The sum of the angles of a triangle - what is it equal to? Theorem on the sum of the angles of a triangle From the last two properties it follows that each angle in an equilateral

Theorem. The sum of the interior angles of a triangle is equal to two right angles.

Take some triangle ABC (Fig. 208). Let us denote its interior angles by 1, 2 and 3. Let us prove that

∠1 + ∠2 + ∠3 = 180°.

Let us draw through some vertex of the triangle, for example B, the line MN parallel to AC.

At vertex B, we got three angles: ∠4, ∠2 and ∠5. Their sum is a straight angle, therefore, it is equal to 180 °:

∠4 + ∠2 + ∠5 = 180°.

But ∠4 \u003d ∠1 are internal cross-lying angles with parallel lines MN and AC and a secant AB.

∠5 = ∠3 are internal cross lying angles with parallel lines MN and AC and secant BC.

Hence, ∠4 and ∠5 can be replaced by their equals ∠1 and ∠3.

Therefore, ∠1 + ∠2 + ∠3 = 180°. The theorem is proven.

2. Property of the external angle of a triangle.

Indeed, in triangle ABC (Fig. 209) ∠1 + ∠2 = 180° - ∠3, but also ∠BCD, the external angle of this triangle, not adjacent to ∠1 and ∠2, is also equal to 180° - ∠3 .

Thus:

∠1 + ∠2 = 180° - ∠3;

∠BCD = 180° - ∠3.

Therefore, ∠1 + ∠2= ∠BCD.

The derived property of the external angle of a triangle refines the content of the previously proved theorem on the external angle of a triangle, in which it was stated only that the external angle of a triangle is greater than each internal angle of the triangle that is not adjacent to it; now it is established that the external angle is equal to the sum of both internal angles not adjacent to it.

3. Property of a right triangle with an angle of 30°.

Let the angle B be equal to 30° in a right-angled triangle ACB (Fig. 210). Then its other acute angle will be 60°.

Let us prove that the leg AC is equal to half of the hypotenuse AB. We continue the leg AC beyond the vertex of the right angle C and set aside the segment CM, equal to the segment AC. We connect point M with point B. The resulting triangle BCM is equal to triangle DIA. We see that each angle of the triangle AVM is equal to 60°, therefore, this triangle is equilateral.

The AC leg is equal to half of AM, and since AM is equal to AB, the AC leg will be equal to half of the hypotenuse AB.

Following up on yesterday:

We play with a mosaic for a fairy tale in geometry:

There were triangles. So similar that they are just copies of each other.
They stood side by side in a straight line. And since they were all the same height -
then their tops were on the same level, under the ruler:

Triangles loved to roll and stand on their heads. They climbed to the top row and stood on the corner like acrobats.
And we already know - when they stand with their tops exactly in a line,
then their soles are also lined - because if someone is of the same height, then he is upside down with the same height!

In everything they were the same - and the height was the same, and the soles were one to one,
and slides on the sides - one is steeper, the other is more gentle - the same length
and they have the same slope. Well, just twins! (only in different clothes, each has its own piece of the puzzle).

Where do the triangles have the same sides? Where are the corners?

Triangles stood on the head, stood, and decided to slip off and lie down in the bottom row.
Slipped and slid down like a hill; and the slides are the same!
So they fit exactly between the lower triangles, without gaps, and no one pressed anyone.

We looked around the triangles and noticed an interesting feature.
Wherever their corners met together, all three corners certainly met:
the largest is the "angle-head", the sharpest angle and the third, average angle.
They even tied colored ribbons, so that it would be immediately noticeable where it was.

And it turned out that the three corners of the triangle, if you combine them -
make up one big corner, "open corner" - like the cover of an open book,

______________________O ___________________

That's what it's called: a twisted angle.

Any triangle is like a passport: three angles together are equal to a straight angle.
Someone will knock on you: - knock-knock, I'm a triangle, let me spend the night!
And you to him - Show me the sum of the angles in expanded form!
And it is immediately clear whether this is a real triangle or an impostor.
Failed verification - Turn around one hundred and eighty degrees and go home!

When they say "turn 180 °" it means to turn around backwards and
go in the opposite direction.

