The angles of a triangle are always Theorem on the sum of the angles of a triangle. From the last two properties it follows that each angle in an equilateral
A triangle is a polygon with three sides (three corners). Most often, the sides are indicated in small letters, corresponding to capital letters, which represent 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
Distinguish the following types polygon with three vertices:
- acute-angled, in which all corners are sharp;
- rectangular, having one right angle, with its generators, called legs, and the side that is placed opposite 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 we 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 with different parties direct 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 has been 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. Consequently,
∟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 equals 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. ICA triangle with the first sign of equality equal to triangle MNA. 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:
- into which 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 foundation;
- 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.
Targets and goals:
Educational:
- repeat and generalize knowledge about the triangle;
- prove the triangle sum theorem;
- practically verify the correctness of the formulation of the theorem;
- learn to apply the acquired knowledge in solving problems.
Developing:
- to develop geometric thinking, interest in the subject, cognitive and creative activity of students, mathematical speech, the ability to independently acquire knowledge.
Educational:
- develop personal qualities students, such as purposefulness, perseverance, accuracy, ability to work in a team.
Equipment: multimedia projector, triangles made of colored paper, teaching materials "Live Mathematics", computer, screen.
Preparatory stage: The teacher instructs the student to prepare historical reference about the triangle sum of angles theorem.
Lesson type: learning new material.
During the classes
I. Organizational moment
Greetings. Psychological attitude of students to work.
II. Warm up
We met 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 suggestions and the teacher writes them on the blackboard. The results are being discussed:
Picture 1
III. We formulate the task of the lesson
So, we already know a lot about the triangle. But not all. Each of you has triangles and protractors on your desk. What do you think, what task can we 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(contributes to the actualization of knowledge and self-knowledge skills). Measure the angles with a protractor and find their sum. Write down the results in a notebook (listen to the answers received). We find out that the sum of the angles for everyone turned out to be different (this can happen because the protractor was inaccurately applied, the calculation was carelessly performed, etc.).
Fold along the dotted lines and find out what else the sum of the angles of the triangle is equal to:
a) 
Figure 2
b) 
Figure 3
in) 
Figure 4
G) 
Figure 5
e) 
Figure 6
After completing the practical work, the students formulate the answer: The sum of the angles of a triangle is equal to the degree measure of the expanded angle, i.e. 180°.
Teacher: In mathematics practical work only makes it possible to make some kind of statement, but it needs to be proved. 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.
History reference: The property of the sum of the angles of a triangle was established in ancient Egypt. The proof given in modern textbooks is found in Proclus' comments 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 is easy to understand with the help of a drawing (Fig. 7):
Figure 7
Figure 8
Drawings are displayed on the screen through a projector.
The teacher offers to prove the theorem with the help of drawings.
Then the proof is carried out using the CMD "Live Mathematics". The teacher on the computer projects the proof of the theorem.
Triangle sum of angles theorem: "The sum of the angles of a triangle is 180°"
Figure 9
Proof:
a) 
Figure 10
b) 
Figure 11
in) 
Figure 12
The students in the notebook make a brief record of the proof of the theorem:
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. minute.
VI. Explanation of new material (continued)
The consequence of the theorem on the sum of the angles of a triangle is derived by students on their own, this contributes to the development of the ability to formulate their own point of view, express and argue it:
In any triangle, either all angles are acute, or two acute angles, and the third obtuse or right.
If all angles in a triangle are acute, then it is called acute-angled.
If one of the angles of a triangle is obtuse, then it is called obtuse.
If one of the angles of a triangle is right, then it is called rectangular.
The triangle sum theorem allows us to classify triangles not only by sides, but also by angles. (In the course of introducing the types of triangles, students fill out a table)
Table 1
| Triangle view | Isosceles | Equilateral | Versatile |
| Rectangular | 
|
|
|
| obtuse | 
|
|
|
| acute-angled | 
|
|
|
VII. Consolidation of the studied material.
- Solve problems orally:
(The drawings are displayed on the screen through the projector)
(basic abstract)
Visual geometry Grade 7. Reference abstract No. 4 The sum of the angles of a triangle.
Great French scientist of the 17th century Blaise Pascal loved to play with as a child geometric shapes. He was familiar with the protractor and knew how to measure angles. The young researcher noticed that for all triangles the sum of three angles is the same - 180 °. “How can you prove it? thought Pascal. “After all, you can’t check the sum of the angles of all triangles - there are an infinite number of them.” Then he cut off two corners of the triangle with scissors and attached them to the third corner. It turned out a developed angle, which, as you know, is equal to 180 °. It was his first own discovery. The further fate of the boy was already predetermined.
In this topic, you'll learn five features of right triangle equality and perhaps the most popular property of a 30° right triangle. It sounds like this: the leg lying opposite an angle of 30 ° is equal to half the hypotenuse. Dividing an equilateral triangle with a height, we immediately get a proof of this property.
THEOREM. The sum of the angles of a triangle is 180°. To prove this, we draw a line through the vertex parallel to the base. The dark angles are equal and the gray angles are equal as they lie across parallel lines. The dark corner, the gray corner and the corner at the apex form a straight corner, their sum is 180°. It follows from the theorem that the angles of an equilateral triangle are 60° each and that the sum of the acute angles of a right triangle is 90°.
outside corner triangle is called the angle adjacent to the angle of the triangle. Therefore, sometimes the angles of the triangle itself are called internal angles.
THEOREM on the exterior angle of a triangle. The external angle of a triangle is equal to the sum of two internal angles that are not adjacent to it. Indeed, an outer corner and two inner ones not adjacent to it complete the filled corner up to 180°. It follows from the theorem that an exterior angle is greater than any interior angle not adjacent to it.
