What is the sum of angles in a triangle. Sum of triangle angles

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

Proof

Let "ABC" be an arbitrary triangle. Let us draw a straight line through vertex B, parallel to straight line AC (such a straight line is called the Euclidean straight line). Mark point D on it so that points A and D lie on opposite sides of straight line BC. Angles DBC and ACB are equal as interior lying crosswise, formed by the transversal BC with parallel lines AC and BD. Therefore, the sum of the angles of the triangle at vertices B and C is equal to the angle ABD. The sum of all three angles of the triangle is equal to the sum of the angles ABD and BAC Since these are one-sided interior angles for parallel AC and BD. secant AB, then their sum is equal to 180°.
2) The exterior angle of a triangle at a given vertex is the angle adjacent to the angle of the triangle at this vertex.

Theorem: 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 ABC be the given triangle. By the theorem on the sum of angles in a triangle
∠ ABC + ∠ BCA + ∠ CAB = 180º.
It follows
∠ ABC + ∠ CAB = 180 º - ∠ BCA = ∠ BCD
The theorem has been proven.

From the theorem it follows:
An exterior angle of a triangle is greater than any angle of the triangle that is not adjacent to it.
3) Sum of triangle angles = 180 degrees. If one of the angles is right (90 degrees) the other two are also 90. This means that each of them is less than 90, that is, they are acute. if one of the angles is obtuse, then the other two are less than 90, that is, they are clearly acute.
4) obtuse - more than 90 degrees
acute - less than 90 degrees
5) a. A triangle in which one of the angles is 90 degrees.
b. Legs and hypotenuse
6) 6°. In every triangle, the larger angle lies opposite the larger side and the larger angle lies opposite the larger angle. Any segment has one and only one midpoint.
7) According to the Pythagorean theorem: the square of the hypotenuse is equal to the sum of the squares of the legs, which means the hypotenuse is greater than each of the legs
8) --- same as 7
9) The sum of the angles of a triangle is 180 degrees. and if each side of the triangle were greater than the sum of the other two sides, then the sum of the angles would be greater than 180, which is impossible. Therefore, each side of the triangle is less than the sum of the other two sides.
10) The sum of the angles of any triangle is 180 degrees.
Since this triangle is right-angled, one of its angles is right, i.e. equal to 90 degrees.
Therefore, the sum of the other two acute angles is 180-90=90 degrees.
11) 1. Consider a right triangle ABC in which angle A is a right angle, angle B = 30 degrees and angle C = 60. Let us attach to triangle ABC an equal triangle ABD. We get triangles BCD in which angle B = angle D = 60 degrees, therefore DC = BC. But according to the construction, AC is 1/2 BC, which is what needed to be proven.2. If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is equal to 30 degrees. Let’s prove this. Let’s consider a right triangle ABC, whose leg AC is equal to half the hypotenuse AC. Let us attach to triangle ABC an equal triangle ABD. Gets an equilateral triangle BCD. The angles of an equilateral triangle are equal to each other (since equal angles lie opposite equal sides), so each of them = 60 degrees. But angle DBC = 2 angles ABC, therefore angle ABC = 30 degrees, which is what needed to be proven.

This theorem is also formulated in the textbook by L.S. Atanasyan. , and in the textbook Pogorelov A.V. . The proofs of this theorem in these textbooks do not differ significantly, and therefore we present its proof, for example, from the textbook by A.V. Pogorelov.

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

Proof. Let ABC be the given triangle. Let us draw a line through vertex B parallel to line AC. Let's mark point D on it so that points A and D lie on opposite sides of straight line BC (Fig. 6).

Angles DBC and ACB are equal as internal cross-lying ones, formed by the secant BC with parallel straight lines AC and BD. Therefore, the sum of the angles of a triangle at vertices B and C is equal to angle ABD. And the sum of all three angles of a triangle is equal to the sum of angles ABD and BAC. Since these are one-sided interior angles for parallel AC and BD and secant AB, their sum is 180°. The theorem has been proven.

The idea of ​​this proof is to draw a parallel line and indicate that the required angles are equal. Let us reconstruct the idea of ​​such an additional construction by proving this theorem using the concept of a thought experiment. Proof of the theorem using a thought experiment. So, the subject of our thought experiment is the angles of a triangle. Let us place him mentally in conditions in which his essence can be revealed with particular certainty (stage 1).

Such conditions will be such an arrangement of the corners of the triangle in which all three of their vertices will be combined at one point. Such a combination is possible if we allow the possibility of “moving” the corners by moving the sides of the triangle without changing the angle of inclination (Fig. 1). Such movements are essentially subsequent mental transformations (stage 2).