The same in more familiar expressions, without "they lived":

Let's make a parallel translation of the triangle ABC along the axis OX
per vector AB equal to the length of the base AB.
Line DF passing through vertices C and C 1 of triangles
parallel to the OX axis, due to the fact that perpendicular to the OX axis
the segments h and h 1 (heights of equal triangles) are equal.
Thus, the base of the triangle A 2 B 2 C 2 is parallel to the base AB
and equal to it in length (because the top C 1 is shifted relative to C by the amount AB).
Triangles A 2 B 2 C 2 and ABC are equal on three sides.
And so the angles ∠A 1 ∠B ∠C 2 , forming a developed angle, are equal to the angles of the triangle ABC.
=> The sum of the angles of a triangle is 180°

With movements - "broadcasts" the so-called proof is shorter and clearer,
on the pieces of the puzzle, even a baby can understand.

But the traditional school:

based on the equality of internal cross-lying angles cut off on parallel lines

valuable in that it gives an idea of ​​why this is so,
Why the sum of the angles of a triangle is equal to the angle?

Because otherwise parallel lines would not have the properties familiar to our world.

Theorems work both ways. From the axiom of parallel lines it follows
equality of crosswise lying and vertical angles, and of them - the sum of the angles of a triangle.

But the opposite is also true: as long as the angles of the triangle are 180 ° - there are parallel lines
(such that through a point not lying on a line it is possible to draw a unique line || given).
If one day a triangle appears in the world, in which the sum of the angles is not equal to the straight angle -
then the parallel ones will cease to be parallel, the whole world will be twisted and skewed.

If the stripes with an ornament of triangles are placed one above the other -
you can cover the entire field with a repeating pattern, like a floor with tiles:

you can trace different shapes on such a grid - hexagons, rhombuses,
star polygons and get a variety of parquets

Tiling a plane with parquet is not only an entertaining game, but also an actual mathematical problem:

________________________________________ _______________________-------__________ ________________________________________ ______________
/\__||_/\__||_/\__||_/\__||_/\__|)0(|_/\__||_/\__||_/\__||_/\__||_/\=/\__||_/ \__||_/\__||_/\__||_/\__|)0(|_/\__||_/\__||_/\__||_/\__||_/\

Since each quadrilateral is a rectangle, square, rhombus, etc.,
can be made up of two triangles,
respectively, the sum of the angles of the quadrilateral: 180° + 180°= 360°

Identical isosceles triangles are folded into squares in different ways.
Small square in 2 parts. Medium of 4. And the largest of the 8.
How many figures in the drawing, consisting of 6 triangles?

Sections: Mathematics

Presentation . (Slide 1)

Lesson type: lesson learning new material.

Lesson Objectives:

• Educational:
  • consider the sum of triangle angles theorem,
  • show the application of the theorem in solving problems.
• Educational:
  • fostering a positive attitude of students to knowledge,
  • instill confidence in students by means of a lesson.
• Educational:
  • development of analytical thinking,
  • development of "skills to learn": to use knowledge, skills and abilities in the educational process,
  • development of logical thinking, the ability to clearly articulate their thoughts.

Equipment: interactive board, presentation, cards.

DURING THE CLASSES

I. Organizational moment

- Today in the lesson we will remember the definitions of right-angled, isosceles, equilateral triangles. Let's repeat the properties of the angles of triangles. Using the properties of internal one-sided and internal cross-lying angles, we will prove the theorem on the sum of the angles of a triangle and learn how to apply it in solving problems.

II. Orally(Slide 2)

1) Find right-angled, isosceles, equilateral triangles in the figures.
2) Define these triangles.
3) Formulate the properties of the angles of an equilateral and isosceles triangle.

4) In the figure KE II NH. (slide 3)

– Specify secants for these lines
– Find internal one-sided angles, internal cross-lying angles, name their properties

III. Explanation of new material

Theorem. The sum of the angles of a triangle is 180 o

According to the formulation of the theorem, the guys build a drawing, write down the condition, conclusion. Answering the questions, independently prove the theorem.