THEOREM on the relationships between sides and angles of a triangle. In a triangle, the larger side is opposite the larger angle, and the larger side is opposite the larger angle. It follows from this: 1) The leg is less than the hypotenuse. 2) The perpendicular is less than the slope.
Distance from point to line . Since the perpendicular is less than any oblique drawn from the same point, its length is taken as the distance from the point to the line.
triangle inequality . The length of any side of a triangle is less than the sum of its other two sides, i.e. a< b + с , b< а + с , With< а + b . Consequence. The length of the polyline is greater than the segment connecting its ends.
SIGNS OF EQUALITY
RECTANGULAR TRIANGLES
On two legs. If two legs of one right triangle are respectively equal to two legs of another triangle, then such triangles are congruent.
Along the leg and adjacent acute angle. If the leg and the acute angle adjacent to it of one right-angled triangle are respectively equal to the leg and the acute angle adjacent to it of another triangle, then such triangles are congruent.
Along the leg and opposite acute angle. If the leg and the opposite acute angle of one right triangle are respectively equal to the leg and the opposite acute angle of another triangle, then such triangles are congruent.
By hypotenuse and acute angle. If the hypotenuse and acute angle of one right triangle are respectively equal to the hypotenuse and acute angle of another triangle, then such triangles are congruent.
The proof of these criteria immediately reduces to one of the criteria for the equality of triangles.
By leg and hypotenuse. If the leg and hypotenuse of one right triangle are respectively equal to the leg and hypotenuse of another right triangle, then such triangles are congruent.
Proof. We apply triangles with equal legs. We get an isosceles triangle. Its height drawn from the top will also be the median. Then the second legs of the triangles are equal, and the triangles are equal on three sides.
THEOREM on the property of a leg lying opposite an angle of 30°. The leg opposite the 30° angle is equal to half the hypotenuse. It is proved by completing the triangle to an equilateral one.
THEOREM on the property of angle bisector points. Any point on the bisector of an angle is equidistant from its sides. If a point is equidistant from the sides of an angle, then it lies on the bisector of the angle. Proved by drawing two perpendiculars to the sides of the angle and considering right triangles.
Second great point . The bisectors of a triangle intersect at one point.
Distance between parallel lines. THEOREM. All points of each of two parallel lines are at the same distance from the other line. The definition of the distance between parallel lines follows from the theorem.
Definition. The distance between two parallel lines is the distance from any point on one of the parallel lines to the other line.
Detailed proofs of theorems
This is the reference abstract No. 4 in geometry in grade 7. Choose next steps:
The sum of the angles of a triangle important but sufficient simple theme, which take place in the 7th grade on geometry. The topic consists of a theorem, a short proof and several logical consequences. Knowledge of this topic helps in solving geometric problems in the subsequent study of the subject.
Theorem - what are the angles of an arbitrary triangle folded together?
The theorem says - if you take any triangle, regardless of its type, the sum of all angles will invariably be 180 degrees. This is proved as follows:
- for example, take the triangle ABC, draw a straight line through the point B located at the top and designate it as “a”, while the straight line “a” is strictly parallel to the side AC;
- between the straight line "a" and the sides AB and BC designate the angles, marking them with the numbers 1 and 2;
- angle 1 is recognized as equal to angle A, and angle 2 is equal to angle C, since these angles are considered to be lying crosswise;
- thus, the sum between angles 1, 2 and 3 (which is indicated in place of angle B) is recognized as equal to the expanded angle with vertex B - and is 180 degrees.
If the sum of the angles indicated by the numbers is 180 degrees, then the sum of the angles A, B and C is recognized as equal to 180 degrees. This rule is true for any triangle.
What follows from the geometric theorem
It is customary to single out several corollaries from the above theorem.
- If the problem considers a triangle with a right angle, then one of its angles will default to 90 degrees, and the sum of acute angles will also be 90 degrees.
- If a we are talking about a right-angled isosceles triangle, then its acute angles, in the sum of 90 degrees, will individually be equal to 45 degrees.
- An equilateral triangle consists of three equal angles, respectively, each of them will be equal to 60 degrees, and in total they will be 180 degrees.
- The exterior angle of any triangle will be equal to the sum between the two interior angles not adjacent to it.
We can deduce the following rule - in any of the triangles there are at least two acute angles. In some cases, the triangle consists of three acute angles, and if there are only two, then the third angle will be obtuse or right.
Proof:
- Triangle ABC is given.
- Draw a line DK through the vertex B parallel to the base AC.
- \angle CBK= \angle C as internal crosswise lying with parallel DK and AC, and secant BC.
- \angle DBA = \angle A internal crosswise lying at DK \parallel AC and secant AB. Angle DBK is straight and equal to
- \angle DBK = \angle DBA + \angle B + \angle CBK
- Since the straight angle is 180 ^\circ , and \angle CBK = \angle C and \angle DBA = \angle A , we get 180 ^\circ = \angle A + \angle B + \angle C.
Theorem proven
Consequences from the theorem on the sum of angles of a triangle:
- The sum of the acute angles of a right triangle is 90°.
- In an isosceles right triangle, each acute angle is 45°.
- In an equilateral triangle, each angle is 60°.
- In any triangle, either all angles are acute, or two angles are acute, and the third is obtuse or right.
- An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.
Triangle exterior angle theorem
An exterior angle of a triangle is equal to the sum of the two remaining angles of the triangle that are not adjacent to that exterior angle.
Proof:
- Triangle ABC is given, where BCD is the exterior angle.
- \angle BAC + \angle ABC +\angle BCA = 180^0
- From the equalities, the angle \angle BCD + \angle BCA = 180^0
- We get \angle BCD = \angle BAC+\angle ABC.
