When studying mathematics, students begin to get acquainted with various types of geometric shapes. Today we will talk about different types of triangles.
Definition
Geometric figures that consist of three points that are not on the same straight line are called triangles.
The line segments connecting the points are called sides, and the points are called vertices. Vertices are denoted by capital Latin letters, for example: A, B, C.
The sides are indicated by the names of the two points of 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 according to angles and sides. Each type of triangle has its own properties.
There are three types of triangles in the corners:
- acute-angled;
- rectangular;
- obtuse.
All angles acute-angled triangles are acute, that is, the degree measure of each is no more than 90 0.
Rectangular the triangle contains a right angle. The other two angles will always be acute, 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 greater 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 in 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 compose simple problems in geometry. Therefore, remember: if two sides of a triangle are 3, then the third one 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-angled.
Isosceles a triangle is a triangle with only two equal sides. These sides are called lateral, and the third - 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 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 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 a 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 2:2:1.
Task:
Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?
Decision:
To solve this task, 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 when solving problems.
The division of triangles into acute, right and obtuse triangles. 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 versatile at the same time.
When determining the type by the type of corners, be very careful. An obtuse-angled triangle will be called such a triangle, in which one of the angles is, that is, it 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 an acute triangle, you will need to make sure that all three of its angles are acute.
Defining the view triangle by aspect ratio, first you have to find out the lengths of all three sides. However, if by condition the lengths of the sides are not given to you, the angles can help you. A triangle will be versatile, all three sides of which 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-angled or acute-angled.
A triangle is isosceles if two of its three sides are equal. If the lengths of the sides are not given to you, be guided by two equal angles. An isosceles triangle, like a scalene triangle, can be obtuse, right-angled and acute-angled.
An equilateral triangle can only be such that all three sides of which 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-angled.
Advice 2: How to identify an obtuse and acute triangle
The simplest of the polygons is the triangle. It is formed with the help of three points lying in the same plane, but not lying on the same straight line, connected in pairs by segments. However, triangles come in different types, which means they have different properties.
Instruction
It is customary to distinguish three types: obtuse, acute and rectangular. It's like the corners. An obtuse triangle is a triangle in which one of the angles is obtuse. An obtuse angle is one 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, so the triangle is obtuse.
An acute triangle is a triangle in which all angles are acute. An acute angle is one that is less than ninety 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°, so it is a triangle. If you know that a triangle has all sides equal, it means that all its angles are also equal to each other, while being equal to sixty degrees. Accordingly, all angles in such a triangle are less than ninety degrees, and therefore such a triangle is acute-angled.
If in a triangle one of the angles is equal to ninety degrees, this means that it does not belong to either the wide-angle type or the acute-angle type. This is a right triangle.
If the type of triangle is determined by the aspect ratio, they will be equilateral, scalene and isosceles. In an equilateral triangle, all sides are equal, and this, as you found out, indicates that the triangle is acute. If a triangle has only two equal sides or if the sides are not equal to each other, it can be obtuse, right-angled, or acute-angled. So, in these cases, it is necessary to calculate or measure the angles and draw conclusions, according to paragraphs 1, 2 or 3.
Related videos
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, simple pencil, protractor, ruler.
Instruction
Open your seventh grade geometry textbook to the paragraph on the signs of the equality 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 tested are arbitrary, then there are three main equality criteria for them. If some additional information about triangles is known, then the main three signs are supplemented by several more. This applies, for example, to the case of equality of right triangles.
Read the first rule about the equality of triangles. As is known, it allows us to consider triangles equal if it can be proved that any one angle and two adjacent sides of two triangles are equal. In order to understand this law, draw on a sheet of paper with a protractor two identical definite angles formed by two rays emanating from one point. Measure with a ruler the same sides from the top of the drawn corner in both cases. Using a protractor, measure the angles of the two formed triangles, make sure they are equal.
In order not to resort to such practical measures to understand the criterion for the equality of triangles, read the proof of the first criterion for equality. The fact is that each rule about the equality of triangles has a strict theoretical proof, it's just not convenient to use it in order to memorize the rules.
Read the second sign of equality of triangles. It says that two triangles will be congruent if any one side and two adjacent angles of two such triangles are congruent. In order to remember this rule, imagine the drawn side of the triangle and two corners adjacent to it. Imagine that the lengths of the sides of the corners gradually increase. Eventually, they will intersect, forming a third angle. In this mental task, it is important that the point of intersection of the sides that are mentally increased, as well as the resulting angle, are uniquely determined by the third side and two angles adjacent to it.
If you are not given any information about the angles of the triangles under study, then use the third test 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 that they uniquely determine the triangle itself.