Given: Prove:

Proof:

1. Draw a line BD II AC through the vertex B of the triangle.
2. Specify secants for parallel lines.
3. What can be said about the angles CBD and ACB? (make a record)
4. What do we know about angles CAB and ABD? (make a record)
5. Replace angle CBD with angle ACB
6. Make a conclusion.

IV. Finish the offer.(Slide 4)

1. The sum of the angles of a triangle is ...
2. In a triangle, one of the angles is equal, the other, the third angle of the triangle is equal to ...
3. The sum of the acute angles of a right triangle is ...
4. The angles of an isosceles right triangle are equal to ...
5. The angles of an equilateral triangle are equal ...
6. If the angle between the sides of an isosceles triangle is 1000, then the angles at the base are ...

V. A bit of history.(Slides 5-7)

A triangle is a polygon with three sides (three corners). Most often, the sides are denoted by small letters, corresponding to the capital letters that denote opposite vertices. In this article, we will get acquainted with the types of these geometric shapes, a theorem that determines what the sum of the angles of a triangle is.

Types by the size of the angles

There are the following types of polygon with three vertices:

• acute-angled, in which all corners are sharp;
• rectangular, having one right angle, with its generators, are called legs, and the side that is located opposite the right angle is called the hypotenuse;
• obtuse when alone;
• isosceles, in which two sides are equal, and they are called lateral, and the third is the base of the triangle;
• equilateral, having all three equal sides.

Properties

Allocate the main properties that are characteristic of each type of triangle:

• opposite the larger side there is always a larger angle, and vice versa;
• opposite sides of equal size are equal angles, and vice versa;
• any triangle has two acute angles;
• an exterior angle is larger compared to any interior angle not adjacent to it;
• the sum of any two angles is always less than 180 degrees;
• An exterior angle is equal to the sum of the other two angles that do not intersect with it.

Triangle sum of angles theorem

The theorem states that if you add up all the angles of a given geometric figure, which is located on the Euclidean plane, then their sum will be 180 degrees. Let's try to prove this theorem.

Let us have an arbitrary triangle with vertices of KMN.

Draw a KN through the vertex M (this line is also called the Euclidean line). We mark point A on it in such a way that points K and A are located on different sides of the straight line MN. We get equal angles AMN and KNM, which, like internal ones, lie crosswise and are formed by the secant MN together with straight lines KH and MA, which are parallel. From this it follows that the sum of the angles of the triangle located at the vertices M and H is equal to the size of the angle KMA. All three angles make up the sum, which is equal to the sum of the angles KMA and MKN. Since these angles are internal one-sided with respect to parallel straight lines KN and MA with a secant KM, their sum is 180 degrees. The theorem is proven.

Consequence

The following corollary follows from the theorem proved above: any triangle has two acute angles. To prove this, let us assume that a given geometric figure has only one acute angle. It can also be assumed that none of the angles is acute. In this case, there must be at least two angles that are equal to or greater than 90 degrees. But then the sum of the angles will be greater than 180 degrees. But this cannot be, because according to the theorem, the sum of the angles of a triangle is 180 ° - no more and no less. This is what had to be proven.

External corner property

What is the sum of the angles of a triangle that are external? This question can be answered in one of two ways. The first is that it is necessary to find the sum of the angles, which are taken one at each vertex, that is, three angles. The second implies that you need to find the sum of all six angles at the vertices. First, let's deal with the first option. So, the triangle contains six external corners - two at each vertex.

Each pair has equal angles because they are vertical:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

In addition, it is known that the external angle of a triangle is equal to the sum of two internal ones that do not intersect with it. Hence,

∟1 = ∟A + ∟C, ∟2 = ∟A + ∟B, ∟3 = ∟B + ∟C.

From this it turns out that the sum of the external angles, which are taken one at a time near each vertex, will be equal to:

∟1 + ∟2 + ∟3 = ∟A + ∟C + ∟A + ∟B + ∟B + ∟C = 2 x (∟A + ∟B + ∟C).

Given that the sum of the angles is 180 degrees, it can be argued that ∟A + ∟B + ∟C = 180°. And this means that ∟1 + ∟2 + ∟3 = 2 x 180° = 360°. If the second option is used, then the sum of the six angles will be, respectively, twice as large. That is, the sum of the external angles of the triangle will be:

∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 = 2 x (∟1 + ∟2 + ∟2) = 720°.

