Sum of angles of a triangle. Complete lessons - Knowledge Hypermarket. The sum of the angles of a triangle - what is it equal to? Types by angle size

In 8th grade, during geometry lessons at school, students are first introduced to the concept of a convex polygon. Very soon they will learn that this figure has a very interesting property. No matter how complex it may be, the sum of all internal and external angles of a convex polygon takes on a strictly defined value. In this article, a mathematics and physics tutor talks about what the sum of the angles of a convex polygon is equal to.

Sum of interior angles of a convex polygon

How to prove this formula?

Before moving on to the proof of this statement, let us remember which polygon is called convex. A convex polygon is a polygon that lies entirely on one side of a line containing any of its sides. For example, the one shown in this figure:

If the polygon does not satisfy the specified condition, then it is called non-convex. For example, like this:

The sum of the interior angles of a convex polygon is equal to , where is the number of sides of the polygon.

The proof of this fact is based on the theorem on the sum of angles in a triangle, well known to all schoolchildren. I am sure that this theorem is familiar to you too. The sum of the interior angles of a triangle is .

The idea is to split a convex polygon into several triangles. This can be done in different ways. Depending on which method we choose, the evidence will be slightly different.

1. Divide the convex polygon into triangles using all possible diagonals drawn from some vertex. It is easy to understand that then our n-gon will be divided into triangles:

Moreover, the sum of all the angles of all the resulting triangles is equal to the sum of the angles of our n-gon. After all, each angle in the resulting triangles is a partial angle in our convex polygon. That is, the required amount is equal to .

2. You can also select a point inside the convex polygon and connect it to all the vertices. Then our n-gon will be divided into triangles:

Moreover, the sum of the angles of our polygon in this case will be equal to the sum of all the angles of all these triangles minus the central angle, which is equal to . That is, the required amount is again equal to .

Sum of exterior angles of a convex polygon

Let us now ask the question: “What is the sum of the external angles of a convex polygon?” This question can be answered as follows. Each external corner is adjacent to the corresponding internal one. Therefore it is equal to:

Then the sum of all external angles is equal to . That is, it is equal.

That is, a very funny result is obtained. If we plot all the external angles of any convex n-gon sequentially one after another, then the result will be exactly the entire plane.

This interesting fact can be illustrated as follows. Let's proportionally reduce all sides of some convex polygon until it merges into a point. After this happens, all external angles will be laid aside from one another and thus fill the entire plane.

Interesting fact, isn't it? And there are a lot of such facts in geometry. So learn geometry, dear schoolchildren!

The material on what the sum of the angles of a convex polygon is equal to was prepared by Sergey Valerievich

Sum of triangle angles- an important, but fairly simple topic that is taught in 7th grade geometry. The topic consists of a theorem, a short proof and several logical consequences. Knowledge of this topic helps in solving geometric problems in subsequent study of the subject.

Theorem - what are the angles of an arbitrary triangle added together?

The theorem states that if you take any triangle, regardless of its type, the sum of all angles will invariably be 180 degrees. This is proven as follows:

for example, take triangle ABC, draw a straight line through point B located at the apex and designate it as “a”, straight line “a” is strictly parallel to side AC;
between straight line “a” and sides AB and BC, angles are designated, marking them with numbers 1 and 2;
angle 1 is considered equal to angle A, and angle 2 is considered equal to angle C, since these angles are considered to lie crosswise;
thus, the sum between angles 1, 2 and 3 (which is designated in place of angle B) is recognized as equal to the unfolded angle with vertex B - and is 180 degrees.

If the sum of the angles indicated by numbers is 180 degrees, then the sum of angles A, B and C is recognized as equal to 180 degrees. This rule is true for any triangle.

What follows from the geometric theorem

It is customary to highlight several corollaries from the above theorem.

If the problem considers a triangle with a right angle, then one of its angles will be equal to 90 degrees by default, and the sum of the acute angles will also be 90 degrees.
If we are talking about a right isosceles triangle, then its acute angles, which add up to 90 degrees, will individually be equal to 45 degrees.
An equilateral triangle consists of three equal angles, respectively, each of them will be equal to 60 degrees, and in total they will be 180 degrees.
The exterior angle of any triangle will be equal to the sum between two interior angles not adjacent to it.

The following rule can be derived: any triangle has at least two acute angles. In some cases, a triangle consists of three acute angles, and if there are only two, then the third angle will be obtuse or right.

(background summary)

Visual geometry 7th grade. Supporting note No. 4 Sum of angles of a triangle.

Great French scientist of the 17th century Blaise Pascal As a child, I loved to tinker with geometric shapes. He was familiar with the protractor and knew how to measure angles. The young researcher noticed that for all triangles the sum of the three angles is the same - 180°. “How can we prove this? - Pascal thought. “After all, it’s impossible to check the sum of the angles of all triangles - there are an infinite number of them.” Then he cut off two corners of the triangle with scissors and attached them to the third corner. The result is a rotated angle, which, as is known, is equal to 180°. This was his first own discovery. The boy's future fate was already predetermined.

In this topic, you will learn five properties of congruence of right triangles and, perhaps, the most popular property of a right triangle with an angle of 30°. It sounds like this: the leg lying opposite the angle of 30° is equal to half the hypotenuse. By dividing an equilateral triangle by height, we immediately obtain a proof of this property.

THEOREM. The sum of the angles of a triangle is 180°. To prove this, draw a line through the top parallel to the base. Dark angles are equal and gray angles are equal as if they lie crosswise on parallel lines. The dark angle, the gray angle and the apex angle form an extended angle, their sum is 180°. From the theorem it follows that the angles of an equilateral triangle are equal to 60° and that the sum of the acute angles of a right triangle is equal to 90°.

