Note. This material contains the theorem and its proof, as well as a number of problems illustrating the application of the theorem on the sum of angles of a convex polygon on practical examples.

Convex polygon angle sum theorem

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Proof.

To prove the theorem on the sum of angles of a convex polygon, we use the already proven theorem that the sum of the angles of a triangle is 180 degrees.

Let A 1 A 2... A n be given convex polygon, and n > 3. Draw all the diagonals of the polygon from the vertex A 1. They divide it into n – 2 triangles: Δ A 1 A 2 A 3, Δ A 1 A 3 A 4, ... , Δ A 1 A n – 1 A n . The sum of the angles of the polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 180°, and the number of triangles is (n - 2). Therefore, the sum of the angles of a convex n-gon A 1 A 2... A n is 180° (n – 2).

A task.

In a convex polygon, three angles are 80 degrees and the rest are 150 degrees. How many corners are in a convex polygon?

Solution.

The theorem says: For a convex n-gon, the sum of the angles is 180°(n-2) .

So for our case:

180(n-2)=3*80+x*150, where

3 angles of 80 degrees are given to us according to the condition of the problem, and the number of other angles is still unknown to us, so we denote their number as x.

However, from the entry on the left side, we determined the number of corners of the polygon as n, since we know the values ​​of three of them from the condition of the problem, it is obvious that x=n-3.

So the equation will look like this:

180(n-2)=240+150(n-3)

We solve the resulting equation

180n - 360 = 240 + 150n - 450

180n - 150n = 240 + 360 - 450

Answer: 5 peaks

A task.

How many vertices can a polygon have if each angle is less than 120 degrees?

Solution.

To solve this problem, we use the theorem on the sum of angles of a convex polygon.

The theorem says: For a convex n-gon, the sum of all angles is 180°(n-2) .

Hence, for our case, it is necessary to first estimate the boundary conditions of the problem. That is, make the assumption that each of the angles is equal to 120 degrees. We get:

180n - 360 = 120n

180n - 120n = 360 (we will consider this expression separately below)

Based on the equation obtained, we conclude: when the angles are less than 120 degrees, the number of corners of the polygon is less than six.

Explanation:

Based on the expression 180n - 120n = 360 , provided that the subtracted right side is less than 120n, the difference should be more than 60n. Thus, the quotient of division will always be less than six.

Answer: the number of polygon vertices will be less than six.

A task

A polygon has three angles of 113 degrees, and the rest are equal to each other and their degree measure is an integer. Find the number of vertices of the polygon.

Solution.

To solve this problem, we use the theorem on the sum of the external angles of a convex polygon.

The theorem says: For a convex n-gon, the sum of all exterior angles is 360° .

In this way,

3*(180-113)+(n-3)x=360

the right side of the expression is the sum of the external angles, on the left side the sum of the three angles is known by condition, and the degree measure of the rest (their number, respectively, n-3, since three angles are known) is denoted as x.

159 is decomposed only into two factors 53 and 3, and 53 is a prime number. That is, there are no other pairs of factors.

Thus, n-3 = 3, n=6, that is, the number of corners of the polygon is six.

Answer: six corners

A task

Prove that a convex polygon can have at most three sharp corners.

Solution

As you know, the sum of the external angles of a convex polygon is 360 0 . Let us prove by contradiction. If a convex polygon has at least four acute internal corners, therefore, among its external angles there are at least four obtuse ones, which implies that the sum of all external angles of the polygon is greater than 4*90 0 = 360 0 . We have a contradiction. The assertion has been proven.

The sum of the angles of an n-gon Theorem. The sum of the angles of a convex n-gon is 180 o (n-2). Proof. From some vertex of a convex n-gon we draw all its diagonals. Then the n-gon will break into n-2 triangles. In each triangle, the sum of the angles is 180 o, and these angles make up the angles of the n-gon. Therefore, the sum of the angles of an n-gon is 180 o (n-2).

