Theorem on the sum of the angles of a triangle. Sum of angles of a triangle. Complete lessons – Knowledge Hypermarket

The fact that “The sum of the angles of any triangle in Euclidean geometry is 180 degrees” can simply be remembered. If it’s not easy to remember, you can conduct a couple of experiments for better memorization.

Experiment one

Draw several arbitrary triangles on a piece of paper, for example:

• with arbitrary sides;
• isosceles triangle;
• right triangle.

Be sure to use a ruler. Now you need to cut out the resulting triangles, doing it exactly along the drawn lines. Color in the corners of each triangle with a colored pencil or marker. For example, in the first triangle all corners will be red, in the second - blue, in the third - green. http://bit.ly/2gY4Yfz

From the first triangle, cut off all 3 corners and connect them at one point with their vertices, so that the nearest sides of each corner are connected. As you can see, the three corners of the triangle formed an extended angle, which is equal to 180 degrees. Do the same with the other two triangles - the result will be the same. http://bit.ly/2zurCrd

Experiment two

Draw an arbitrary triangle ABC. We select any vertex (for example, C) and draw a straight line DE through it, parallel to the opposite side (AB). http://bit.ly/2zbYNzq

We get the following:

1. Angles BAC and ACD are equal as internal angles perpendicular to AC;
2. Angles ABC and BCE are equal as internal angles perpendicular to BC;
3. We see that angles 1, 2 and 3 are the angles of a triangle, connected at one point to form a developed angle DCE, which is equal to 180 degrees.

The triangle angle sum theorem states that the sum of all internal corners of any triangle is 180°.

Let the interior angles of a triangle be a, b and c, then:

a + b + c = 180°.

From this theory we can conclude that the sum of all external angles of any triangle is equal to 360°. Since an external angle is adjacent to an internal angle, their sum is 180°. Let the interior angles of a triangle be a, b and c, then the exterior angles at these angles are 180° - a, 180° - b and 180° - c.

Let's find the sum of the external angles of a triangle:

180° - a + 180° - b + 180° - c = 540° - (a + b + c) = 540° - 180° = 360°.

Answer: the sum of the interior angles of a triangle is 180°; the sum of the external angles of a triangle is 360°.

Preliminary information

First, let's look directly at the concept of a triangle.

Definition 1

We will call a triangle a geometric figure that is made up of three points connected to each other by segments (Fig. 1).

Definition 2

Within the framework of Definition 1, we will call the points the vertices of the triangle.

Definition 3

Within the framework of Definition 1, the segments will be called sides of the triangle.

Obviously, any triangle will have 3 vertices, as well as three sides.

Theorem on the sum of angles in a triangle

Let us introduce and prove one of the main theorems related to triangles, namely the theorem on the sum of angles in a triangle.

Theorem 1

The sum of the angles in any arbitrary triangle is $180^\circ$.

Proof.

Consider the triangle $EGF$. Let us prove that the sum of the angles in this triangle is equal to $180^\circ$. Let's make an additional construction: draw the straight line $XY||EG$ (Fig. 2)

Since the lines $XY$ and $EG$ are parallel, then $∠E=∠XFE$ lie crosswise at the secant $FE$, and $∠G=∠YFG$ lie crosswise at the secant $FG$

Angle $XFY$ will be reversed and therefore equals $180^\circ$.

$∠XFY=∠XFE+∠F+∠YFG=180^\circ$

Hence

$∠E+∠F+∠G=180^\circ$

The theorem has been proven.

Triangle Exterior Angle Theorem

Another theorem on the sum of angles for a triangle can be considered the theorem on the external angle. First, let's introduce this concept.

Definition 4

We will call an external angle of a triangle an angle that will be adjacent to any angle of the triangle (Fig. 3).

Let us now consider the theorem directly.

Theorem 2

An exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.

Proof.

Consider an arbitrary triangle $EFG$. Let it have an external angle of the triangle $FGQ$ (Fig. 3).

