What is the name of the triangle in which. Obtuse triangle: length of sides, sum of angles. Described obtuse triangle. Properties common to all triangles

A triangle (from the point of view of Euclid's space) is a geometric figure that is formed by three segments connecting three points that do not lie on one straight line. The three points that form a triangle are called its vertices, and the segments connecting the vertices are called the sides of the triangle. What are the triangles?

Equal triangles

There are three signs that triangles are equal. What triangles are called equal? These are those with:

  • two sides and the angle between these sides are equal;
  • equal to one side and two adjacent corners;
  • all three sides are equal.

Right-angled triangles have the following equality signs:

  • acute angle and hypotenuse;
  • along an acute angle and leg;
  • on two legs;
  • along the hypotenuse and leg.

What are the triangles

By the number of equal sides, a triangle can be:

  • Equilateral. It is a triangle with three equal sides. All angles in an equilateral triangle are 60 degrees. In addition, the centers of the circumcircle and the incircle coincide.
  • Unilateral. A triangle that has no equal sides.
  • Isosceles. It is a triangle with two equal sides. Two identical sides are the sides, and the third side is the base. In such a triangle, the bisector, median and height coincide if they are lowered to the base.

In terms of angles, a triangle can be:

  1. Obtuse - when one of the angles is more than 90 degrees, that is, when it is obtuse.
  2. Acute - if all three angles in the triangle are acute, that is, they are less than 90 degrees.
  3. Which triangle is called rectangular? This is one that has one right angle equal to 90 degrees. The legs in it will be called the two sides that form this angle, and the hypotenuse is the side opposite to the right angle.

Basic properties of triangles

  1. A smaller angle always lies against the smaller side, and the larger angle always lies against the larger side.
  2. Equal angles always lie opposite equal sides, and different angles always lie opposite different sides. In particular, in an equilateral triangle, all angles have the same value.
  3. In any triangle, the angles add up to 180 degrees.
  4. The outer angle can be obtained by extending one of its sides near the triangle. The value of the outer corner will be equal to the sum of non-adjacent inner corners.
  5. The side of a triangle is greater than the difference between its other two sides, but less than their sum.

In the spatial geometry of Lobachevsky, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere, this value is greater than 180 degrees. The difference between 180 degrees and the sum of the angles of the triangle is called a defect.

When studying mathematics, students begin to become familiar with different types of geometric shapes. Today we will focus on different types of triangles.

Definition

Geometric shapes that are made up of three points that are not on the same straight line are called triangles.

The lines that connect the points are called sides, and the points are called vertices. Vertices are indicated by capital Latin letters, for example: A, B, C.

The sides are designated by the names of the two points of which they are composed - AB, BC, AC. Crossing, the sides form corners. The bottom side is considered the base of the figure.

Rice. 1. Triangle ABC.

Types of triangles

Triangles are classified by angles and sides. Each type of triangle has its own properties.

There are three types of corner triangles:

  • acute-angled;
  • rectangular;
  • obtuse.

All angles acute-angled triangles are sharp, that is, the degree measure of each is no more than 90 0.

Rectangular a triangle contains a right angle. The other two angles will always be sharp, because otherwise the sum of the angles of the triangle will exceed 180 degrees, which is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two legs. The hypotenuse is always larger than the leg.

Obtuse a triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two corners in such a triangle will be sharp.

Rice. 2. Types of triangles in the corners.

A Pythagorean triangle is a rectangle whose sides are equal to 3, 4, 5.

Moreover, the big side is the hypotenuse.

Such triangles are often used to compose simple problems in geometry. Therefore, remember: if the two sides of the triangle are equal to 3, then the third will necessarily be 5. This will simplify the calculations.

Types of triangles on the sides:

  • equilateral;
  • isosceles;
  • versatile.

Equilateral a triangle is a triangle in which all sides are equal. All angles of such a triangle are equal to 60 0, that is, it is always acute-angled.

Isosceles triangle - a triangle in which only two sides are equal. These sides are called side, and the third is called the base. In addition, the angles at the base of an isosceles triangle are equal and always sharp.

Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.

If in the problem there are no clarifications about the figure, then it is considered that we are talking about an arbitrary triangle.

Rice. 3. Types of triangles on the sides.

The sum of all the angles of a triangle, regardless of its type, is 1800.

Opposite the larger corner is the larger side. And also the length of any side is always less than the sum of its two other sides. These properties are confirmed by the triangle inequality theorem.

There is the concept of the golden triangle. This is an isosceles triangle, in which two sides are proportional to the base and equal to a certain number. In such a figure, the angles are proportional to the ratio of 2: 2: 1.

Task:

Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?

Solution:

To solve this problem, you need to use the inequality a

What have we learned?

From this material from the 5th grade mathematics course, we learned that triangles are classified by sides and angles. Triangles have certain properties that can be used to solve problems.

Some triangle, in which all sides are not the same length, is usually called versatile.