The second method of proof Theorem. The sum of the angles of a convex n-gon is 180 o (n-2). Proof 2. Let O be some inner point convex n-gon A 1 …A n. Connect it to the vertices of this polygon. Then the n-gon will be divided into n triangles. In each triangle, the sum of the angles is 180 o. These angles make up the angles of the n-gon and another 360 o. Therefore, the sum of the angles of an n-gon is 180 o (n-2).

Exercise 3 Prove that the sum of the exterior angles of a convex n-gon is 360 o. Proof. The exterior angle of a convex polygon is 180° minus the corresponding interior angle. Therefore, the sum of the exterior angles of a convex n-gon is 180 o n minus the sum of the interior angles. Since the sum of the internal angles of a convex n-gon is 180 o (n-2), then the sum of the external angles will be 180 o n o (n-2) = 360 o.

Exercise 4 What are the angles of a regular: a) triangle; b) quadrilateral; c) a pentagon; d) hexagon; e) an octagon; e) decagon; g) a dodecagon? Answer: a) 60 o; b) 90 o; c) 108 o; d) 120 o; e) 135 o; f) 144 o; g) 150 o.

Exercise 12* What largest number can a convex n-gon have sharp corners? Solution. Since the sum of the external angles of a convex polygon is 360 o, then a convex polygon cannot have more than three obtuse corners, therefore, it cannot have more than three internal acute angles. Answer. 3.

These geometric shapes surround us everywhere. Convex polygons are natural, such as honeycombs, or artificial (man-made). These figures are used in the production various kinds coatings, in painting, architecture, decoration, etc. Convex polygons have the property that all their points are on the same side of a line that passes through a pair of adjacent vertices of this line. geometric figure. There are other definitions as well. A polygon is called convex if it is located in a single half-plane with respect to any straight line containing one of its sides.

In the course of elementary geometry, only simple polygons are always considered. To understand all the properties of such, it is necessary to understand their nature. To begin with, it should be understood that any line is called closed, the ends of which coincide. Moreover, the figure formed by it can have a variety of configurations. A polygon is a simple closed broken line, in which neighboring links are not located on the same straight line. Its links and vertices are, respectively, the sides and vertices of this geometric figure. A simple polyline must not have self-intersections.

The vertices of a polygon are called adjacent if they represent the ends of one of its sides. A geometric figure that has nth number vertices, and hence nth quantity sides is called an n-gon. The broken line itself is called the border or contour of this geometric figure. A polygonal plane or a flat polygon is called the end part of any plane bounded by it. The adjacent sides of this geometric figure are called segments of a broken line emanating from one vertex. They will not be adjacent if they come from different vertices of the polygon.

Other definitions of convex polygons

In elementary geometry, there are several more equivalent definitions indicating which polygon is called convex. Moreover, all these expressions the same degree are true. A convex polygon is one that has:

Every line segment that connects any two points within it lies entirely within it;

All its diagonals lie inside it;

Any internal angle does not exceed 180°.

A polygon always splits a plane into 2 parts. One of them is limited (it can be enclosed in a circle), and the other is unlimited. The first is called the inner region, and the second is the outer region of this geometric figure. This polygon is an intersection (in other words, a common component) of several half-planes. Moreover, each segment that has ends at points that belong to the polygon completely belongs to it.

Varieties of convex polygons

The definition of a convex polygon does not indicate that there are many kinds of them. And each of them has certain criteria. So, convex polygons that have an interior angle of 180° are called weakly convex. A convex geometric figure that has three vertices is called a triangle, four - a quadrilateral, five - a pentagon, etc. Each of the convex n-gons corresponds to the following essential requirement: n must be equal to or greater than 3. Each of the triangles is convex. Geometric figure of this type, in which all vertices are located on the same circle, is called inscribed in the circle. A convex polygon is called circumscribed if all its sides near the circle touch it. Two polygons are said to be equal only if they can be superimposed by superposition. Flat polygon called a polygonal plane (part of the plane), which is limited by this geometric figure.

