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.
Views by angles
The following types of polygon with three vertices are distinguished:
- acute-angled, in which all corners are sharp;
- rectangular, having one right angle, while the sides that form it are called legs, and the side that is placed opposite to the right angle is called the hypotenuse;
- obtuse when one corner is obtuse;
- 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
They highlight 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 outside corner is larger than any inside corner not adjacent to it;
- the sum of any two angles is always less than 180 degrees;
- outer corner equals the sum of the other two corners 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's have an arbitrary triangle with vertices of KMN.
Through the vertex M draw a straight line parallel to the straight line KN (this line is also called the Euclidean straight 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 KN 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 interior one-sided with respect toparallel straight lines KN and MA with a secant KM, their sum is 180 degrees. Theorem 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.
Exterior 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.
Besides, it is known that the external angle of a triangle is equal to the sum of two internal angles that do not intersect with it. Therefore, ∟1=∟A + ∟C, ∟2=∟A + ∟B, ∟3=∟B + ∟C.
From this it turns out that the sum of externalcorners, which are taken one at 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 acute angles of a right triangle? 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 its veracity.
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°. That's what we had to prove.
In addition to the above properties of a right triangle, you can add the following:
- 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;
- lega triangle that lies opposite an angle of 30 degrees is half the hypotenuse, that 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 isosceles is a polygon with three vertices, containing two equal sides. 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 ∟К=∟Н. 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 proved.
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 triangle sum theorem itself was proved earlier.
Except as discussedproperties about the angles of a triangle, there are also such important statements:
- in an isosceles triangle, the height that was lowered to the base is both the median, the bisector of the angle that is between equal sides, as well as the axis of symmetry of 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, it is the triangle with all sides equal. Therefore, the angles are also equal. Each one is 60 degrees. Let's prove this property.
Assume that we have a triangle KMN. 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 statement is proved.
As you can see from the above proof based on the theorem, the sum of the angles of an equilateral triangle, 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:
- 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 beequals (a x √3): 3;
- if you inscribe a circle into 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.
Obt-angled triangle
According to the definition of an obtuse triangle, 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.