External corner of a triangle is the angle adjacent to the angle of the triangle. Therefore, sometimes the angles of the triangle itself are called interior angles.

THEOREM about the external angle of a triangle. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it. Indeed, the outer corner and two inner, not adjacent to it, complement the shaded angle up to 180°. It follows from the theorem that an exterior angle is greater than any interior angle not adjacent to it.

THEOREM about the relationships between the sides and angles of a triangle. In a triangle, the larger angle is opposite the larger side, and the larger angle is opposite the larger angle. It follows: 1) The leg is less than the hypotenuse. 2) The perpendicular is less than the inclined one.

Distance from point to line . Since the perpendicular is less than any inclined line drawn from the same point, its length is taken as the distance from the point to the straight line.

Triangle inequality . The length of any side of a triangle is less than the sum of its two other sides, i.e. A< b + с , b< а + с , With< а + b . Consequence. The length of the broken line is greater than the segment connecting its ends.

SIGNS OF EQUALITY
RECTANGULAR TRIANGLES

On two sides. If two legs of one right triangle are respectively equal to two legs of another triangle, then such triangles are congruent.

Along the leg and adjacent acute angle. If the leg and the adjacent acute angle of one right triangle are respectively equal to the leg and the adjacent acute angle of another triangle, then such triangles are congruent.

Along the leg and the opposite acute angle. If the leg and the acute angle opposite it of one right triangle are respectively equal to the leg and the acute angle opposite it of another triangle, then such triangles are congruent.

By hypotenuse and acute angle. If the hypotenuse and acute angle of one right triangle are respectively equal to the hypotenuse and acute angle of another triangle, then such triangles are congruent.

The proof of these signs immediately reduces to one of the tests for the equality of triangles.

By leg and hypotenuse. If the leg and hypotenuse of one right triangle are respectively equal to the leg and hypotenuse of another right triangle, then such triangles are congruent.

Proof. Let's attach triangles with equal legs. We get an isosceles triangle. Its height drawn from the vertex will also be the median. Then the triangles have equal second legs, and the triangles are equal on three sides.

THEOREM about the property of a leg lying opposite an angle of 30°. The leg opposite the 30° angle is equal to half the hypotenuse. Proved by completing the triangle to an equilateral one.

THEOREM about the property of angle bisector points. Any point on the bisector of an angle is equidistant from its sides. If a point is equidistant from the sides of an angle, then it lies on the bisector of the angle. Proved by drawing two perpendiculars to the sides of the angle and considering right triangles.

Second great point . The bisectors of a triangle intersect at one point.

Distance between parallel lines. THEOREM. All points of each of two parallel lines are at equal distances from the other line. The theorem implies the definition of the distance between parallel lines.

Definition. The distance between two parallel lines is the distance from any point of one of the parallel lines to the other line.

Detailed proofs of the theorems

This is reference note No. 4 on geometry in 7th grade. Select next steps:

The sum of the interior angles of a triangle is 180 0. This is one of the fundamental axioms of Euclid's geometry. This is the geometry that schoolchildren study. Geometry is defined as the science that studies the spatial forms of the real world.

What motivated the ancient Greeks to develop geometry? The need to measure fields, meadows - areas of the earth's surface. At the same time, the ancient Greeks accepted that the surface of the Earth was horizontal and flat. Taking this assumption into account, Euclid’s axioms were created, including the sum of the internal angles of a triangle of 180 0.

An axiom is a proposition that does not require proof. How should this be understood? A wish is expressed that suits the person, and then it is confirmed by illustrations. But everything that has not been proven is fiction, something that does not exist in reality.

Taking the earth's surface horizontal, the ancient Greeks automatically accepted the shape of the Earth as flat, but it is different - spherical. There are no horizontal planes or straight lines in nature at all, because gravity bends space. Straight lines and horizontal planes are found only in the human brain.

Therefore, Euclid's geometry, which explains the spatial forms of the fictional world, is a simulacrum - a copy that has no original.

One of Euclid's axioms states that the sum of the interior angles of a triangle is 180 0. In fact, in real curved space, or on the spherical surface of the Earth, the sum of the internal angles of a triangle is always greater than 180 0.

Let's think like this. Any meridian on the globe intersects with the equator at an angle of 90 0. To get a triangle, you need to move another meridian away from the meridian. The sum of the angles of the triangle between the meridians and the side of the equator will be 180 0. But there will still be an angle at the pole. As a result, the sum of all angles will be more than 180 0.

If the sides intersect at an angle of 90 0 at the pole, then the sum of the internal angles of such a triangle will be 270 0. Two meridians intersecting the equator at right angles in this triangle will be parallel to each other, and at the pole intersecting each other at an angle of 90 0 will become perpendiculars. It turns out that two parallel lines on the same plane not only intersect, but can also be perpendiculars at the pole.

Of course, the sides of such a triangle will not be straight lines, but convex, repeating the spherical shape of the globe. But this is exactly the real world of space.

The geometry of real space, taking into account its curvature in the middle of the 19th century. developed by the German mathematician B. Riemann (1820-1866). But schoolchildren are not told about this.

So, Euclidean geometry, which takes the form of the Earth as flat with a horizontal surface, which in fact it is not, is a simulacrum. Nootic is Riemannian geometry that takes into account the curvature of space. The sum of the interior angles of the triangle in it is greater than 180 0.

>>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:

Triangle;
Theorem on the sum of the angles of a triangle;
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: