Theorem on the sum of the angles of a triangle. Sum of angles of a triangle. Theorem on the sum of the angles of a triangle The sum of its angles is 180 degrees

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

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

∠1 + ∠2 + ∠3 = 180°.

Let us draw through some vertex of the triangle, for example B, a straight 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 = ∠1 are internal crosswise angles with parallel lines MN and AC and secant AB.

∠5 = ∠3 - these are internal crosswise angles with parallel lines MN and AC and secant BC.

This means that ∠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.

Theorem. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.

In fact, in triangle ABC (Fig. 209) ∠1 + ∠2 = 180° - ∠3, but also ∠ВСD, 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 exterior angle of a triangle clarifies the content of the previously proven theorem on the exterior angle of a triangle, which stated only that the exterior angle of a triangle is greater than each interior angle of a triangle 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°.

Theorem. A leg of a right triangle lying opposite an angle of 30° is equal to half the hypotenuse.

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

Let us prove that leg AC is equal to half the hypotenuse AB. Let's extend the leg AC beyond the vertex of the right angle C and set aside a segment CM equal to the segment AC. Let's connect point M to point B. The resulting triangle ВСМ is equal to triangle ACB. We see that each angle of triangle ABM is equal to 60°, therefore this triangle is an equilateral triangle.

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

Can you prove that the sum of the angles in a triangle is equal to 180 degrees? and got the best answer

Answer from Top_ed[guru]
Why prove something that has already been proven a very, very long time ago.
The triangle angle sum theorem, a classical theorem of Euclidean geometry, states that
The sum of the angles of a triangle is 180°.
Let ABC be an arbitrary triangle. Let us draw a line through vertex B parallel to line AC. Let us mark point D on it so that points A and D lie on opposite sides of line BC.
Angles DBC and ACB are congruent as internal cross-lying ones formed by the transversal BC with parallel lines AC and BD. Therefore, the sum of the angles of a triangle at vertices B and C is equal to angle ABD.
The sum of all three angles of a triangle is equal to the sum of angles ABD and BAC. Since these are one-sided interior angles for parallel AC and BD and secant AB, their sum is 180°. The theorem is proven.

Reply from Boriska(c)[guru]
I can, but I don’t remember how))

Reply from Murashkina[guru]
Can. Is it urgent for you? ? Are you taking the fifth grade exam? ? :))

Reply from Oriy Semykin[guru]
1. It depends on the geometry of the space. On the Riemann plane > 180, on the square. Lobachevsky< 180. На Эвклидовой - равенство.
2. Draw a line through the vertex parallel to one of the sides and examine the crosswise angles formed by the two sides and the additional line. The resulting angle (180) is equal to the sum of the three angles of the triangle.

The proof essentially relies on the fact that only one parallel line can be drawn. There are a lot of geometries where this is not the case.

Reply from Yuri[guru]
Why prove what has been proven?)) Cut the square into two parts if you want something new))

Reply from Nikolay Evgenievich[guru]
I can't.

Reply from Alex Brichka[expert]
Yes, there’s nothing to prove here, you just need to add angles to each other and that’s it.

Reply from 2 answers[guru]

Hello! Here is a selection of topics with answers to your question: Can you prove that the sum of the angles in a triangle is equal to 180 degrees?

Following on from yesterday:

Let's play with a mosaic based on a geometry fairy tale:

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

Triangles loved to tumble 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 also follow a ruler - because if someone is the same height, then they are also the same height upside down!

They were the same in everything - the same height, and the same soles,
and the slides on the sides - one steeper, the other flatter - are the same in length
and they have the same slope. Well, just twins! (only in different clothes, each with their own piece of the puzzle).

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

The triangles stood on their heads, stood there, and decided to slide off and lie down in the bottom row.
They slid and slid down a hill; but their slides are the same!
So they fit exactly between the lower triangles, without gaps, and no one pushed anyone aside.

We looked around the triangles and noticed an interesting feature.
Wherever their angles come together, all three angles will certainly meet:
the largest is the “head angle”, the most acute angle and the third, medium largest angle.
They even tied colored ribbons so that it would be immediately obvious which was which.

And it turned out that the three angles of the triangle, if you combine them -
make up one large angle, an “open corner” - like the cover of an open book,

______________________O ___________________

it's called a turned angle.

