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.
Related videos
Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Examine the geometric shapes and find the “extra” among them (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 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 the same straight line, and three segments connecting these points in pairs.
The points are called triangle vertices, segments - his parties. The sides of the triangle form There are three angles at the vertices of a triangle.
The main features of a triangle are three sides and three corners. Triangles are classified according to the angle acute, rectangular and obtuse.
A triangle is called acute-angled if all three of its angles are acute, that is, less than 90 ° (Fig. 4).
Rice. 4. Acute triangle
A triangle is called right-angled if one of its angles is 90° (Fig. 5).
Rice. 5. Right Triangle
A triangle is called obtuse if one of its angles is obtuse, i.e. greater than 90° (Fig. 6).
Rice. 6. Obtuse Triangle
According to the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is a triangle in which two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These sides are called lateral, the third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is called, 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. Scalene triangle
Complete the task. Divide these triangles into three groups (Fig. 11).
Rice. 11. Illustration for the task
First, let's distribute according to the size of the angles.
Acute triangles: No. 1, No. 3.
Right triangles: #2, #6.
Obtuse triangles: #4, #5.
These triangles are divided into groups according to the number of equal sides.
Scalene triangles: No. 4, No. 6.
Isosceles triangles: No. 2, No. 3, No. 5.
Equilateral Triangle: No. 1.
Review the drawings.
Think about what piece of wire each triangle is made of (fig. 12).
Rice. 12. Illustration for the task
You can argue like this.
The first piece of wire is divided into three equal parts, so you can make an equilateral triangle out of it. It is shown third in the figure.
The second piece of wire is divided into three different parts, so you can make a scalene triangle out of it. It is shown first in the picture.
The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.
Today in the lesson we got acquainted with different types of triangles.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Finish the phrases.
a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.
b) The points are called … , segments - his … . The sides of a triangle form at the vertices of a triangle ….
c) According to the size of the angle, triangles are ..., ..., ....
d) According to the number of equal sides, triangles are ..., ..., ....
2. Draw
a) a right triangle
b) an acute triangle;
c) an obtuse triangle;
d) an equilateral triangle;
e) scalene triangle;
e) an isosceles triangle.
3. Make a task on the topic of the lesson for your comrades.
The simplest polygon that is studied at school is a triangle. It is more understandable for students and encounters fewer difficulties. Despite the fact that there are different types of triangles that have special properties.
What shape is called a triangle?
Formed by three points and line segments. The former are called vertices, the latter are called sides. Moreover, all three segments must be connected so that corners form between them. Hence the name of the figure "triangle".
Differences in the names in the corners
Since they can be sharp, obtuse and straight, the types of triangles are determined by these names. Accordingly, there are three groups of such figures.
- First. If all the angles of a triangle are acute, then it will be called an acute triangle. Everything is logical.
- Second. One of the angles is obtuse, so the triangle is obtuse. Easier nowhere.
- Third. There is an angle equal to 90 degrees, which is called a right angle. The triangle becomes rectangular.
Differences in names on the sides
Depending on the features of the sides, the following types of triangles are distinguished:
the general case is versatile, in which all sides have an arbitrary length;
isosceles, two sides of which have the same numerical values;
equilateral, the lengths of all its sides are the same.
If the task does not specify a specific type of triangle, then you need to draw an arbitrary one. In which all angles are acute, and the sides have different lengths.
Properties common to all triangles
- If you add up all the angles of a triangle, you get a number equal to 180º. And it doesn't matter what kind it is. This rule always applies.
- The numerical value of any side of the triangle is less than the other two added together. Moreover, it is greater than their difference.
- Each outer corner has a value that is obtained by adding two inner corners that are not adjacent to it. Moreover, it is always larger than the adjacent internal one.
- The smallest side of a triangle is always opposite the smallest angle. Conversely, if the side is large, then the angle will be the largest.
These properties are always valid, no matter what types of triangles are considered in problems. All the rest follow from specific features.
Properties of an isosceles triangle
- The angles adjacent to the base are equal.
- The height that is drawn to the base is also the median and the bisector.
- The heights, medians and bisectors, which are built to the sides of the triangle, are respectively equal to each other.
Properties of an equilateral triangle
If there is such a figure, then all the properties described a little above will be true. Because an equilateral will always be an isosceles one. But not vice versa, an isosceles triangle will not necessarily be equilateral.
- All its angles are equal to each other and have a value of 60º.
- Any median of an equilateral triangle is its height and bisector. And they are all equal to each other. To determine their values, there is a formula that consists of the product of the side and the square root of 3 divided by 2.
Properties of a right triangle
- Two acute angles add up to 90º.
- The length of the hypotenuse is always greater than that of any of the legs.
- The numerical value of the median drawn to the hypotenuse is equal to half of it.
- The leg is equal to the same value if it lies opposite an angle of 30º.
- The height, which is drawn from the top with a value of 90º, has a certain mathematical dependence on the legs: 1 / n 2 \u003d 1 / a 2 + 1 / in 2. Here: a, c - legs, n - height.
Problems with different types of triangles
No. 1. Given an isosceles triangle. Its perimeter is known and is equal to 90 cm. It is required to know its sides. As an additional condition: the lateral side is 1.2 times smaller than the base.
The value of the perimeter directly depends on the quantities that need to be found. The sum of all three sides will give 90 cm. Now you need to remember the sign of a triangle, according to which it is isosceles. That is, the two sides are equal. You can make an equation with two unknowns: 2a + b \u003d 90. Here a is the side, b is the base.
It's time for an additional condition. Following it, the second equation is obtained: b \u003d 1.2a. You can substitute this expression into the first one. It turns out: 2a + 1.2a \u003d 90. After transformations: 3.2a \u003d 90. Hence a \u003d 28.125 (cm). Now it's easy to find out the reason. It is best to do this from the second condition: v \u003d 1.2 * 28.125 \u003d 33.75 (cm).
To check, you can add three values: 28.125 * 2 + 33.75 = 90 (cm). All right.
Answer: the sides of the triangle are 28.125 cm, 28.125 cm, 33.75 cm.
No. 2. The side of an equilateral triangle is 12 cm. You need to calculate its height.
Decision. To search for an answer, it is enough to return to the moment where the properties of the triangle were described. This is the formula for finding the height, median and bisector of an equilateral triangle.
n \u003d a * √3 / 2, where n is the height, a is the side.
Substitution and calculation give the following result: n = 6 √3 (cm).
This formula does not need to be memorized. Suffice it to recall that the height divides the triangle into two rectangular ones. Moreover, it turns out to be a leg, and the hypotenuse in it is the side of the original one, the second leg is half of the known side. Now you need to write down the Pythagorean theorem and derive a formula for the height.
Answer: the height is 6 √3 cm.
No. 3. MKR is given - a triangle, 90 degrees in which makes an angle K. The sides MP and KR are known, they are equal to 30 and 15 cm, respectively. You need to find out the value of the angle P.
Decision. If you make a drawing, it becomes clear that MP is the hypotenuse. Moreover, it is twice as large as the leg of the CD. Again, you need to turn to the properties. One of them is just related to the corners. From it it is clear that the angle of the KMR is 30º. So the desired angle P will be equal to 60º. This follows from another property which states that the sum of two acute angles must equal 90º.
Answer: angle R is 60º.
No. 4. You need to find all the angles of an isosceles triangle. It is known about him that the external angle from the angle at the base is 110º.
Decision. Since only the outer corner is given, this should be used. It forms with an internal angle developed. So they add up to 180º. That is, the angle at the base of the triangle will be equal to 70º. Since it is isosceles, the second angle has the same value. It remains to calculate the third angle. By a property common to all triangles, the sum of the angles is 180º. So the third is defined as 180º - 70º - 70º = 40º.
Answer: the angles are 70º, 70º, 40º.
No. 5. It is known that in an isosceles triangle the angle opposite the base is 90º. A dot is marked on the base. The segment connecting it with a right angle divides it in a ratio of 1 to 4. You need to know all the angles of the smaller triangle.
Decision. One of the corners can be determined immediately. Since the triangle is right-angled and isosceles, those that lie at its base will be 45º, that is, 90º / 2.
The second of them will help to find the relation known in the condition. Since it is equal to 1 to 4, then the parts into which it is divided are only 5. So, to find out the smaller angle of the triangle, you need 90º / 5 = 18º. It remains to find out the third. To do this, from 180º (the sum of all the angles of a triangle), you need to subtract 45º and 18º. The calculations are simple, and it turns out: 117º.
Tasks:
1. Introduce students to different types of triangles depending on the type of angles (rectangular, acute-angled, obtuse-angled). Learn to find triangles and their types in the drawings. To fix the basic geometric concepts and their properties: straight line, segment, ray, angle.