By Theorem 1, we will have that $∠E+∠F+∠G=180^\circ$, therefore,

$∠G=180^\circ-(∠E+∠F)$

Since the angle $FGQ$ is external, it is adjacent to the angle $∠G$, then

$∠FGQ=180^\circ-∠G=180^\circ-180^\circ+(∠E+∠F)=∠E+∠F$

The theorem has been proven.

Sample tasks

Example 1

Find all angles of a triangle if it is equilateral.

Since all the sides of an equilateral triangle are equal, we will have that all the angles in it are also equal to each other. Let us denote their degree measures by $α$.

Then, by Theorem 1 we get

$α+α+α=180^\circ$

Answer: all angles equal $60^\circ$.

Example 2

Find all angles isosceles triangle, if one of its angles is equal to $100^\circ$.

Let us introduce the following notation for angles in an isosceles triangle:

Since we are not given in the condition exactly what angle $100^\circ$ is equal to, then two cases are possible:

An angle equal to $100^\circ$ is the angle at the base of the triangle.

Using the theorem on angles at the base of an isosceles triangle, we obtain

$∠2=∠3=100^\circ$

But then only their sum will be greater than $180^\circ$, which contradicts the conditions of Theorem 1. This means that this case does not occur.

An angle equal to $100^\circ$ is the angle between equal sides, that is

Goals and objectives:

Educational:

• repeat and generalize knowledge about the triangle;
• prove the theorem on the sum of the angles of a triangle;
• practically verify the correctness of the formulation of the theorem;
• learn to apply acquired knowledge when solving problems.

Educational:

• develop geometric thinking, interest in the subject, cognitive and creative activity students, mathematical speech, the ability to independently obtain knowledge.

Educational:

• develop personal qualities students, such as determination, perseverance, accuracy, ability to work in a team.

Equipment: multimedia projector, triangles made of colored paper, educational complex “Living Mathematics”, computer, screen.

Preparatory stage: The teacher gives the student the task to prepare historical information about the theorem “Sum of the angles of a triangle.”

Lesson type: learning new material.

During the classes

I. Organizational moment

Greetings. Psychological attitude of students to work.

II. Warm-up

WITH geometric figure“triangle” we met in previous lessons. Let's repeat what we know about the triangle?

Students work in groups. They are given the opportunity to communicate with each other, each to independently build the process of cognition.

What happened? Each group makes their proposals, the teacher writes them on the board. The results are discussed:

Picture 1

III. Formulating the lesson objective

So, we already know quite a lot about the triangle. But not all. Each of you has triangles and protractors on your desk. What kind of problem do you think we can formulate?

Students formulate the task of the lesson - to find the sum of the angles of a triangle.

IV. Explanation of new material

Practical part(promotes updating knowledge and self-knowledge skills). Measure the angles using a protractor and find their sum. Write down the results in your notebook (listen to the answers received). We find out that the sum of the angles is different for everyone (this can happen because the protractor was not applied accurately, the calculation was carried out carelessly, etc.).

Fold along the dotted lines and find out what else the sum of the angles of a triangle is equal to:

A)
Figure 2

b)
Figure 3

V)
Figure 4

G)
Figure 5

d)
Figure 6

After completing the practical work, students formulate the answer: The sum of the angles of a triangle is equal to the degree measure of the unfolded angle, i.e. 180°.

Teacher: In mathematics practical work It only makes it possible to make some kind of statement, but it needs to be proven. A statement whose validity is established by proof is called a theorem. What theorem can we formulate and prove?

Students: The sum of the angles of a triangle is 180 degrees.

Historical reference: The property of the sum of the angles of a triangle was established in Ancient Egypt. The proof, set out in modern textbooks, is contained in Proclus's commentary on Euclid's Elements. Proclus claims that this proof (Fig. 8) was discovered by the Pythagoreans (5th century BC). In the first book of the Elements, Euclid sets out another proof of the theorem on the sum of the angles of a triangle, which can be easily understood with the help of a drawing (Fig. 7):


Figure 7


Figure 8

The drawings are displayed on the screen through a projector.

