting of 7 levels. Each level is an answer to a question.

Answer the questions:

1) Guys, what did you write down “types of triangles”? (The topic of our lesson)

2)What was our goal? (Find out what all 3 types of triangles are called, learn to identify these types by measuring the lengths of the sides.)

3) What types of triangles did you recognize? (scalene, isosceles, equilateral)

4) Why are they called that?

( Equilateral - a triangle with all sides equal.

Isosceles - a triangle with at least two equal sides, including an equilateral triangle, because it has two equal sides.)

Versatile - a triangle with all sides different.)

5) Have you learned to schematically depict all types of triangles? (Yes, at independent work.)

6) What discoveries did you make today? (New types of triangles, their names.)

7) Guys, can you determine the type of triangle based on its dimensions? (Yes) I will now tell you the measurements, and you will raise up a card with the name of the type of triangle (cards are issued additionally - 3 cards each.)

1. 2cm, 3cm, 5cm - versatile

2. 4cm, 4cm, 2cm - isosceles

3.6cm, 6cm,6cm - equilateral, isosceles

Raise your hands, who has reached the pinnacle of this knowledge today? (raise)

Raise your hands if 1 or 2 levels are not enough. (They pick it up.)

(The teacher analyzes the “pyramids of knowledge in children, draws conclusions - which level falls and in the next lesson begins updating knowledge from this.)

When studying mathematics, students begin to become familiar with different types of geometric shapes. Today we will talk about different types of triangles.

Definition

Geometric figures that consist of three points that are not on the same line are called triangles.

The segments connecting the points are called sides, and the points are called vertices. Vertices are designated in capital letters, for example: A, B, C.

The sides are designated by the names of the two points from which they consist - AB, BC, AC. Intersecting, the sides form angles. The bottom side is considered the base of the figure.

Rice. 1. Triangle ABC.

Types of triangles

Triangles are classified by angles and sides. Each type of triangle has its own properties.

There are three types of triangles at the corners:

acute-angled;
rectangular;
obtuse-angled.

All angles acute-angled triangles are acute, that is, the degree measure of each is no more than 90 0.

Rectangular a triangle contains a right angle. The other two angles will always be acute, since otherwise the sum of the angles of the triangle will exceed 180 degrees, and this is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two are called the legs. The hypotenuse is always larger than the leg.

Obtuse the triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two angles in such a triangle will be acute.

Rice. 2. Types of triangles at the corners.

A Pythagorean triangle is a rectangle whose sides are 3, 4, 5.

Moreover, the larger side is the hypotenuse.

Such triangles are often used to construct simple problems in geometry. Therefore, remember: if two sides of a triangle are equal to 3, then the third will definitely be 5. This will simplify the calculations.

Types of triangles on the sides:

equilateral;
isosceles;
versatile.

Equilateral a triangle is a triangle in which all sides are equal. All angles of such a triangle are equal to 60 0, that is, it is always acute.

Isosceles triangle - a triangle with only two sides equal. These sides are called lateral, and the third is called the base. In addition, the angles at the base of an isosceles triangle are equal and always acute.

Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.

If the problem does not contain any clarifications about the figure, then it is generally accepted that we are talking about an arbitrary triangle.

Rice. 3. Types of triangles on the sides.

The sum of all angles of a triangle, regardless of its type, is 1800.

Opposite the larger angle is the larger side. And also the length of any side is always less than the sum of its other two sides. These properties are confirmed by the triangle inequality theorem.

There is a concept of the golden triangle. This is an isosceles triangle in which two sides are proportional to the base and equal to a certain number. In such a figure, the angles are proportional to the ratio 2:2:1.

Task:

Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?