Regular convex polygons

Regular polygons are geometric shapes with equal angles and parties. Inside them there is a point 0, which is at the same distance from each of its vertices. It is called the center of this geometric figure. The segments connecting the center with the vertices of this geometric figure are called apothems, and those that connect the point 0 with the sides are called radii.

A regular quadrilateral is a square. right triangle called equilateral. For such figures, there is the following rule: each angle of a convex polygon is 180° * (n-2)/ n,

where n is the number of vertices of this convex geometric figure.

The area of ​​any regular polygon determined by the formula:

where p is equal to half the sum of all sides of the given polygon, and h is equal to the length of the apothem.

Properties of convex polygons

Convex polygons have certain properties. So, a segment that connects any 2 points of such a geometric figure is necessarily located in it. Proof:

Suppose P is a given convex polygon. We take 2 arbitrary points, for example, A, B, which belong to R. By existing definition of a convex polygon, these points are located on one side of the line, which contains any side P. Therefore, AB also has this property and is contained in P. A convex polygon can always be divided into several triangles by absolutely all the diagonals drawn from one of its vertices.

Angles of convex geometric shapes

The corners of a convex polygon are the corners that are formed by its sides. The interior corners are in inner region this geometric figure. The angle that is formed by its sides that converge at one vertex is called the angle of a convex polygon. with internal angles of a given geometric figure are called external. Each corner of a convex polygon located inside it is equal to:

where x is the value of the external angle. This simple formula applies to any geometric shapes of this type.

AT general case, for external corners there exists following rule: each angle of a convex polygon is equal to the difference between 180° and the value of the interior angle. It can have values ​​ranging from -180° to 180°. Therefore, when the inside angle is 120°, the outside angle will be 60°.

Sum of angles of convex polygons

The sum of the interior angles of a convex polygon is determined by the formula:

where n is the number of vertices of the n-gon.

The sum of the angles of a convex polygon is quite easy to calculate. Consider any such geometric figure. To determine the sum of angles inside a convex polygon, one of its vertices must be connected to other vertices. As a result of this action, (n-2) triangles are obtained. We know that the sum of the angles of any triangle is always 180°. Since their number in any polygon is (n-2), the sum of the interior angles of such a figure is 180° x (n-2).

The sum of the angles of a convex polygon, namely any two internal and adjacent external angles, for a given convex geometric figure will always be 180°. Based on this, you can determine the sum of all its angles:

The sum of the interior angles is 180° * (n-2). Based on this, the sum of all external angles of a given figure is determined by the formula:

180° * n-180°-(n-2)= 360°.

The sum of the exterior angles of any convex polygon will always be 360° (regardless of the number of sides).

The exterior angle of a convex polygon is generally represented by the difference between 180° and the interior angle.

Other properties of a convex polygon

In addition to the basic properties of these geometric shapes, they have others that arise when manipulating them. So, any of the polygons can be divided into several convex n-gons. To do this, it is necessary to continue each of its sides and cut this geometric figure along these straight lines. It is also possible to split any polygon into several convex parts in such a way that the vertices of each of the pieces coincide with all its vertices. From such a geometric figure, triangles can be very simply made by drawing all the diagonals from one vertex. Thus, any polygon can eventually be divided into a certain number of triangles, which turns out to be very useful in solving various tasks associated with such geometric shapes.

Perimeter of a convex polygon

The segments of a broken line, called the sides of a polygon, are most often indicated by the following letters: ab, bc, cd, de, ea. These are the sides of a geometric figure with vertices a, b, c, d, e. The sum of the lengths of all sides of this convex polygon is called its perimeter.

Polygon circle

Convex polygons can be inscribed and circumscribed. A circle that touches all sides of this geometric figure is called inscribed in it. Such a polygon is called circumscribed. The center of a circle that is inscribed in a polygon is the intersection point of the bisectors of all angles within a given geometric figure. The area of ​​such a polygon is:

where r is the radius of the inscribed circle and p is the semi-perimeter of the given polygon.