Right triangle

What is the sum of the angles of a right triangle that are acute? The answer to this question, again, follows from the theorem, which states that the angles in a triangle add up to 180 degrees. And our statement (property) sounds like this: in a right triangle, acute angles add up to 90 degrees. Let's prove it to be true.

Let us be given a triangle KMN, in which ∟Н = 90°. It is necessary to prove that ∟K + ∟M = 90°.

So, according to the angle sum theorem, ∟К + ∟М + ∟Н = 180°. Our condition says that ∟Н = 90°. So it turns out, ∟K + ∟M + 90° = 180°. That is, ∟K + ∟M = 180° - 90° = 90°. This is exactly what we had to prove.

In addition to the above properties of a right triangle, you can add the following:

• the angles that lie against the legs are sharp;
• the hypotenuse is triangular more than any of the legs;
• the sum of the legs is greater than the hypotenuse;
• the leg of the triangle, which lies opposite the angle of 30 degrees, is half the hypotenuse, that is, it is equal to half of it.

As another property of this geometric figure, the Pythagorean theorem can be distinguished. She states that in a triangle with an angle of 90 degrees (rectangular), the sum of the squares of the legs is equal to the square of the hypotenuse.

The sum of the angles of an isosceles triangle

Earlier we said that a polygon with three vertices and two equal sides is called isosceles. This property of a given geometric figure is known: the angles at its base are equal. Let's prove it.

Take the triangle KMN, which is isosceles, KN ​​is its base.

We are required to prove that ∟K = ∟H. So, let's say that MA is the bisector of our triangle KMN. The MCA triangle, taking into account the first sign of equality, is equal to the MCA triangle. Namely, by condition it is given that KM = NM, MA is a common side, ∟1 = ∟2, since MA is a bisector. Using the fact that these two triangles are equal, we can state that ∟K = ∟Н. So the theorem is proven.

But we are interested in what is the sum of the angles of a triangle (isosceles). Since in this respect it does not have its own peculiarities, we will start from the theorem considered earlier. That is, we can say that ∟K + ∟M + ∟H = 180°, or 2 x ∟K + ∟M = 180° (since ∟K = ∟H). We will not prove this property, since the theorem on the sum of angles of a triangle itself was proved earlier.

In addition to the considered properties about the angles of a triangle, there are also such important statements:

• in which it was lowered to the base, is at the same time the median, the bisector of the angle that is between equal sides, as well as its base;
• medians (bisectors, heights) that are drawn to the sides of such a geometric figure are equal.

Equilateral triangle

It is also called right, this is the triangle in which all sides are equal. Therefore, the angles are also equal. Each one is 60 degrees. Let's prove this property.

Let's say we have a KMN triangle. We know that KM = NM = KN. And this means that according to the property of the angles located at the base in an isosceles triangle, ∟К = ∟М = ∟Н. Since, according to the theorem, the sum of the angles of a triangle is ∟К + ∟М + ∟Н = 180°, then 3 x ∟К = 180° or ∟К = 60°, ∟М = 60°, ∟Н = 60°. Thus, the assertion is proved.

As can be seen from the above proof based on the theorem, the sum of the angles, like the sum of the angles of any other triangle, is 180 degrees. There is no need to prove this theorem again.

There are also such properties characteristic of an equilateral triangle:

• the median, bisector, height in such a geometric figure are the same, and their length is calculated as (a x √3): 2;
• if you describe a circle around a given polygon, then its radius will be equal to (a x √3): 3;
• if you inscribe a circle in an equilateral triangle, then its radius will be (a x √3): 6;
• the area of ​​this geometric figure is calculated by the formula: (a2 x √3): 4.

obtuse triangle

By definition, one of its angles is between 90 and 180 degrees. But given that the other two angles of this geometric figure are acute, we can conclude that they do not exceed 90 degrees. Therefore, the triangle sum of angles theorem works when calculating the sum of angles in an obtuse triangle. It turns out that we can safely say, based on the aforementioned theorem, that the sum of the angles of an obtuse triangle is 180 degrees. Again, this theorem does not need to be re-proved.

