What can you say about the angles of a triangle? Properties of a triangle. Including equality and similarity, congruent triangles, sides of a triangle, angles of a triangle, area of ​​a triangle - calculation formulas, right triangle, isosceles

Triangle is a polygon with three sides (or three angles). The sides of a triangle are often indicated by small letters (a, b, c), which correspond to capital letters indicating opposite vertices (A, B, C).

If all three angles of a triangle are acute, then it is acute triangle .

If one of the angles in a triangle is right, then it is right triangle. The sides forming a right angle are called legs. The side opposite the right angle is called hypotenuse.

If one of the angles in a triangle is obtuse, then it is obtuse triangle.

Isosceles triangle, if its two sides are equal; these equal sides are called lateral, and the third side is called the base of the triangle.

Equilateral triangle, if all its sides are equal.

Basic properties of triangles

In any triangle:

1. Opposite the larger side lies the larger angle, and vice versa.

2. Equal angles lie opposite equal sides, and vice versa.
In particular, all angles in an equilateral triangle are equal.

3. The sum of the angles of a triangle is 180º.
From the last two properties it follows that every angle in an equilateral
triangle is 60º.

4. Continuing one of the sides of the triangle, we get the outer
corner. The external angle of a triangle is equal to the sum of the internal angles,
not adjacent to it.

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

Signs of equality of triangles.

Triangles are congruent 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 triangles.

Two right triangles are congruent 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 acute angle of the other.

Triangle height is a perpendicular dropped from any vertex to the opposite side (or its continuation). This side is called the base of the triangle. The three altitudes of a triangle always intersect at one point called orthocenter of the triangle. The orthocenter of an acute triangle is located inside the triangle, and the orthocenter of an obtuse triangle is outside; The orthocenter of a right triangle coincides with the vertex right angle.

Median is a segment connecting any vertex of a triangle with the middle of the opposite side. Three medians of a triangle intersect at one point, which always lies inside the triangle and is its center of gravity. This point divides each median in a ratio of 2:1, counting from the vertex.

Property of the median of an isosceles triangle. In an isosceles triangle, the median drawn to the base is the bisector and the altitude.

Bisector- this is the bisector segment of the angle from the vertex to the point of intersection with the opposite side. The three bisectors of a triangle intersect at one point, which always lies inside the triangle and is center of the inscribed circle. The bisector divides the opposite side into parts proportional to the adjacent sides.

Median perpendicular is a perpendicular drawn from the midpoint of a segment (side). The three perpendicular medians of a triangle intersect at one point, which is the center of the circumscribed circle. In an acute triangle, this point lies inside the triangle; in an obtuse angle - outside; in a rectangular one - in the middle of the hypotenuse. The orthocenter, center of gravity, circumcenter and inscribed circle coincide only in an equilateral triangle.

Middle line of the triangle is a segment connecting the midpoints of its two sides.

Property of the midline of a triangle. The middle line of the triangle, connecting the midpoints of two given sides, is parallel to the third side and equal to half of it.

Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c 2 = a 2 + b 2 .

Proofs of the Pythagorean theorem you can see Here.

Theorem of sines. The sides of a triangle are proportional to the sines of the opposite angles .

Cosine theorem. The square of any side of a triangle is equal to the sum of the squares of the other two sides without twice the product of these sides by the cosine of the angle between them .

Proofs of the sine theorem and the cosine theorem you can see Here.

Theorem on the sum of angles in a triangle. The sum of the interior angles of a triangle is 180°.

Triangle Exterior Angle Theorem. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.

Types of Triangles

Let's consider three points that do not lie on the same line, and three segments connecting these points (Fig. 1).

A triangle is a part of the plane bounded by these segments, the segments are called the sides of the triangle, and the ends of the segments (three points that do not lie on the same straight line) are the vertices of the triangle.

Table 1 lists all possible types of triangles depending on the size of their angles .