A circle containing the vertices of a polygon is called circumscribed around it. Moreover, this convex geometric figure is called inscribed. The center of the circle, which is circumscribed about such a polygon, is the intersection point of the so-called perpendicular bisectors of all sides.

Diagonals of convex geometric shapes

The diagonals of a convex polygon are line segments that connect neighboring vertices. Each of them lies inside this geometric figure. The number of diagonals of such an n-gon is determined by the formula:

N = n (n - 3) / 2.

The number of diagonals of a convex polygon is important role in elementary geometry. The number of triangles (K) into which each convex polygon can be divided is calculated by the following formula:

The number of diagonals of a convex polygon always depends on the number of its vertices.

Splitting a convex polygon

In some cases, to solve geometric problems it is necessary to split a convex polygon into several triangles with non-intersecting diagonals. This problem can be solved by deriving a specific formula.

Definition of the problem: let's call a correct partition of a convex n-gon into several triangles by diagonals that intersect only at the vertices of this geometric figure.

Solution: Suppose that Р1, Р2, Р3 …, Pn are vertices of this n-gon. The number Xn is the number of its partitions. Let us carefully consider the resulting diagonal of the geometric figure Pi Pn. In any of the regular partitions P1 Pn belongs to a certain triangle P1 Pi Pn, which has 1

Let i = 2 be one group of regular partitions always containing the diagonal Р2 Pn. The number of partitions included in it coincides with the number of partitions of the (n-1)-gon Р2 Р3 Р4… Pn. In other words, it equals Xn-1.

If i = 3, then this other group of partitions will always contain the diagonals P3 P1 and P3 Pn. In this case, the number of regular partitions contained in this group will coincide with the number of partitions of the (n-2)-gon Р3 Р4… Pn. In other words, it will equal Xn-2.

Let i = 4, then among the triangles a regular partition will certainly contain a triangle P1 P4 Pn, to which the quadrilateral P1 P2 P3 P4, (n-3)-gon P4 P5 ... Pn will adjoin. The number of regular partitions of such a quadrilateral is X4, and the number of partitions of an (n-3)-gon is Xn-3. Based on the foregoing, we can say that the total number of correct partitions contained in this group is Xn-3 X4. Other groups for which i = 4, 5, 6, 7… will contain Xn-4 X5, Xn-5 X6, Xn-6 X7 … regular partitions.

Let i = n-2, then the number of correct partitions in this group will be the same as the number of partitions in the group where i=2 (in other words, equals Xn-1).

Since X1 = X2 = 0, X3=1, X4=2…, then the number of all partitions of a convex polygon is equal to:

Xn = Xn-1 + Xn-2 + Xn-3 X4 + Xn-4 X5 + ... + X 5 Xn-4 + X4 Xn-3 + Xn-2 + Xn-1.

X5 = X4 + X3 + X4 = 5

X6 = X5 + X4 + X4 + X5 = 14

X7 = X6 + X5 + X4 * X4 + X5 + X6 = 42

X8 = X7 + X6 + X5 * X4 + X4 * X5 + X6 + X7 = 132

The number of regular partitions intersecting one diagonal inside

When checking special cases, one can come to the assumption that the number of diagonals of convex n-gons is equal to the product of all partitions of this figure by (n-3).

Proof of this assumption: imagine that P1n = Xn * (n-3), then any n-gon can be divided into (n-2)-triangles. Moreover, an (n-3)-quadrilateral can be composed of them. Along with this, each quadrilateral will have a diagonal. Since two diagonals can be drawn in this convex geometric figure, this means that additional (n-3) diagonals can be drawn in any (n-3)-quadrilaterals. Based on this, we can conclude that in any regular partition it is possible to draw (n-3)-diagonals that meet the conditions of this problem.