A triangle with two identical sides is denoted as isosceles... It is customary to call the same parties lateral, third party - basis. This definition will be equally true base of a triangle is the side of an isosceles triangle that is not equal to the other two sides.

V isosceles triangle base angles are equal. Height, median, bisector of an isosceles triangle, drawn to its base, are aligned.

Triangle, with all the same sides, is denoted as equilateral or correct... In an equilateral triangle, all angles are 60 °, and the centers of the inscribed and circumscribed circles are aligned.

Types of triangles depending on the parameters of the angles.

A triangle in which only the corners are less than 90 0 (acute) is called acute-angled.

The triangle in which the angle 90 0 is represented is called rectangular... The sides of a triangle forming a right angle are usually denoted legs, and the side opposite the right angle is hypotenuse.

Today we go to the country of Geometry, where we will get acquainted with different types of triangles.

Consider the geometric shapes and find among them "superfluous" (Fig. 1).

Rice. 1. Illustration for example

We see that the figures # 1, 2, 3, 5 are quadrangles. Each of them has its own name (Fig. 2).

Rice. 2. Quadrangles

This means that the "extra" figure is a triangle (Fig. 3).

Rice. 3. Illustration for example

A triangle is a figure that consists of three points that do not lie on one straight line, and three segments that connect these points in pairs.

The points are called the vertices of the triangle, segments - it parties... The sides of the triangle form there are three corners at the vertices of the triangle.

The main signs of a triangle are three sides and three corners. In terms of angle, triangles are acute-angled, rectangular and obtuse-angled.

A triangle is called acute-angled if all three corners are acute, that is, less than 90 ° (Fig. 4).

Rice. 4. Acute-angled triangle

A triangle is called rectangular if one of its corners is 90 ° (Fig. 5).

Rice. 5. Right-angled triangle

A triangle is called obtuse if one of its corners is obtuse, that is, more than 90 ° (Fig. 6).

Rice. 6. Obtuse triangle

According to the number of equal sides, triangles are equilateral, isosceles, versatile.

An isosceles triangle is a triangle whose two sides are equal (Fig. 7).

Rice. 7. Isosceles triangle

These parties are called lateral, Third side - basis. In an isosceles triangle, the angles at the base are equal.

Isosceles triangles are acute-angled and obtuse-angled(fig. 8) .

Rice. 8. Acute and obtuse isosceles triangles

An equilateral triangle is a triangle in which all three sides are equal (Fig. 9).

Rice. 9. Equilateral triangle

In an equilateral triangle all angles are equal. Equilateral triangles always acute-angled.

A triangle is called versatile, in which all three sides have different lengths (Fig. 10).

Rice. 10. Versatile triangle

Complete the task. Divide these triangles into three groups (fig. 11).

Rice. 11. Illustration for the task

First, we distribute by the magnitude of the angles.

Acute triangles: No. 1, No. 3.

Rectangular triangles: No. 2, No. 6.

Obtuse triangles: No. 4, No. 5.

We will distribute the same triangles into groups according to the number of equal sides.

Versatile triangles: No. 4, No. 6.

Isosceles triangles: No. 2, No. 3, No. 5.

Equilateral triangle: No. 1.

Consider the drawings.

Think about which piece of wire you made each triangle (fig. 12).

Rice. 12. Illustration for the task

You can reason like this.

The first piece of wire is divided into three equal parts, so an equilateral triangle can be made from it. In the figure, he is shown as the third.

The second piece of wire is divided into three different parts, so you can make a versatile triangle out of it. He is shown first in the figure.

The third piece of wire is divided into three parts, where the two parts are the same length, which means that an isosceles triangle can be made from it. In the figure, he is shown as the second.

Today in the lesson we got acquainted with the different types of triangles.

Bibliography

  1. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Education", 2012.
  2. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Education", 2012.
  3. M.I. Moreau. Mathematics Lessons: Guidelines for Teachers. Grade 3. - M .: Education, 2012.
  4. Normative legal document. Monitoring and evaluation of learning outcomes. - M .: "Education", 2011.
  5. "School of Russia": Programs for elementary school. - M .: "Education", 2011.
  6. S.I. Volkova. Mathematics: Verification work. Grade 3. - M .: Education, 2012.
  7. V.N. Rudnitskaya. Tests. - M .: "Exam", 2012.
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Homework

1. Complete the phrases.

a) A triangle is a figure that consists of ..., not lying on one straight line, and ..., connecting these points in pairs.

b) Points are called … , segments - it … ... The sides of the triangle form at the vertices of the triangle ….

c) In terms of angle, triangles are…,…,….

d) According to the number of equal sides, triangles are…,…,….

2. Draw

a) a right-angled triangle;

b) acute-angled triangle;

c) obtuse triangle;

d) an equilateral triangle;

e) versatile triangle;

f) isosceles triangle.

3. Make an assignment on the topic of the lesson for your peers.

Triangle . Acute-angled, obtuse-angled and right-angled triangles.

Legs and hypotenuse. Isosceles and equilateral triangle.

The sum of the angles of a triangle.