2. Development of thinking, imagination, mathematical speech.
3. Education of attention, activity.
During the classes
I. Organizational moment.
How much do we need guys?
For our skillful hands?
Draw two squares
And they have a big circle.
And then some more circles
Triangle cap.
So it came out very, very
Cheerful Weird.
II. Announcement of the topic of the lesson.
Today in the lesson we will make a trip around the city of Geometry and visit the Triangles microdistrict (that is, we will get acquainted with different types of triangles depending on their angles, we will learn to find these triangles in the drawings.) We will conduct a lesson in the form of a “competition game” by commands.
1 team - “Segment”.
2 team - "Ray".
Team 3 - "Corner".
And the guests will represent the jury.
The jury will guide us along the way
And will not leave without attention. (Evaluate by points 5,4,3,...).
And on what will we travel around the city of Geometry? Remember what types of passenger transport are in the city? There are so many of us, which one shall we choose? (Bus).
Bus. Clearly, briefly. Boarding begins.
Let's get comfortable and start our journey. Team captains get tickets.
But these tickets are not easy, and the tickets are “tasks”.
III. Repetition of the material covered.
First stop"Repeat."
Question for all teams.
Find a straight line in the drawing and name its properties.
Without end and edge, the line is straight!
At least a hundred years go along it,
You won't find the end of the road!
- The straight line has neither beginning nor end - it is infinite, so it cannot be measured.
Let's start our competition.
Protecting your team names.
(All teams read the first questions and discuss. In turn, the team captains read out the questions, 1 team reads 1 question).
1. Show a segment in the drawing. What is called a cut. Name its properties.
- The part of a straight line bounded by two points is called a line segment. A line segment has a beginning and an end, so it can be measured with a ruler.
(Team 2 reads 1 question).
1. Show the beam in the drawing. What is called a beam. Name its properties.
- If you mark a point and draw a part of a straight line from it, you get an image of a beam. The point from which a part of the line is drawn is called the beginning of the ray.
The beam has no end, so it cannot be measured.
(Team 3 reads 1 question).
1. Show the angle on the drawing. What is called an angle. Name its properties.
- Drawing two rays from one point, a geometric figure is obtained, which is called an angle. An angle has a vertex, and the rays themselves are called sides of the angle. Angles are measured in degrees using a protractor.
Fizkultminutka (to the music).
IV. Preparing to study new material.
Second stop"Fabulous".
On a walk, the Pencil met different angles. I wanted to say hello to them, but I forgot the name of each of them. Pencil will have to help.
(The angles of the study are checked using the model of a right angle).
Assignment to teams. Read questions #2 and discuss.
Team 1 reads question 2.
2. Find a right angle, give a definition.
- An angle of 90° is called a right angle.
Team 2 reads question 2.
2. Find an acute angle, give a definition.
- An angle less than a right angle is called an acute angle.
Team 3 reads question 2.
2. Find an obtuse angle, give a definition.
An angle greater than a right angle is called obtuse.
In the microdistrict where Pencil liked to walk, all the corners differed from other residents in that the three of us always walked, the three of us drank tea, and the three of us went to the cinema. And the Pencil could not understand what kind of geometric figure three angles together make up?
A poem will give you a clue.
You on me, you on him
Look at all of us.
We have everything, we have everything
We only have three!
Which shape is being referred to?
- About the triangle.
What shape is called a triangle?
- A triangle is a geometric figure that has three vertices, three angles, and three sides.
(Learners show a triangle in the drawing, name the vertices, angles and sides).
Vertices: A, B, C (points)
Angles: BAC, ABC, BCA.
Sides: AB, BC, CA (segments).
V. Physical education:
stomp your foot 8 times,
Clap your hands 9 times
we will squat 10 times,
and bend over 6 times
we'll jump straight
so many (triangle display)
Hey, yes, count! Game and more!
VI. Learning new material.
Soon the corners became friends and became inseparable.
And now we will call the microdistrict: the Triangles microdistrict.
The third stop is “Znayka”.
What are the names of these triangles?
Let's give them names. And let's try to formulate the definition ourselves.
2. Find triangles of different types
1 team will find and show obtuse triangles.
2 command will find and show right triangles.
3 command will find and show acute triangles.
VIII. The next stop is Thinking.
Assignment to all teams.
After shifting 6 sticks, make 4 equal triangles from the lantern.
What kind of angles are triangles? (Acute-angled).
IX. Summary of the lesson.
What neighborhood did we visit?
What types of triangles are you familiar with?
