A triangle consists of. Triangle
One could probably write a whole book on the topic “Triangle”. But it takes too long to read the whole book, right? Therefore, here we will consider only facts that relate to any triangle in general, and all sorts of special topics, such as, etc. separated into separate topics - read the book in pieces. Well, as for any triangle.
1. Sum of angles of a triangle. External corner.
Remember firmly and do not forget. We will not prove this (see the following levels of theory).
The only thing that may confuse you in our formulation is the word “internal”.
Why is it here? But precisely to emphasize that we are talking about the angles that are inside the triangle. Are there really any other corners outside? Just imagine, they do happen. The triangle still has external corners. And the most important consequence of the fact that the amount internal corners triangle is equal to, touches just the outer triangle. So let's find out what this outer angle of the triangle is.
Look at the picture: take a triangle and (let’s say) continue one side.
Of course, we could leave the side and continue the side. Like this:
But you can’t say that about the angle under any circumstances. it is forbidden!
So not every angle outside a triangle has the right to be called an external angle, but only the one formed one side and a continuation of the other side.
So what should we know about external angles?
Look, in our picture this means that.
How does this relate to the sum of the angles of a triangle?
Let's figure it out. The sum of interior angles is
but - because and - are adjacent.
Well, here it comes: .
Do you see how simple it is?! But very important. So remember:
The sum of the interior angles of a triangle is equal, and the exterior angle of a triangle is equal to the sum two internal ones, not adjacent to it.
2. Triangle inequality
The next fact concerns not the angles, but the sides of the triangle.
It means that
Have you already guessed why this fact is called the triangle inequality?
Well, where can this triangle inequality be useful?
Imagine that you have three friends: Kolya, Petya and Sergei. And so, Kolya says: “From my house to Petya’s in a straight line.” And Petya: “From my house to Sergei’s house, meters in a straight line.” And Sergei: “It’s good for you, but from my house to Kolinoye it’s a straight line.” Well, here you have to say: “Stop, stop! Some of you are telling lies!”
Why? Yes, because if from Kolya to Petya there are m, and from Petya to Sergei there are m, then from Kolya to Sergei there must definitely be less () meters - otherwise the same triangle inequality is violated. Well, common sense is definitely, naturally, violated: after all, everyone knows from childhood that the path to a straight line () should be shorter than the path to a point. (). So the triangle inequality simply reflects this well-known fact. Well, now you know how to answer, say, a question:
Does a triangle have sides?
You must check whether it is true that any two of these three numbers add up to more than the third. Let’s check: that means there is no such thing as a triangle with sides! But with the sides - it happens, because
3. Equality of triangles
Well, what if there is not one, but two or more triangles. How can you check if they are equal? Actually, by definition:
But... this is a terribly inconvenient definition! How, pray tell, can one overlap two triangles even in a notebook?! But fortunately for us there is signs of equality of triangles, which allow you to act with your mind without putting your notebooks at risk.
And besides, throwing away frivolous jokes, I’ll tell you a secret: for a mathematician, the word “superimposing triangles” does not mean cutting them out and superimposing them at all, but saying many, many, many words that will prove that two triangles will coincide when superimposed. So, in no case should you write in your work “I checked - the triangles coincide when applied” - they will not count it towards you, and they will be right, because no one guarantees that you did not make a mistake when applying, say, a quarter of a millimeter.
So, some mathematicians said a bunch of words, we will not repeat these words after them (except perhaps in the last level of the theory), but we will actively use three signs of equality of triangles.
In everyday (mathematical) use, such shortened formulations are accepted - they are easier to remember and apply.
- The first sign is on two sides and the angle between them;
- The second sign is on two corners and the adjacent side;
- The third sign is on three sides.
TRIANGLE. BRIEFLY ABOUT THE MAIN THINGS
A triangle is a geometric figure formed by three segments that connect three points that do not lie on the same straight line.
Basic concepts.
Basic properties:
- The sum of the interior angles of any triangle is equal, i.e.
- The external angle of a triangle is equal to the sum of two internal angles that are not adjacent to it, i.e.
or - The sum of the lengths of any two sides of a triangle is greater than the length of its third side, i.e.
- In a triangle, the larger side lies opposite the larger angle, and the larger angle lies opposite the larger side, i.e.
if, then, and vice versa,
if, then.
Signs of equality of triangles.
1. First sign- on two sides and the angle between them.
2. Second sign- on two corners and the adjacent side.
3. Third sign- on three sides.
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successful passing the Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who received a good education, earn much more than those who did not receive it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
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The science of geometry tells us what a triangle, square, and cube are. IN modern world it is studied in schools by everyone without exception. Also, the science that studies directly what a triangle is and what properties it has is trigonometry. She explores in detail all phenomena related to data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems associated with them.
What is a triangle? Definition
This is a flat polygon. It has three corners, as is clear from its name. It also has three sides and three vertices, the first of them are segments, the second are points. Knowing what two angles are equal to, you can find the third by subtracting the sum of the first two from the number 180.
What types of triangles are there?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The former have acute angles, that is, those that are equal to less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.
Right triangle
It is impossible not to talk about what a right triangle is.
Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The remaining two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. Using it, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can also recall an isosceles triangle. This is one in which two of the sides are equal, and two angles are also equal.
What are leg and hypotenuse?
A leg is one of the sides of a triangle that forms an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. You can lower a perpendicular from it onto the leg. The ratio of the adjacent side to the hypotenuse is called cosine, and the opposite side is called sine.