By designating the angles and sides of a triangle (Fig. 2), the angles obtained by “moving,” we thereby mentally form the environment, the system of connections in which we place our subject of thought (stage 3).

Line AB, “moving” along line BC and without changing the angle of inclination to it, transfers angle 1 to angle 5, and “moving” along line AC, transfers angle 2 to angle 4. Since with such a “movement” line AB does not change the angle of inclination to lines AC and BC, then the conclusion is obvious: rays a and a1 are parallel to AB and transform into each other, and rays b and b1 are a continuation of sides BC and AC, respectively. Since angle 3 and the angle between rays b and b1 are vertical, they are equal. The sum of these angles is equal to the rotated angle aa1 - which means 180°.

CONCLUSION

In the thesis, “constructed” proofs of some school geometric theorems were carried out, using the structure of a thought experiment, which confirmed the formulated hypothesis.

The presented evidence was based on such visual and sensory idealizations: “compression”, “stretching”, “sliding”, which made it possible to transform the original geometric object in a special way and highlight its essential characteristics, which is typical for a thought experiment. In this case, the thought experiment acts as a certain “creative tool” that contributes to the emergence of geometric knowledge (for example, about the midline of a trapezoid or the angles of a triangle). Such idealizations make it possible to grasp the whole idea of ​​proof, the idea of ​​carrying out “additional construction,” which allows us to talk about the possibility of a more conscious understanding by schoolchildren of the process of formal deductive proof of geometric theorems.

A thought experiment is one of the basic methods for obtaining and discovering geometric theorems. It is necessary to develop a methodology for transferring the method to the student. The question remains open about the age of a student acceptable for “accepting” the method, about the “side effects” of the evidence presented in this way.

These issues require further study. But in any case, one thing is certain: a thought experiment develops theoretical thinking in schoolchildren, is its basis and, therefore, the ability for thought experimentation needs to be developed.

Theorem. The sum of the interior angles of a triangle is equal to two right angles.

Let's take some triangle ABC (Fig. 208). Let us denote its interior angles by numbers 1, 2 and 3. Let us prove that

∠1 + ∠2 + ∠3 = 180°.

Let us draw through some vertex of the triangle, for example B, a straight line MN parallel to AC.

At vertex B we got three angles: ∠4, ∠2 and ∠5. Their sum is a straight angle, therefore it is equal to 180°:

∠4 + ∠2 + ∠5 = 180°.

But ∠4 = ∠1 are internal crosswise angles with parallel lines MN and AC and secant AB.

∠5 = ∠3 - these are internal crosswise angles with parallel lines MN and AC and secant BC.

This means that ∠4 and ∠5 can be replaced by their equals ∠1 and ∠3.

Therefore, ∠1 + ∠2 + ∠3 = 180°. The theorem has been proven.

2. Property of the external angle of a triangle.

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

In fact, in triangle ABC (Fig. 209) ∠1 + ∠2 = 180° - ∠3, but also ∠ВСD, the external angle of this triangle, not adjacent to ∠1 and ∠2, is also equal to 180° - ∠3 .

Thus:

∠1 + ∠2 = 180° - ∠3;

∠BCD = 180° - ∠3.

Therefore, ∠1 + ∠2= ∠BCD.

The derived property of the exterior angle of a triangle clarifies the content of the previously proven theorem on the exterior angle of a triangle, which stated only that the exterior angle of a triangle is greater than each interior angle of a triangle not adjacent to it; now it is established that the external angle is equal to the sum of both internal angles not adjacent to it.

3. Property of a right triangle with an angle of 30°.

Theorem. A leg of a right triangle lying opposite an angle of 30° is equal to half the hypotenuse.

Let angle B in the right triangle ACB be equal to 30° (Fig. 210). Then its other acute angle will be equal to 60°.

Let us prove that leg AC is equal to half the hypotenuse AB. Let's extend the leg AC beyond the vertex of the right angle C and set aside a segment CM equal to the segment AC. Let's connect point M to point B. The resulting triangle ВСМ is equal to triangle ACB. We see that each angle of triangle ABM is equal to 60°, therefore this triangle is an equilateral triangle.

Leg AC is equal to half of AM, and since AM is equal to AB, leg AC will be equal to half of the hypotenuse AB.

>>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;
• Using historical material to develop students’ cognitive activity;
• 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 ||.
The living connection between generations is not interrupted for a moment; 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?