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Perhaps the most basic, simple and interesting figure in geometry is a triangle. In a secondary school course, its basic properties are studied, but sometimes knowledge on this topic is formed incomplete. The types of triangles initially determine their properties. But this view remains mixed. So now let's take a closer look at this topic.
The types of triangles depend on the degree measure of the angles. These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute-angled. If at least one angle of the triangle is 90 degrees, then you are dealing with a rectangular subspecies. Accordingly, in all other cases, the considered one is called obtuse-angled.
There are many tasks for acute-angled subspecies. A distinctive feature is the internal location of the intersection points of the bisectors, medians and heights. In other cases, this condition may not be met. Determining the type of figure "triangle" is not difficult. It is enough to know, for example, the cosine of each angle. If any values are less than zero, then the triangle is obtuse in any case. In the case of a zero exponent, the figure has a right angle. All positive values are guaranteed to tell you that you have an acute-angled view.
It is impossible not to say about the right triangle. This is the most ideal view, where all the intersection points of medians, bisectors and heights coincide. The center of the inscribed and circumscribed circles also lies in the same place. To solve problems, you need to know only one side, since the angles are initially set for you, and the other two sides are known. That is, the figure is given by only one parameter. There are Their main feature - the equality of the two sides and angles at the base.
Sometimes there is a question about whether there is a triangle with given sides. What you are really asking is whether this description fits the main species. For example, if the sum of two sides is less than the third, then in reality such a figure does not exist at all. If the task asks to find the cosines of the angles of a triangle with sides 3,5,9, then here the obvious can be explained without complex mathematical tricks. Suppose you want to get from point A to point B. The distance in a straight line is 9 kilometers. However, you remembered that you need to go to point C in the store. The distance from A to C is 3 kilometers, and from C to B - 5. Thus, it turns out that when moving through the store, you will walk one kilometer less. But since point C is not located on line AB, you will have to go an extra distance. Here a contradiction arises. This is, of course, a hypothetical explanation. Mathematics knows more than one way to prove that all kinds of triangles obey the basic identity. It says that the sum of two sides is greater than the length of the third.
Each type has the following properties:
1) The sum of all angles is 180 degrees.
2) There is always an orthocenter - the point of intersection of all three heights.
3) All three medians drawn from the vertices of the interior angles intersect in one place.
4) A circle can be circumscribed around any triangle. It is also possible to inscribe a circle so that it has only three points of contact and does not go beyond the outer sides.
Now you have got acquainted with the basic properties that different types of triangles have. In the future, it is important to understand what you are dealing with when solving a problem.
Of all the polygons triangles have the least number of angles and sides.
Triangles can be distinguished by the shape of their angles.
If all angles of a triangle are acute, then it is called an acute triangle.(Fig. 113, a).
If one of the angles of a triangle is right, then it is called a right triangle.(Fig. 113, b).
If one of the angles of a triangle is obtuse, then it is called an obtuse triangle.(Fig. 113, c).
They say that we classified triangles according to their angles.
Triangles can be classified not only by the type of angles, but also by the number of equal sides.
If two sides of a triangle are equal, then it is called an isosceles triangle.
Figure 114, a shows an isosceles triangle ABC, in which AB \u003d BC. In the figure, equal sides are marked with an equal number of dashes. Equal sides AB and BC are called sides, and the side AC − basis isosceles triangle ABC.
If the sides of a triangle are equal, then it is called an equilateral triangle.
The triangle shown in Figure 114b is equilateral, it has MN = NE = EM.
A triangle with three sides of different lengths is called a scalene triangle.
The triangles shown in Figure 113 are scalene. If the side of an equilateral triangle is a, then its perimeter is calculated by the formula:
P = 3a
Example 1 . Using a ruler and a protractor, construct a triangle whose two sides are 3 cm and 2 cm and the angle between them is 50°.
Using a protractor, we will construct an angle A, the degree measure of which is 50 ° (Fig. 115). On the sides of this angle from its top, using a ruler, set aside a segment AB 3 cm long and a segment AC 2 cm long ( fig. 116). Connecting points B and C with a segment, we get the desired triangle ABC ( fig. 117).
Example 2 . Using a ruler and a protractor, construct a triangle ABC whose side AB is 2 cm and whose angles CAB and CBA are respectively 40° and 110°.
Decision. Using a ruler, we build a segment AB 2 cm long ( fig. 118). From the beam AB with the help of a protractor we set aside an angle with a vertex at point A, the degree measure of which is 40 °. From the ray BA in the same direction from the straight line AB, in which the first angle was plotted, we lay off the angle with the vertex at point B, the degree measure of which is 110 ° (Fig. 119).
Having found the point C of the intersection of the sides of the angles A and B, we obtain the desired triangle ABC (Fig. 120).