Table 1 - Types of triangles depending on the size of the angles

Drawing	Triangle type	Definition
	Acute triangle	A triangle with all angles are sharp , called acute-angled
	Right triangle	A triangle with one of the angles is right , called rectangular
	Obtuse triangle	A triangle with one of the angles is obtuse , called obtuse

Acute triangle

Definition: A triangle with all angles are sharp , called acute-angled

Right triangle

Definition: A triangle with one of the angles is right , called rectangular

Obtuse triangle

Definition: A triangle with one of the angles is obtuse , called obtuse

Depending on the lengths of the sides There are two important types of triangles.

Table 2 - Isosceles and equilateral triangles

Drawing	Triangle type	Definition
	Isosceles triangle	sides, and the third side is called the base of an isosceles triangle
	Equilateral (correct) triangle	A triangle in which all three sides are equal is called an equilateral or regular triangle.

Isosceles triangle

Definition: A triangle whose two sides are equal is called an isosceles triangle. In this case, two equal sides are called sides, and the third side is called the base of an isosceles triangle

Equilateral (right) triangle

Definition: A triangle in which all three sides are equal is called an equilateral or regular triangle.

Signs of equality of triangles

Triangles are said to be equal if they can be combined by overlay .

Table 3 shows signs of equality of triangles.

Table 3 – Signs of equality of triangles

Drawing	Feature name	Attribute wording
	By two sides and the angle between them
	Test for equivalence of triangles By side and two adjacent angles
	Test for equivalence of triangles By three parties

Signs of equality of right triangles

The following names are commonly used for the sides of right triangles.

The hypotenuse is the side of a right triangle lying opposite the right angle (Fig. 2), the other two sides are called legs.

Table 4 – Signs of equality of right triangles

Drawing	Feature name	Attribute wording
	By two sides
	Equality test for right triangles By leg and adjacent acute angle
	Equality test for right triangles By 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 right triangle, then such right triangles are congruent
	Equality test for right triangles 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 right triangle, then such right triangles are congruent
	Equality test for right 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 right triangles are congruent

Dividing triangles into acute, rectangular and obtuse. Classification by aspect ratio divides triangles into scalene, equilateral and isosceles. Moreover, each triangle simultaneously belongs to two. For example, it can be rectangular and scalene at the same time.

When determining the type by the type of angles, be very careful. An obtuse triangle will be called a triangle in which one of the angles is , that is, more than 90 degrees. A right triangle can be calculated by having one right (equal to 90 degrees) angle. However, to classify a triangle as acute, you will need to make sure that all three of its angles are acute.

Defining the species triangle according to the aspect ratio, first you will have to find out the lengths of all three sides. However, if, according to the condition, the lengths of the sides are not given to you, the angles can help you. A scalene triangle is one whose three sides have different lengths. If the lengths of the sides are unknown, then a triangle can be classified as scalene if all three of its angles are different. A scalene triangle can be obtuse, right, or acute.

An isosceles triangle is one in which two of its three sides are equal to each other. If the lengths of the sides are not given to you, use two equal angles as a guide. An isosceles triangle, like a scalene triangle, can be obtuse, rectangular or acute.

Only a triangle can be equilateral if all three sides have the same length. All its angles are also equal to each other, and each of them is equal to 60 degrees. From this it is clear that equilateral triangles are always acute.

Tip 2: How to determine an obtuse and acute triangle

The simplest of polygons is a triangle. It is formed using three points lying in the same plane, but not on the same straight line, connected in pairs by segments. However, there are triangles different types, which means they have different properties.

Instructions

It is customary to distinguish three types: obtuse-angled, acute-angled and rectangular. It's like corners. An obtuse triangle is a triangle in which one of the angles is obtuse. An obtuse angle is an angle that is greater than ninety degrees but less than one hundred and eighty. For example, in triangle ABC, angle ABC is 65°, angle BCA is 95°, and angle CAB is 20°. Angles ABC and CAB are less than 90°, but angle BCA is greater, which means the triangle is obtuse.