The outer corner of the triangle. Signs of equality of triangles.

Wonderful lines and points in the triangle: heights, medians,

bisectors, median e perpendiculars, orthocenter,

center of gravity, center of an inscribed circle, center of an inscribed circle.

Pythagorean theorem. Aspect ratio in an arbitrary triangle.

Triangle Is a polygon with three sides (or three corners). The sides of a triangle are often denoted by small letters, which correspond to capital letters denoting opposite vertices.

If all three corners are sharp (Fig. 20), then this acute-angled triangle ... If one of the corners is straight(C, fig. 21), that is right triangle; partiesa, bforming a right angle are called legs; sidecopposite to the right angle is called hypotenuse... If one of obtuse angles (B, fig. 22), that is obtuse triangle.


Triangle ABC (fig. 23) - isosceles, if two its sides are equal (a= c); these equal sides are called lateral, the third party is called basis triangle. Triangle ABC (fig. 24) - equilateral, if all its sides are equal (a = b = c). In general ( a ≠ b ≠ c) we have scalene triangle .

Basic properties of triangles. In any triangle:

1. There is a larger angle against the larger side, and vice versa.

2. Equal angles lie opposite equal sides, and vice versa.

In particular, all angles in equilateral triangle are equal.

3. The angles of a triangle add up to 180 º .

From the last two properties it follows that each angle in an equilateral

triangle is 60 º.

4. Continuing one of the sides of the triangle (AC, fig. 25), we get external

angle BCD . The outer angle of the triangle is equal to the sum of the inner angles,

not adjacent to him : BCD = A + B.

5. Any side of a triangle is less than the sum of the other two sides and more

their differences (a < b + c, a > b – c;b < a + c, b > a – c;c < a + b,c > a – b).

Signs of equality of triangles.

Triangles are equal if they are respectively equal:

a ) two sides and the angle between them;

b ) two corners and the side adjacent to them;

c) three sides.

Signs of equality of right-angled triangles.

Two rectangular triangles are equal if one of the following conditions is true:

1) their legs are equal;

2) the leg and hypotenuse of one triangle are equal to the leg and hypotenuse of the other;

3) the hypotenuse and acute angle of one triangle are equal to the hypotenuse and acute angle of the other;

4) the leg and the adjacent acute angle of one triangle are equal to the leg and the adjacent acute angle of the other;

5) the leg and the opposite acute angle of one triangle are equal to the leg and the opposite sharp corner of the other.

Wonderful lines and points in the triangle.

Height triangle isperpendicular,dropped from any vertex to the opposite side ( or its continuation). This side is calledbase of triangle . Three heights of a triangle always intersectat one pointcalled orthocenter triangle. Orthocenter of an acute-angled triangle (point O , fig. 26) is located inside the triangle, andthe orthocenter of an obtuse triangle (point O , fig. 27) – outside; the orthocenter of a right-angled triangle coincides with the apex of the right angle.

Median - this is section connecting any vertex of the triangle to the midpoint of the opposite side. Three medians of a triangle (AD, BE, CF, fig. 28) intersect at one point O always lying inside the triangle and being his center of gravity. This point divides each median by a 2: 1 ratio from the top.

Bisector - this is bisector segment corner from apex to a point intersection with the opposite side. Three bisectors of a triangle (AD, BE, CF, fig. 29) intersect at one point Oh always lying inside a triangle and being the center of the inscribed circle(see the section "Inscribedand the described polygons ").

The bisector divides the opposite side into parts proportional to the adjacent sides ; for example, in fig. 29 AE: CE = AB: BC.

Median perpendicular Is a perpendicular drawn from the middle segment points (sides). Three median perpendiculars of triangle ABC(KO, MO, NO, fig. 30 ) intersect at one point O, which is center the outlined circle (points K, M, N - the midpoints of the sides of the triangle ABC).

In an acute-angled triangle, this point lies inside the triangle; in obtuse - outside; in rectangular - in the middle of the hypotenuse. Orthocenter, center of gravity, center of circumscribed and center of inscribed circle coincide only in an equilateral triangle.

Pythagorean theorem. In a right triangle, the square of lengththe hypotenuse is equal to the sum of the squares of the lengths of the legs.

The proof of the Pythagorean theorem clearly follows from Fig. 31. Consider a right triangle ABC with legs a, b and hypotenuse c.

Let's build a square AKMB using the hypotenuse AB as a side. Thenextend the sides of the right triangle ABC so to get a square CDEF whose side is equal toa + b.It is now clear that the area of ​​a square CDEF is ( a + b) 2 ... On the other hand, this area is equal to the sum squares four right triangles and square AKMB, that is

c 2 + 4 (ab / 2) = c 2 + 2 ab,

from here,

c 2 + 2 ab= (a + b) 2 ,

and finally we have:

c 2 =a 2 + b 2 .

Aspect ratio in an arbitrary triangle.

In the general case (for an arbitrary triangle) we have:

c 2 =a 2 + b 2 – 2ab· cos C,

where C - the angle between the sidesa and b .

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