- what are its features?
It's rectangular. Its legs are three and four, and its hypotenuse is five. If you see that the legs of a given triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, using this principle, you can easily determine that the leg will be equal to three if the second is equal to four, and the hypotenuse is equal to five. To prove this statement, you can apply the Pythagorean theorem. If two legs are equal to 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is equal to 5. An Egyptian triangle is also a right triangle whose sides are equal to 6, 8 and 10; 9, 12 and 15 and other numbers with the ratio 3:4:5.
What else could a triangle be?
Triangles can also be inscribed or circumscribed. The figure around which the circle is described is called inscribed; all its vertices are points lying on the circle. A circumscribed triangle is one into which a circle is inscribed. All its sides come into contact with it at certain points.
How is it located?
The area of any figure is measured in square units (sq. meters, sq. millimeters, sq. centimeters, sq. decimeters, etc.) This value can be calculated in a variety of ways, depending on the type of triangle. The area of any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite corner, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle located between these sides, and divide this result by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the resulting four values. Next, find from the number that came out. The area of an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by that circumscribed around it, multiplied by four.
The area of a circumscribed triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: square the side, multiply the resulting figure by the root of three, then divide this number by four. In a similar way, you can calculate the height of a triangle in which all sides are equal; to do this, you need to multiply one of them by the root of three, and then divide this number by two.
Theorems related to triangle
The main theorems that are associated with this figure are the Pythagorean theorem described above and cosines. The second (of sines) is that if you divide any side by the sine of the angle opposite it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosines) is that if from the sum of the squares of the two sides we subtract their product, multiplied by two and the cosine of the angle located between them, then we get the square of the third side.
Dali Triangle - what is it?
Many, when faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. Dali's triangle is common name three places that are closely related to life famous artist. Its “peaks” are the house in which Salvador Dali lived, the castle that he gave to his wife, as well as the museum of surrealist paintings. You can learn a lot during a tour of these places. interesting facts about this unique creative artist known throughout the world.
Two triangles are said to be congruent if they can be brought together by overlapping. Figure 1 shows equal triangles ABC and A 1 B 1 C 1. Each of these triangles can be superimposed on the other so that they are completely compatible, that is, their vertices and sides are compatible in pairs. It is clear that the angles of these triangles will also match in pairs.
Thus, if two triangles are congruent, then the elements (i.e. sides and angles) of one triangle are respectively equal to the elements of the other triangle. Note that in equal triangles against correspondingly equal sides(i.e., overlapping when superimposed) equal angles lie and back: Equal sides lie opposite respectively equal angles.
So, for example, in equal triangles ABC and A 1 B 1 C 1, shown in Figure 1, opposite equal sides AB and A 1 B 1, respectively, lie equal angles C and C 1. We will denote the equality of triangles ABC and A 1 B 1 C 1 as follows: Δ ABC = Δ A 1 B 1 C 1. It turns out that the equality of two triangles can be established by comparing some of their elements.
Theorem 1. The first sign of equality of triangles. If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent (Fig. 2).
Proof. Consider triangles ABC and A 1 B 1 C 1, in which AB = A 1 B 1, AC = A 1 C 1 ∠ A = ∠ A 1 (see Fig. 2). Let us prove that Δ ABC = Δ A 1 B 1 C 1 .
Since ∠ A = ∠ A 1, then triangle ABC can be superimposed on triangle A 1 B 1 C 1 so that vertex A is aligned with vertex A 1, and sides AB and AC are respectively superimposed on rays A 1 B 1 and A 1 C 1 . Since AB = A 1 B 1, AC = A 1 C 1, then side AB will align with side A 1 B 1 and side AC will align with side A 1 C 1; in particular, points B and B 1, C and C 1 will coincide. Consequently, sides BC and B 1 C 1 will align. So, triangles ABC and A 1 B 1 C 1 are completely compatible, which means they are equal.
Theorem 2 is proved in a similar way using the superposition method.
Theorem 2. The second sign of equality of triangles. If a side and two adjacent angles of one triangle are respectively equal to the side and two adjacent angles of another triangle, then such triangles are congruent (Fig. 34).
Comment. Based on Theorem 2, Theorem 3 is established.
Theorem 3. The sum of any two interior angles of a triangle is less than 180°.
Theorem 4 follows from the last theorem.
Theorem 4. An exterior angle of a triangle is greater than any interior angle not adjacent to it.
Theorem 5. The third sign of equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent ().
Example 1. In triangles ABC and DEF (Fig. 4)
∠ A = ∠ E, AB = 20 cm, AC = 18 cm, DE = 18 cm, EF = 20 cm. Compare triangles ABC and DEF. What angle in triangle DEF is equal to angle B?
Solution. These triangles are equal according to the first sign. Angle F of triangle DEF is equal to angle B of triangle ABC, since these angles lie opposite respectively equal sides DE and AC.
Example 2. Segments AB and CD (Fig. 5) intersect at point O, which is the middle of each of them. What is the length of segment BD if segment AC is 6 m?
Solution.
Triangles AOC and BOD are equal (according to the first criterion): ∠ AOC = ∠ BOD (vertical), AO = OB, CO = OD (by condition).
From the equality of these triangles it follows that their sides are equal, i.e. AC = BD. But since according to the condition AC = 6 m, then BD = 6 m.