An acute triangle is a triangle in which all angles are acute. An acute angle is an angle that is less than ninety degrees and greater than zero degrees. For example, in triangle ABC, angle ABC is 60°, angle BCA is 70°, and angle CAB is 50°. All three angles are less than 90°, which means it is a triangle. If you know that a triangle has all sides equal, this means that all its angles are also equal to each other, and they are equal to sixty degrees. Accordingly, all angles in such a triangle are less than ninety degrees, and therefore such a triangle is acute.

If one of the angles in a triangle is ninety degrees, this means that it is neither a wide-angle nor an acute-angle type. This is a right triangle.

If the type of triangle is determined by the ratio of the sides, they will be equilateral, scalene and isosceles. In an equilateral triangle, all sides are equal, and this, as you found out, means that the triangle is acute. If a triangle has only two sides equal or the sides are not equal, it can be obtuse, rectangular, or acute. This means that in these cases it is necessary to calculate or measure the angles and draw conclusions according to points 1, 2 or 3.

Video on the topic

Sources:

• obtuse triangle

The equality of two or more triangles corresponds to the case when all sides and angles of these triangles are equal. However, there are a number of simpler criteria for proving this equality.

You will need

• Geometry textbook, sheet of paper, pencil, protractor, ruler.

Instructions

Open your seventh grade geometry textbook to the section on the criteria for congruence of triangles. You will see that there are a number of basic signs that prove the equality of two triangles. If the two triangles whose equality is being checked are arbitrary, then for them there are three main signs of equality. If some additional information about triangles is known, then the main three features are supplemented with several more. This applies, for example, to the case of equality of right triangles.

Read the first rule about congruence of triangles. As is known, it allows us to consider triangles equal if it can be proven that any one angle and two adjacent sides of two triangles are equal. In order to understand this law, draw on a piece of paper using a protractor two identical specific angles formed by two rays emanating from one point. Using a ruler, measure the same sides from the top of the drawn angle in both cases. Using a protractor, measure the resulting angles of the two triangles formed, making sure they are equal.

In order not to resort to such practical measures to understand the test for equality of triangles, read the proof of the first test for equality. The fact is that every rule about the equality of triangles has a strict theoretical proof, it’s just not convenient to use for the purpose of memorizing the rules.

Read the second test for congruence of triangles. It states that two triangles will be equal if any one side and two adjacent angles of two such triangles are equal. In order to remember this rule, imagine the drawn side of the triangle and its two adjacent angles. Imagine that the lengths of the sides of the corners gradually increase. Eventually they will intersect, forming a third corner. In this mental task, it is important that the intersection point of the sides that are mentally increased, as well as the resulting angle, are uniquely determined by the third side and two adjacent angles.

If you are not given any information about the angles of the triangles being studied, then use the third criterion for the equality of triangles. According to this rule, two triangles are considered equal if all three sides of one of them are equal to the corresponding three sides of the other. Thus, this rule says that the lengths of the sides of a triangle uniquely determine all the angles of the triangle, which means they uniquely determine the triangle itself.

Video on the topic

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

Definition

Geometric figures that consist of three points that are not on the same line are called triangles.

The segments connecting the points are called sides, and the points are called vertices. Vertices are designated in capital letters, for example: A, B, C.

The sides are designated by the names of the two points from which they consist - AB, BC, AC. Intersecting, the sides form angles. 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 triangles at the corners:

• acute-angled;
• rectangular;
• obtuse-angled.

All angles acute-angled triangles are acute, 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 acute, since otherwise the sum of the angles of the triangle will exceed 180 degrees, and this is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two are called the legs. The hypotenuse is always larger than the leg.

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

Rice. 2. Types of triangles at the corners.

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

Moreover, the larger side is the hypotenuse.

Such triangles are often used to make simple tasks in geometry. Therefore, remember: if two sides of a triangle are equal to 3, then the third will definitely 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.

Isosceles triangle - a triangle with only two sides equal. These sides are called lateral, and the third is called the base. In addition, the angles at the base of an isosceles triangle are equal and always acute.