RESEARCH

ON THE TOPIC OF:

"Does the sum of the angles of a triangle always equal 180˚?"

Completed:

7b grade student

MBOU Inza secondary school №2

Inza, Ulyanovsk region

Malyshev Yan

Scientific adviser:

Bolshakova Ludmila Yurievna

TABLE OF CONTENTS

Introduction…………………………………………………..3 page

Main part……………………………………………4

search for information

experiences

conclusion

Conclusion………………………………………………..12

INTRODUCTION

This year I started to study a new subject - geometry. This science studies the properties of geometric shapes. In one of the lessons we studied the triangle sum theorem. And with the help of the proof, they concluded: the sum of the angles of a triangle is 180˚.

I thought, are there any triangles in which the sum of the angles will not be equal to 180˚?

Then I set myselfTARGET :

Find out when the sum of the angles of a triangle is not equal to 180˚?

Put the followingTASKS :

Learn about the history of geometry

Get acquainted with the geometry of Euclid, Roman, Lobachevsky;

Prove empirically that the sum of the angles of a triangle may not be equal to 180˚.

MAIN PART

Geometry arose and developed in connection with the needs of practical human activity. When building even the most primitive structures, it is necessary to be able to calculate how much material will be spent on construction, calculate the distances between points in space and the angles between planes. The development of trade and navigation required the ability to navigate in time and space.

Scientists of ancient Greece did a lot for the development of geometry. The first proofs of geometric facts are associated with the nameThales of Miletus.

One of the most famous schools was the Pythagorean, named after its founder, the author of the proofs of many theorems,Pythagoras.

Geometry, which is studied at school, is called Euclidean, named afterEuclid - Ancient Greek scientist.

Euclid lived in Alexandria. He wrote the famous book "Beginnings". Consistency and rigor made this work a source of geometric knowledge in many countries of the world for more than two millennia. Until recently, almost all school textbooks were in many ways similar to the "Beginnings".

But in the 19th century it was shown that Euclid's axioms are not universal and are not true in all circumstances. The main discoveries of a geometric system in which Euclid's axioms are not true were made by Georg Riemann and Nikolai Lobachevsky. They are spoken of as the creators of non-Euclidean geometry.

And now, relying on the teachings of Euclid, Riemann and Lobachevsky, let's try to answer the question: is the sum of the angles of a triangle always equal to 180˚?

EXPERIENCES

Consider a triangle in terms of geometryEuclid.

To do this, take a triangle.

Let's paint its corners with red, green and blue colors.

Let's draw a straight line. This is a straight angle, it is equal to 180 ˚.

Let's cut off the corners of our triangle and attach them to the unfolded corner. We see that the sum of the three angles is 180˚.

One of the stages in the development of geometry was elliptical geometryRiemann. A special case of this elliptic geometry is the geometry on the sphere. In Riemannian geometry, the sum of the angles of a triangle is greater than 180˚.

So this is a sphere.

Inside this sphere, the meridians and the equator form a triangle. Take this triangle, paint over its corners.

Let's cut them off and attach them to a straight line. We see that the sum of the three angles is greater than 180˚.

In geometryLobachevsky the sum of the angles of a triangle is less than 180˚.

This geometry is viewed on the surface of a hyperbolic paraboloid (it is a concave surface resembling a saddle).

Examples of paraboloids can be found in architecture.

And even Pringle chips are an example of a paraboloid.

Let's check the sum of angles on the hyperbolic paraboloid model.

A triangle is formed on the surface.

Let's take this triangle, paint over its corners, cut them off and attach them to a straight line. Now we see that the sum of the three angles is less than 180˚.

CONCLUSION

Thus, we have proved that the sum of the angles of a triangle is not always equal to 180˚.

It can be more or less.

CONCLUSION

In conclusion of my work, I want to say that it was interesting to work on this topic. I learned a lot of new things for myself and, in the future, I will be happy to study this interesting geometry.

INFORMATION SOURCES

en.wikipedia.org

e-osnova.ru

vestishki.ru

yun.moluch.ru