Any triangle is like a passport: three angles together are equal to the unfolded angle.
Someone knocks on your door: - knock-knock, I'm a triangle, let me spend the night!
And you tell him - Show me the sum of the angles in expanded form!
And it’s 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 thing in more familiar expressions, without “once upon a time”:

Let us perform a parallel translation of triangle ABC along the OX axis
to 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
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 (since the vertex 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.
Therefore, the angles ∠A 1 ∠B ∠C 2 forming a straight angle are equal to the angles of triangle ABC.
=> The sum of the angles of a triangle is 180°

With movements - “translations”, the so-called proof is shorter and clearer,
even a child can understand the pieces of the mosaic.

But 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 reverse angle?

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

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

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

If stripes with triangle patterns 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 a relevant mathematical problem:

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

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

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

Preliminary information

First, let's look directly at the concept of a triangle.

Definition 1

We will call a triangle a geometric figure that is made up of three points connected to each other by segments (Fig. 1).

Definition 2

Within the framework of Definition 1, we will call the points the vertices of the triangle.

Definition 3

Within the framework of Definition 1, the segments will be called sides of the triangle.

Obviously, any triangle will have 3 vertices, as well as three sides.

Theorem on the sum of angles in a triangle

Let us introduce and prove one of the main theorems related to triangles, namely the theorem on the sum of angles in a triangle.

Theorem 1

The sum of the angles in any arbitrary triangle is $180^\circ$.

Proof.

Consider the triangle $EGF$. Let us prove that the sum of the angles in this triangle is equal to $180^\circ$. Let's make an additional construction: draw the straight line $XY||EG$ (Fig. 2)

Since the lines $XY$ and $EG$ are parallel, then $∠E=∠XFE$ lie crosswise at the secant $FE$, and $∠G=∠YFG$ lie crosswise at the secant $FG$

Angle $XFY$ will be reversed and therefore equals $180^\circ$.

$∠XFY=∠XFE+∠F+∠YFG=180^\circ$

Hence

$∠E+∠F+∠G=180^\circ$

The theorem is proven.

Triangle Exterior Angle Theorem

Another theorem on the sum of angles for a triangle can be considered the theorem on the external angle. First, let's introduce this concept.

Definition 4

We will call an external angle of a triangle an angle that is adjacent to any angle of the triangle (Fig. 3).

Let us now consider the theorem directly.

Theorem 2

An exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.

Proof.

Consider an arbitrary triangle $EFG$. Let it have an external angle of the triangle $FGQ$ (Fig. 3).

By Theorem 1, we will have that $∠E+∠F+∠G=180^\circ$, therefore,

$∠G=180^\circ-(∠E+∠F)$

Since the angle $FGQ$ is external, it is adjacent to the angle $∠G$, then

$∠FGQ=180^\circ-∠G=180^\circ-180^\circ+(∠E+∠F)=∠E+∠F$

The theorem is proven.

Sample tasks

Example 1

Find all angles of a triangle if it is equilateral.

Since all the sides of an equilateral triangle are equal, we will have that all the angles in it are also equal to each other. Let us denote their degree measures by $α$.

Then, by Theorem 1 we get

$α+α+α=180^\circ$

Answer: all angles equal $60^\circ$.

Example 2

Find all angles of an isosceles triangle if one of its angles is equal to $100^\circ$.

Let us introduce the following notation for angles in an isosceles triangle:

Since we are not given in the condition exactly what angle $100^\circ$ is equal to, then two cases are possible:

An angle equal to $100^\circ$ is the angle at the base of the triangle.

Using the theorem on angles at the base of an isosceles triangle, we obtain

$∠2=∠3=100^\circ$

But then only their sum will be greater than $180^\circ$, which contradicts the conditions of Theorem 1. This means that this case does not occur.

An angle equal to $100^\circ$ is the angle between equal sides, that is

Goals and objectives:

Educational:

repeat and generalize knowledge about the triangle;
prove the theorem on the sum of the angles of a triangle;
practically verify the correctness of the formulation of the theorem;
learn to apply acquired knowledge when solving problems.

Educational:

develop geometric thinking, interest in the subject, cognitive and creative activity of students, mathematical speech, and the ability to independently obtain knowledge.

Educational:

develop students’ personal qualities, such as determination, perseverance, accuracy, and the ability to work in a team.