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

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

Rice. 3. Types of triangles on the sides.

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

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

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

Task:

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

Solution:

To solve this task you need to use the inequality a

What have we learned?

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

Triangle - definition and general concepts

A triangle is a simple polygon consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.

All vertices of any triangle, regardless of its type, are designated by capital Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, only not in capital letters, but in small ones. So, for example, a triangle with vertices labeled A, B and C has sides a, b, c.

If we consider a triangle in Euclidean space, then this is such geometric figure, which was formed using three segments connecting three points that do not lie on the same straight line.

Look carefully at the picture shown above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms angles inside it.

Types of triangles

According to the size of the angles of triangles, they are divided into such varieties as: Rectangular;
Acute angular;
Obtuse.

Rectangular triangles include those that have one right angle and the other two have acute angles.

Acute triangles are those in which all its angles are acute.

And if a triangle has one obtuse angle and the other two acute angles, then such a triangle is classified as obtuse.

Each of you understands perfectly well that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:

Isosceles;
Equilateral;
Versatile.

Task: Draw different types triangles. Define them. What difference do you see between them?

Basic properties of triangles

Although these simple polygons may differ from each other in the size of their angles or sides, each triangle has the basic properties that are characteristic of this figure.

In any triangle:

The total sum of all its angles is 180º.
If it belongs to equilaterals, then each of its angles is 60º.
An equilateral triangle has equal and equal angles.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are equal angles, and vice versa.
If we take a triangle and extend its side, we end up with an external angle. It is equal to the sum of the internal angles.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:

1.a< b + c, a >b–c;
2.b< a + c, b >a–c;
3.c< a + b, c >a–b.

Exercise

The table shows the already known two angles of the triangle. Knowing total amount of all angles, find what the third angle of the triangle is equal to and put it in the table:

1. How many degrees does the third angle have?
2. What type of triangle does it belong to?

Tests for equivalence of triangles

I sign

II sign

III sign

Height, bisector and median of a triangle

The altitude of a triangle - the perpendicular drawn from the vertex of the figure to its opposite side is called the altitude of the triangle. All altitudes of a triangle intersect at one point. The point of intersection of all 3 altitudes of a triangle is its orthocenter.

A segment drawn from a given vertex and connecting it at the middle of the opposite side is the median. Medians, as well as altitudes of a triangle, have one common point of intersection, the so-called center of gravity of the triangle or centroid.

The bisector of a triangle is a segment connecting the vertex of an angle and a point on the opposite side, and also dividing this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.

The segment that connects the midpoints of 2 sides of a triangle is called the midline.

Historical reference

A figure such as a triangle was known back in Ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron’s formula, the study of the properties of a triangle moved to more high level, but still, this happened more than two thousand years ago.

In XV – 16th centuries They began to conduct a lot of research on the properties of a triangle, and as a result such a science as planimetry arose, which was called “New Triangle Geometry”.

Russian scientist N.I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application in mathematics, physics and cybernetics.

Thanks to knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when drawing up maps, measuring areas, and even when designing various mechanisms.

What is the most famous triangle you know? This is of course the Bermuda Triangle! It got its name in the 50s because geographical location points (vertices of the triangle), within which, according to the existing theory, associated anomalies arose. The vertices of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.

Assignment: What theories about Bermuda Triangle did you hear?

Did you know that in Lobachevsky’s theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemann's geometry, the sum of all the angles of a triangle is greater than 180º, and in the works of Euclid it is equal to 180 degrees.

Homework

Solve a crossword puzzle on a given topic

Questions for the crossword:

1. What is the name of the perpendicular that is drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the largest side of the triangle?
6. What is the name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90°?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, the capital letter A is...?
11. What is the name of the segment dividing the angle of a triangle in half?

Questions on the topic of triangles:

1. Define it.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called remarkable.
7. What device can you use to measure the angle?
8. If the clock hands show 21 o'clock. What angle do the hour hands make?
9. At what angle does a person turn if he is given the command “left”, “circle”?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?