A triangle is a polygon having three sides (three angles). Most often, the sides are indicated in small letters corresponding to capital letters, which indicate opposite vertices. In this article we will get acquainted with the types of these geometric figures, a theorem that determines what the sum of the angles of a triangle equals.
Types of the largest angles
The following types of polygon with three vertices are distinguished:
- acute-angled, in which all angles are sharp;
- rectangular, having one right angle, while the sides forming it are called legs, and the side that is opposite 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 - the base of the triangle;
- equilateral, having all three equal sides.
The properties
The main properties that are characteristic of each type of triangle are distinguished:
- opposite the larger side there is always a larger angle, and vice versa;
- opposite angles of equal size are equal angles, and vice versa;
- any triangle has two sharp corners;
- the outer corner is larger than any inner corner that is not adjacent to it;
- the sum of any two angles is always less than 180 degrees;
- the external angle is equal to the sum of the remaining two angles that do not interfere with it.
The sum of triangle angles theorem
The theorem claims 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 us try to prove this theorem.
Let us have an arbitrary triangle with the vertices of the KMN.
Through the vertex M we draw a
straight line parallel to the straight line KN (this line is also called the Euclidean line). On it we mark point A 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 a secant MN together with straight 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 KMA angle. 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 relative to parallel straight lines of KH and MA with a secant KM, their sum is 180 degrees. The theorem is proved.
Consequence
The following corollary follows from the theorem proved above: any triangle has two acute angles. To prove this, suppose that a given geometric figure has only one acute angle. It can also be assumed that none of the angles is sharp. In this case, there must be at least two angles, the value of which is equal to or greater than 90 degrees. But then the sum of the angles will be more 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 was what had to be proved.
Outside Corner Property
What is the sum of the angles of the triangle that are external? The answer to this question can be obtained by applying one of two methods. The first is that it is necessary to find the sum of the angles that 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. To begin with, we’ll deal with the first option. So, the triangle contains six outer corners - two at each vertex.
Each pair has equal angles, since 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 interfere with it. Consequently,
∟1 = ∟ + ∟, ∟2 = ∟ + ∟, ∟3 = ∟ + ∟.
From this it turns out that the sum of the outer corners, which are taken one by one near each vertex, will be equal to:
∟1 + ∟2 + ∟3 = ∟ + ∟ + ∟ + ∟ + ∟ + ∟ = 2 (∟ + ∟ + ∟).
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 applied, then the sum of the six corners 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-angled triangle that are sharp? The answer to this question, again, follows from a theorem that states that the angles in a triangle add up to 180 degrees. And our statement (property) sounds like this: in a right-angled triangle, acute angles add up to 90 degrees. Let us prove its veracity.
Let us be given the KMN triangle, in which ∟ = 90 °. It is necessary to prove that ∟ + ∟ = 90 °.
So, according to the theorem on the sum of the angles ∟ + ∟ + ∟ = 180 °. Our condition says that ∟ = 90 °. So it turns out, ∟ + ∟ + 90 ° = 180 °. That is, ∟ + ∟ = 180 ° - 90 ° = 90 °. That is what we should have proved.
In addition to the above properties of a right triangle, you can add the following:
- the angles that lie against the legs are sharp;
- triangular hypotenuse is larger 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, we can distinguish the Pythagorean theorem. She argues 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
We said earlier that a polygon with three vertices, containing two equal sides, is isosceles. This property of this geometric figure is known: the angles at its base are equal. Let us prove it.
Take the KMN triangle, which is isosceles, KN - its base.
We are required to prove that ∟ = ∟. So, let’s say that MA is the bisector of our KMN triangle. The triangle of the MCA, taking into account the first sign of equality, is equal to the triangle of the 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 of the equality of these two triangles, it can be argued that ∟ = ∟. Therefore, the theorem is proved.
But we are interested in what is the sum of the angles of a triangle (isosceles). Since in this respect he does not have his own characteristics, we will proceed from the theorem considered earlier. That is, we can say that ∟ + ∟ + ∟ = 180 °, or 2 ∟ + ∟ = 180 ° (since ∟ = ∟). We will not prove this property, since the theorem on the sum of the angles of a triangle was proved earlier.
In addition to the considered properties 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 at the same time the median, the bisector of the angle that is between equal sides, as well as the axis of symmetry of its base;
- the medians (bisectors, heights) that are drawn to the sides of such a geometric figure are equal.
Equilateral triangle
It is also called regular, this is the triangle in which all sides are equal. Therefore, the angles are also equal. Each of them is 60 degrees. Let us prove this property.
Suppose we have a KMN triangle. We know that KM = NM = KN. And this means that according to the property of angles located at the base in an isosceles triangle, ∟ = ∟ = ∟. Since, according to the theorem, the sum of the angles of the triangle is ∟ + ∟ + ∟ = 180 °, then 3 x ∟ = 180 ° or ∟ = 60 °, ∟ = 60 °, ∟ = 60 °. Thus, the statement is proved.
As can be seen 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:
- the median, bisector, height in such a geometric figure coincide, and their length is calculated as (a x √3): 2;
- if we describe a circle around a given polygon, then its radius will be equal to (a x √3): 3;
- if you enter 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
According to the definition of an obtuse triangle, one of its angles is in the range from 90 to 180 degrees. But given that the other two angles of this geometric figure are sharp, we can conclude that they do not exceed 90 degrees. Therefore, the theorem on the sum of the angles of a triangle works in calculating the sum of the 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 proved again.