Triangle is a polygon having three sides (the three angles).The most common side represent small letters, the corresponding capital letter which designates the opposite vertices.In this article we take a look at these types of geometric shapes, the theorem that determines which equals the sum of the angles of a triangle.
Types largest angles
following types of polygon with three vertices:
- acute-angled in which all sharp angles;
- rectangular having one right angle with the side of his image, called legs, and the side which is placed opposite the right angle is called the hypotenuse;
- obtuse when one angle is obtuse;
- isosceles, which the two sides equal, and they are called lateral, and the third - the base of the triangle;
- equilateral having three equal sides.
Properties
There are basic properties that are characteristic of each type of triangle:
- opposite the larger side always has a large angle, and vice versa;
- opposite sides of equal magnitude are equal angles, and vice versa;
- have any triangle has two acute angles;
- exterior angle is greater than any internal angle is not related to him;
- sum of any two angles is always less than 180 degrees;
- exterior angle equals the sum of the other two corners that are not mezhuyut him.
theorem on the sum of the angles of a triangle
theorem states that if you add up all the corners of the geometric figure, which is located in the Euclidean plane, their sum will be 180 degrees.Let's try to prove this theorem.
Let we have an arbitrary triangle with vertices KMN.Through top M draw a line parallel to the line KN (even this line is called the line of Euclid).It should be noted point A in such a way that the point K and A were located on different sides straight MN.We get the same angle and AMS MUF, which, like the inner lie crosswise to form intersecting MN in cooperation with CN and MA lines that are parallel.From this it follows that the sum of the angles of a triangle located at the vertices of M and N is equal to the size of the angle of the CMA.All three angles consist of a sum equal to the sum of angles CMA and MCS.Since these angles are internal with respect to unilateral parallel lines CN and MA at the cutting KM, their sum is 180 degrees.QED.
investigation
From above this theorem implies the following corollary: every triangle has two acute angles.To prove this, let us assume that this geometrical figure has only one acute angle.Also, it can be assumed that no angle is not acute.In this case, it must be at least two angles, the magnitude of which is equal to or greater than 90 degrees.But then the sum of the angles is greater than 180 degrees.And this can not be, since by Theorem sum of the angles of a triangle is 180 ° - no more and no less.That's what had to be proved.
property outside corners
What is the sum of the angles of a triangle, which are external?The answer to this question can be obtained by using one of two methods.The first is the need 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 the six angles at the vertices.To begin with let's deal with the first one.Thus, the triangle has six exterior angles - at each vertex of the two.Each pair has equal angles to each other, because they are vertical:
∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.
addition, it is known that the outer angle of the triangle is equal to the sum of the two inner, are not mezhuyutsya with it.Therefore,
∟1 = ∟A + ∟S, ∟2 = ∟A + ∟V, ∟3 = ∟V + ∟S.
It turns out that the sum of the external angles are taken one by one near the top of each, will be equal to:
∟1 + ∟2 + ∟3 = ∟A ∟S + + + ∟A ∟V + + ∟V ∟S= 2 x (+ ∟A ∟V + ∟S).
Given the fact that the sum of the angles equals 180 degrees, it can be argued that ∟A + ∟V ∟S = + 180 °.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 correspondingly greater doubled.That is the sum of the exterior angles of a triangle will be:
∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 = 2 x (∟1 + ∟2 + ∟2) = 720 °.
right triangle
What equals the sum of angles of a right triangle is the island?The answer, again, from Theorem, which states that the angles of a triangle add up to 180 degrees.And our assertion sounds (property) as follows: in the right-angled triangle acute angles add up to 90 degrees.We prove its truthfulness.Let there be given a triangle KMN, which ∟N = 90 °.We must prove that ∟K ∟M + = 90 °.
Thus, according to the theorem on the sum of the angles ∟K + ∟M ∟N = + 180 °.In this condition it is said that ∟N = 90 °.It turns out ∟K + ∟M + 90 ° = 180 °.That is ∟K ∟M + = 180 ° - 90 ° = 90 °.That is what we should have to prove.
In addition to the above properties of a right triangle, you can add these:
- angles that lie against the legs are sharp;
- triangular hypotenuse is greater than any of the legs;
- the legs more than the sum of the hypotenuse;
- cathetus of the triangle, which lies opposite the corner 30 degrees, a half of the hypotenuse, i.e. it equals half.
As another property of the geometric shape can be identified Pythagorean theorem.She argues that in a triangle with an angle of 90 degrees (rectangular) is equal to the sum of the squares of the legs to the square of the hypotenuse.
sum of the angles of an isosceles triangle
Earlier we said that an isosceles triangle is called a polygon with three vertices containing two equal sides.This property is known geometrical figure: the angles at its base equal.Let us prove this.
Take triangle KMN, which is isosceles, SC - its base.We are required to prove that ∟K = ∟N.So, let us assume that MA - bisector is our triangle KMN.Triangle MCA with the first sign of a triangle is equal MNA.Namely the condition given that CM = HM, MA is a common side, ∟1 = ∟2, because the AI - a bisector.Using the equality of the two triangles, one could argue that ∟K = ∟N.Hence, the theorem is proved.
But we are interested, what is the sum of the angles of a triangle (isosceles).Since in this regard has no its features, we will start from the theorem discussed above.That is, we can say that ∟K + ∟M ∟N + = 180 °, or 2 x ∟K ∟M + = 180 ° (as ∟K = ∟N).This property will not prove as she theorem sum of the angles of a triangle was proved earlier.
Also considering the properties of the corners of the triangle, there are also such important statements:
- within an equilateral triangle height that has been lowered to the base, is also the median, bisector of the angle which is between equal parties, as well as the axis of symmetry of its foundation;
- median (bisector height), which are held to the sides of a geometric figure are equal.
equilateral triangle
It is also called the right, is the triangle, which are equal to all the parties.And therefore also equal angles.Each of them is 60 degrees.We prove this property.
Let us assume that we have a triangle KMN.We know that KM = NM = CL.This means that according to the property corners, located at the base in an equilateral triangle, ∟K = = ∟M ∟N.Because according to the sum of the angles of a triangle theorem ∟K + ∟M ∟N + = 180 °, the 3 x ∟K = 180 ° or ∟K = 60 °, ∟M = 60 °, ∟N = 60 °.Thus, the statement dokazano.Kak seen from above on the basis of the proof of the theorem, the sum of the angles of an equilateral triangle as the sum of the angles of any other triangle is 180 degrees.Again proving this theorem is not necessary.
There are still some properties characteristic of an equilateral triangle:
- median, bisector, height in such a geometric figure are the same, and their length is calculated as (a × √3): 2;
- if describe a polygon around this circle, then its radius is equal to (a x √3): 3;
- if an equilateral triangle inscribed in a circle, then the radius will be (and x √3): 6;
- area of this geometrical figure is calculated as follows: (a2 x √3): 4.
obtuse triangle
By definition, obtuse-angled triangle, one of its corners is between 90 to 180 degrees.However, given that the angle of the other two geometric shapes are sharp, it can be concluded that they do not exceed 90 degrees.Consequently, the theorem on the sum of the angles of a triangle work in calculating the sum of the angles in an obtuse triangle.So, we can safely say, based on the above theorem that the sum of the angles obtuse triangle is 180 degrees.Again, this theorem does not need to re-proof.