Equipment: multimedia projector, triangles made of colored paper, educational complex “Living Mathematics”, computer, screen.

Preparatory stage: The teacher gives the student the task of preparing a historical note about the theorem “Sum of the angles of a triangle.”

Lesson type: learning new material.

Lesson progress

I. Organizational moment

Greetings. Psychological attitude of students to work.

II. Warm-up

We became familiar with the geometric figure “triangle” in previous lessons. Let's repeat what we know about the triangle?

Students work in groups. They are given the opportunity to communicate with each other, each to independently build the process of cognition.

What happened? Each group makes their proposals, the teacher writes them on the board. The results are discussed:

Figure 1

III. Formulating the lesson objective

So, we already know quite a lot about the triangle. But not all. Each of you has triangles and protractors on your desk. What kind of problem do you think we can formulate?

Students formulate the task of the lesson - to find the sum of the angles of a triangle.

IV. Explanation of new material

Practical part(promotes updating knowledge and self-knowledge skills). Measure the angles using a protractor and find their sum. Write down the results in your notebook (listen to the answers received). We find out that the sum of the angles is different for everyone (this can happen because the protractor was not applied accurately, the calculation was carried out carelessly, etc.).

Fold along the dotted lines and find out what else the sum of the angles of a triangle is equal to:

A)
Figure 2

b)
Figure 3

V)
Figure 4

G)
Figure 5

d)
Figure 6

After completing the practical work, students formulate the answer: The sum of the angles of a triangle is equal to the degree measure of the unfolded angle, i.e. 180°.

Teacher: In mathematics, practical work only makes it possible to make some kind of statement, but it needs to be proven. A statement whose validity is established by proof is called a theorem. What theorem can we formulate and prove?

Students: The sum of the angles of a triangle is 180 degrees.

Historical information: The property of the sum of the angles of a triangle was established in Ancient Egypt. The proof, set out in modern textbooks, is contained in Proclus's commentary on Euclid's Elements. Proclus claims that this proof (Fig. 8) was discovered by the Pythagoreans (5th century BC). In the first book of the Elements, Euclid sets out another proof of the theorem on the sum of the angles of a triangle, which can be easily understood with the help of a drawing (Fig. 7):

Figure 7

Figure 8

The drawings are displayed on the screen through a projector.

The teacher offers to prove the theorem using drawings.

Then the proof is carried out using the teaching and learning complex “Living Mathematics”. The teacher projects the proof of the theorem on the computer.

Theorem on the sum of angles of a triangle: “The sum of the angles of a triangle is 180°”

Figure 9

Proof:

A)

Figure 10

b)

Figure 11

V)

Figure 12

Students make a brief note of the proof of the theorem in their notebooks:

Theorem: The sum of the angles of a triangle is 180°.

Figure 13

Given:Δ ABC

Prove: A + B + C = 180°.

Proof:

What needed to be proven.

V. Phys. just a minute.

VI. Explanation of new material (continued)

The corollary from the theorem about the sum of the angles of a triangle is deduced by students independently, this contributes to the development of the ability to formulate their own point of view, express and argue for it:

In any triangle, either all angles are acute, or two are acute and the third is obtuse or right..

If a triangle has all acute angles, then it is called acute-angled.

If one of the angles of a triangle is obtuse, then it is called obtuse-angled.

If one of the angles of a triangle is right, then it is called rectangular.

The theorem on the sum of triangle angles allows us to classify triangles not only by sides, but also by angles. (As students introduce types of triangles, students fill out the table)

Table 1

Triangle view	Isosceles	Equilateral	Versatile
Rectangular
Obtuse
Acute-angled

VII. Consolidation of the studied material.

Solve problems orally:

(Drawings are displayed on the screen through a projector)

Task 1. Find angle C.

Figure 14

Problem 2. Find the angle F.

Figure 15

Task 3. Find the angles K and N.

Figure 16

Problem 4. Find the angles P and T.

Figure 17

Solve problem No. 223 (b, d) yourself.
Solve the problem on the board and in notebooks, student No. 224.
Questions: Can a triangle have: a) two right angles; b) two obtuse angles; c) one right and one obtuse angle.
(done orally) The cards on each table show various triangles. Determine by eye the type of each triangle.

Figure 18