Obtuse triangle: length of sides, sum of angles. Circumscribed obtuse 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 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 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. Math lessons: Guidelines for the teacher. 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 work. 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. I know high school its basic properties are studied, but sometimes 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. Distinctive feature is the internal location of the intersection points of bisectors, medians and altitudes. 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. They exist main feature- equality of two sides and angles at the base.

Sometimes the question arises as to whether a triangle with given sides exists. In fact you are asked if it is suitable this description under the main types. 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 main properties that they have different kinds triangles. In the future, it is important to understand what you are dealing with when solving a problem.

Tasks:

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

2. Development of thinking, imagination, mathematical speech.

3. Cultivating attention and activity.

During the classes

I. Organizational moment.

How much do we need, guys?
For our skillful hands?
Let's draw two squares,
And there is a huge circle on them.
And then more circles,
Triangle cap.
So it came out very, very
Cheerful Oddball.

II. Announcing the topic of the lesson.

Today in the lesson we will take a trip around the city of Geometry and visit the Triangles microdistrict (i.e. 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 the lesson in the form of a “game-competition” by team.

Team 1 - “Segment”.

Team 2 - “Luch”.

Team 3 - “Angle”.

And the guests will represent the jury.

The jury will guide us along the way

And he won’t leave you without attention. (Evaluate by points 5,4,3,...).

What will we use to travel around the city of Geometry? Remember what types of passenger transport are there in the city? There are a lot of us, which one will we choose? (Bus).

Bus. Clearly, briefly. Boarding begins.

Let's sit back and begin our journey. Team captains will receive tickets.

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

III. Repetition of covered material.

First stop“Repeat.”

Question for all teams.

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

The line is straight without end or edge!
Walk along it for at least a hundred years,
You won't find the end of the road!

• A 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. Team captains take turns reading the questions, 1 team reads 1 question).

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

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

(Team 2 reads 1 question).

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

• If you mark a point and draw part of a straight line from it, you will get an image of a ray. The point from which 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, we get geometric figure, 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.

Physical education session (to music).

IV. Preparing to study new material.

Second stop“Fabulous.”

While walking, Pencil met different angles. I wanted to say hello to them, but I forgot the names of each of them. We'll have to help Pencil.

(Angles are checked using a right angle model).

Assignment to teams. Read questions No. 2, 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 acute.

Team 3 reads question 2.

2. Find an obtuse angle, give a definition.

An angle greater than a right angle is called an obtuse angle.

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

And a poem will be a hint for you.

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

What properties are being discussed about the figure?

• About the triangle.

What figure is called a triangle?

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

(Students 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 minute:

We stamp our foot 8 times,
Let's clap our hands 9 times,
we will sit down 10 times,
and bend over 6 times,
we'll jump straight up
so much (triangle showing)
Oh yeah, count! Game and nothing more!

VI. Learning new material.

Soon the corners became friends and became inseparable.

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

Third stop “Znayka”.

What are the names of these triangles?

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

2. Find different types of triangles

Team 1 will find and display obtuse triangles.

Team 2 will find and display right triangles.

Team 3 will find and show acute triangles.

VIII. Next stop: “Get it figured out.”

Assignment to all teams.

By moving 6 sticks, make 4 equal triangles from the lantern.

What kind of angles did the triangles turn out to be? (Acute angular).

IX. Lesson summary.

What neighborhood did we visit?

What types of triangles did you get acquainted with?

Dividing triangles into acute, rectangular and obtuse. Classification by aspect ratio divides triangles into scalene, equilateral and isosceles. Moreover, each triangle simultaneously belongs to two. For example, it can be rectangular and scalene at the same time.

When determining the type by the type of angles, be very careful. An obtuse triangle will be called a triangle in which one of the angles is , that is, more than 90 degrees. A right triangle can be calculated by having one right (equal to 90 degrees) angle. However, to classify a triangle as acute, you will need to make sure that all three of its angles are acute.

Defining the species triangle according to the aspect ratio, first you will have to find out the lengths of all three sides. However, if, according to the condition, the lengths of the sides are not given to you, the angles can help you. A scalene triangle is one in which all three sides have different lengths. If the lengths of the sides are unknown, then a triangle can be classified as scalene if all three of its angles are different. A scalene triangle can be obtuse, right, or acute.

An isosceles triangle is one in which two of its three sides are equal to each other. If the lengths of the sides are not given to you, use two equal angles as a guide. An isosceles triangle, like a scalene triangle, can be obtuse, rectangular or acute.

Only a triangle can be equilateral if all three sides have the same length. All its angles are also equal to each other, and each of them is equal to 60 degrees. From this it is clear that equilateral triangles are always acute.

Tip 2: How to determine an obtuse and acute triangle

The simplest of polygons is a triangle. It is formed using three points lying in the same plane, but not on the same straight line, connected in pairs by segments. However, there are triangles different types, which means they have different properties.

Instructions

It is customary to distinguish three types: obtuse-angled, acute-angled and rectangular. It's like corners. An obtuse triangle is a triangle in which one of the angles is obtuse. An obtuse angle is an angle that is greater than ninety degrees but less than one hundred and eighty. For example, in triangle ABC, angle ABC is 65°, angle BCA is 95°, and angle CAB is 20°. Angles ABC and CAB are less than 90°, but angle BCA is greater, which means the triangle is obtuse.