The teacher offers to prove the theorem using drawings.

Then the proof is carried out using the teaching and learning complex “Living Mathematics”. The teacher projects the proof of the theorem on the computer.

Theorem on the sum of angles of a triangle: “The sum of the angles of a triangle is 180°”


Figure 9

Proof:

A)

Figure 10

b)

Figure 11

V)

Figure 12

Students make a brief note of the proof of the theorem in their notebooks:

Theorem: The sum of the angles of a triangle is 180°.


Figure 13

Given:Δ ABC

Prove: A + B + C = 180°.

Proof:

What needed to be proven.

V. Phys. just a minute.

VI. Explanation of new material (continued)

The corollary from the theorem on the sum of the angles of a triangle is deduced by students independently, this contributes to the development of the ability to formulate their own point of view, express and argue for it:

In any triangle, either all angles are acute, or two are acute and the third is obtuse or right..

If a triangle has all acute angles, then it is called acute-angled.

If one of the angles of a triangle is obtuse, then it is called obtuse-angled.

If one of the angles of a triangle is right, then it is called rectangular.

The theorem on the sum of angles of a triangle allows us to classify triangles not only by sides, but also by angles. (As students introduce types of triangles, students fill out the table)

Table 1

Triangle view	Isosceles	Equilateral	Versatile
Rectangular
Obtuse
Acute-angled

VII. Consolidation of the studied material.

1. Solve problems orally:

(Drawings are displayed on the screen through a projector)

Task 1. Find angle C.


Figure 14

Problem 2. Find the angle F.


Figure 15

Task 3. Find the angles K and N.

Figure 16

Problem 4. Find the angles P and T.


Figure 17

1. Solve problem No. 223 (b, d) yourself.
2. Solve the problem on the board and in notebooks, student No. 224.
3. Questions: Can a triangle have: a) two right angles; b) two obtuse angles; c) one right and one obtuse angle.
4. (done orally) The cards on each table show various triangles. Determine by eye the type of each triangle.


Figure 18

1. Find the sum of angles 1, 2 and 3.


Figure 19

VIII. Lesson summary.

Teacher: What have we learned? Is the theorem applicable to any triangle?

IX. Reflection.

Tell me your mood, guys! On the reverse side of the triangle, depict your facial expressions.


Figure 20

Homework: paragraph 30 (part 1), question 1 ch. IV page 89 of the textbook; No. 223 (a, c), No. 225.

Theorem on the sum of interior angles of a triangle

The sum of the angles of a triangle is 180°.

Proof:

• Given triangle ABC.
• Through vertex B we draw a straight line DK parallel to the base AC.
• \angle CBK= \angle C as internal crosswise lying with parallel DK and AC, and secant BC.
• \angle DBA = \angle A internal crosswise lying with DK \parallel AC and secant AB. Angle DBK is reversed and equal to
• \angle DBK = \angle DBA + \angle B + \angle CBK
• Since the unfolded angle is equal to 180 ^\circ , and \angle CBK = \angle C and \angle DBA = \angle A , we get 180 ^\circ = \angle A + \angle B + \angle C.

The theorem is proven

Corollaries from the theorem on the sum of angles of a triangle:

1. The sum of the acute angles of a right triangle is equal to 90°.
2. In isosceles right triangle every sharp corner equals 45°.
3. In an equilateral triangle, each angle is equal 60°.
4. In any triangle, either all the angles are acute, or two angles are acute, and the third is obtuse or right.
5. External angle of a triangle equal to the sum two internal angles not adjacent to it.

Triangle Exterior Angle Theorem

An exterior angle of a triangle is equal to the sum of the two remaining angles of the triangle that are not adjacent to this exterior angle

Proof:



• Given a triangle ABC, where BCD is the exterior angle.
• \angle BAC + \angle ABC +\angle BCA = 180^0
• From the equalities the angle \angle BCD + \angle BCA = 180^0
• We get \angle BCD = \angle BAC+\angle ABC.

>>Geometry: Sum of angles of a triangle. Complete lessons

LESSON TOPIC: Sum of angles of a triangle.


Lesson objectives:

• Consolidating and testing students’ knowledge on the topic: “Sum of angles of a triangle”;
• Proof of the properties of the angles of a triangle;
• Application of this property in solving simple problems;
• Usage historical material for the development of cognitive activity of students;
• Instilling the skill of accuracy when constructing drawings.

Lesson objectives:


• Test students' problem-solving skills.

Lesson plan:

1. Triangle;
2. Theorem on the sum of the angles of a triangle;
3. Example tasks.

Triangle.


File:O.gif Triangle- the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.
Three points in space that do not lie on the same straight line correspond to one and only one plane.
Any polygon can be divided into triangles - this process is called triangulation.
There is a section of mathematics entirely devoted to the study of the laws of triangles - Trigonometry.

Theorem on the sum of the angles of a triangle.


File:T.gif The triangle angle sum theorem is a classic theorem of Euclidean geometry that states that the sum of the angles of a triangle is 180°.



Proof" :


Let Δ ABC be given. Let us draw a line parallel to (AC) through vertex B and mark point D on it so that points A and D lie on opposite sides of line BC. Then the angle (DBC) and the angle (ACB) are equal as internal crosswise lying with parallel lines BD and AC and the secant (BC). Then the sum of the angles of the triangle at vertices B and C is equal to angle (ABD). But the angle (ABD) and the angle (BAC) at vertex A of triangle ABC are internal one-sided with parallel lines BD and AC and the secant (AB), and their sum is 180°. Therefore, the sum of the angles of a triangle is 180°. The theorem has been proven.




Consequences.


An exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.


Proof:


Let Δ ABC be given. Point D lies on line AC so that A lies between C and D. Then BAD is external to the angle of the triangle at vertex A and A + BAD = 180°. But A + B + C = 180°, and therefore B + C = 180° – A. Hence BAD = B + C. The corollary is proven.





Consequences.

An exterior angle of a triangle is greater than any angle of the triangle that is not adjacent to it.


Task.


An exterior angle of a triangle is an angle adjacent to any angle of this triangle. Prove that the exterior angle of a triangle is equal to the sum of two angles of the triangle that are not adjacent to it.
(Fig.1)


Solution:


Let in Δ ABC ∠DAС be external (Fig. 1). Then ∠DAC = 180°-∠BAC (by the property of adjacent angles), by the theorem on the sum of the angles of a triangle ∠B+∠C = 180°-∠BAC. From these equalities we obtain ∠DAС=∠В+∠С


Interesting fact:

Sum of the angles of a triangle" :


In Lobachevsky geometry, the sum of the angles of a triangle is always less than 180. In Euclidian geometry it is always equal to 180. In Riemann geometry, the sum of the angles of a triangle is always greater than 180.


From the history of mathematics:


Euclid (3rd century BC) in his work “Elements” gives the following definition: “Parallel lines are lines that are in the same plane and, being extended in both directions indefinitely, do not meet each other on either side.” .
Posidonius (1st century BC) “Two straight lines lying in the same plane, equally spaced from each other”
The ancient Greek scientist Pappus (III century BC) introduced the symbol of parallel lines - the = sign. Subsequently, the English economist Ricardo (1720-1823) used this symbol as an equals sign.
Only in the 18th century did they begin to use the symbol for parallel lines - the sign ||.
Doesn't stop for a moment live connection between generations, every day we learn the experience accumulated by our ancestors. The ancient Greeks, based on observations and practical experience, drew conclusions, expressed hypotheses, and then, at meetings of scientists - symposia (literally “feast”) - they tried to substantiate and prove these hypotheses. At that time, the statement arose: “Truth is born in dispute.”


Questions:


1. What is a triangle?
2. What does the theorem about the sum of the angles of a triangle say?
3. What is the external angle of the triangle?