An acute triangle is a triangle in which all angles are acute. An acute angle is an angle that is less than ninety degrees and greater than zero degrees. For example, in triangle ABC, angle ABC is 60°, angle BCA is 70°, and angle CAB is 50°. All three angles are less than 90°, which means it is a triangle. If you know that a triangle has all sides equal, this means that all its angles are also equal to each other, and they are equal to sixty degrees. Accordingly, all angles in such a triangle are less than ninety degrees, and therefore such a triangle is acute.

If one of the angles in a triangle is ninety degrees, this means that it is neither a wide-angle nor an acute-angle type. This is a right triangle.

If the type of triangle is determined by the ratio of the sides, they will be equilateral, scalene and isosceles. In an equilateral triangle, all sides are equal, and this, as you found out, means that the triangle is acute. If a triangle has only two sides equal or the sides are not equal, it can be obtuse, rectangular, or acute. This means that in these cases it is necessary to calculate or measure the angles and draw conclusions according to points 1, 2 or 3.

Video on the topic

Sources:

• obtuse triangle

The equality of two or more triangles corresponds to the case when all sides and angles of these triangles are equal. However, there are a number of simpler criteria for proving this equality.

You will need

• Geometry textbook, sheet of paper, pencil, protractor, ruler.

Instructions

Open your seventh grade geometry textbook to the section on the criteria for congruence of triangles. You will see that there are a number of basic signs that prove the equality of two triangles. If the two triangles whose equality is being checked are arbitrary, then for them there are three main signs of equality. If any Additional Information about triangles, then the main three features are supplemented by several more. This applies, for example, to the case of equality of right triangles.

Read the first rule about congruence of triangles. As is known, it allows us to consider triangles equal if it can be proven that any one angle and two adjacent sides of two triangles are equal. In order to understand this law, draw on a piece of paper using a protractor two identical specific angles formed by two rays emanating from one point. Using a ruler, measure the same sides from the top of the drawn angle in both cases. Using a protractor, measure the resulting angles of the two triangles formed, making sure they are equal.

In order not to resort to such practical measures to understand the test for equality of triangles, read the proof of the first test for equality. The fact is that every rule about the equality of triangles has a strict theoretical proof, it’s just not convenient to use for the purpose of memorizing the rules.

Read the second test for congruence of triangles. It states that two triangles will be equal if any one side and two adjacent angles of two such triangles are equal. In order to remember this rule, imagine the drawn side of the triangle and its two adjacent angles. Imagine that the lengths of the sides of the corners gradually increase. Eventually they will intersect, forming a third corner. In this mental task, it is important that the intersection point of the sides that are mentally increased, as well as the resulting angle, are uniquely determined by the third side and two adjacent angles.

If you are not given any information about the angles of the triangles being studied, then use the third criterion for the equality of triangles. According to this rule, two triangles are considered equal if all three sides of one of them are equal to the corresponding three sides of the other. Thus, this rule says that the lengths of the sides of a triangle uniquely determine all the angles of the triangle, which means they uniquely determine the triangle itself.

Video on the topic

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

What is a triangle? Definition

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

What types of triangles are there?

They can be classified according to various criteria.

First of all, they are divided into acute-angled, obtuse-angled and rectangular. The former have acute angles, that is, those that are equal to 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 remaining two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. Using it, 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 also recall an isosceles triangle. This is one in which two of the sides are equal, and two angles are also equal.

What are leg and hypotenuse?

A leg is one of the sides of a triangle that forms an angle of 90 degrees. The hypotenuse is the remaining side that is opposite right angle. You can lower a perpendicular from it onto the leg. The ratio of the adjacent side to the hypotenuse is called cosine, and the opposite side is called sine.

- what are its features?

It's rectangular. Its legs are three and four, and its hypotenuse is five. If you see that the legs of a given triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, using this principle, you can easily determine that the leg will be equal to three if the second is equal to four, and the hypotenuse is equal to five. To prove this statement, you can apply the Pythagorean theorem. If two legs are equal to 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is equal to 5. An Egyptian triangle is also a right triangle whose sides are equal to 6, 8 and 10; 9, 12 and 15 and other numbers with the ratio 3:4:5.

What else could a triangle be?

Triangles can also be inscribed or 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 into which a circle is inscribed. All its sides come into contact with it at certain points.

How is it located?

The area of ​​any figure is measured in square units (sq. meters, sq. millimeters, sq. centimeters, sq. 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 corner, 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 located between these sides, and divide this result 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 resulting four values. Next, find from 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 that circumscribed around it, multiplied by four.

The area of ​​a circumscribed 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: square the side, multiply the resulting figure by the root of three, then divide this number by four. In a similar way, you can calculate the height of a triangle in which all sides are equal; to do this, one of them needs to be multiplied by the root of three, and then divided given number by two.

Theorems related to triangle

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

Dali Triangle - what is it?

Many, when faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. Dali's triangle is common name three places that are closely connected with the life of the famous artist. Its “peaks” are the house in which Salvador Dali lived, the castle that he gave to his wife, as well as the museum of surrealist paintings. You can learn a lot during a tour of these places. interesting facts about this unique creative artist known throughout the world.