Area of ​​convex polygons

Often, when solving various problems of elementary geometry, it becomes necessary to determine the area of ​​a convex polygon. Assume that (Xi. Yi), i = 1,2,3… n is the sequence of coordinates of all neighboring vertices of a polygon that does not have self-intersections. In this case, its area is calculated by the following formula:

S = ½ (∑ (X i + X i + 1) (Y i + Y i + 1)),

where (X 1, Y 1) = (X n +1, Y n + 1).

In the basic geometry course, it is proved that the sum of the angles of a convex n-gon is 180° (n-2). It turns out that this statement is also true for non-convex polygons.

Theorem 3. The sum of the angles of an arbitrary n-gon is 180° (n - 2).

Proof. Let's divide the polygon into triangles by drawing diagonals (Fig. 11). The number of such triangles is n-2, and in each triangle the sum of the angles is 180°. Since the angles of the triangles are the angles of the polygon, the sum of the angles of the polygon is 180° (n - 2).

Let us now consider arbitrary closed broken lines, possibly with self-intersections A1A2…AnA1 (Fig. 12, a). Such self-intersecting broken lines will be called star-shaped polygons (Fig. 12, b-d).

Let us fix the direction of counting the angles counterclockwise. Note that the angles formed by a closed polyline depend on the direction in which it is traversed. If the direction of the polyline bypass is reversed, then the angles of the polygon will be the angles that complement the angles of the original polygon up to 360°.

If M is a polygon formed by a simple closed broken line passing in a clockwise direction (Fig. 13, a), then the sum of the angles of this polygon will be equal to 180 ° (n - 2). If the broken line is passed in the counterclockwise direction (Fig. 13, b), then the sum of the angles will be equal to 180 ° (n + 2).

Thus, the general formula for the sum of the angles of a polygon formed by a simple closed polyline has the form = 180 ° (n 2), where is the sum of the angles, n is the number of angles of the polygon, "+" or "-" is taken depending on the direction of bypassing the polyline.

Our task is to derive a formula for the sum of the angles of an arbitrary polygon formed by a closed (possibly self-intersecting) polyline. To do this, we introduce the concept of the degree of a polygon.

The degree of a polygon is the number of revolutions made by a point during a complete sequential bypass of its sides. Moreover, the turns made in the counterclockwise direction are considered with the “+” sign, and the turns in the clockwise direction - with the “-” sign.

It is clear that the degree of a polygon formed by a simple closed broken line is +1 or -1, depending on the direction of the traversal. The degree of the broken line in Figure 12, a is equal to two. The degree of star heptagons (Fig. 12, c, d) is equal to two and three, respectively.

The notion of degree is defined similarly for closed curves in the plane. For example, the degree of the curve shown in Figure 14 is two.

To find the degree of a polygon or curve, you can proceed as follows. Suppose that, moving along the curve (Fig. 15, a), we, starting from some place A1, made a full turn, and ended up at the same point A1. Let's remove the corresponding section from the curve and continue moving along the remaining curve (Fig. 15b). If, starting from some place A2, we again made a full turn and got to the same point, then we delete the corresponding section of the curve and continue moving (Fig. 15, c). Counting the number of remote sections with the signs "+" or "-", depending on their direction of bypass, we obtain the desired degree of the curve.

Theorem 4. For an arbitrary polygon, the formula

180° (n+2m),

where is the sum of the angles, n is the number of angles, m is the degree of the polygon.

Proof. Let the polygon M have degree m and is conventionally shown in Figure 16. M1, …, Mk are simple closed broken lines, passing through which the point makes full turns. A1, …, Ak are the corresponding self-intersection points of the polyline, which are not its vertices. Let us denote the number of vertices of the polygon M that are included in the polygons M1, …, Mk by n1, …, nk, respectively. Since, in addition to the vertices of the polygon M, vertices A1, …, Ak are added to these polygons, the number of vertices of the polygons M1, …, Mk will be equal to n1+1, …, nk+1, respectively. Then the sum of their angles will be equal to 180° (n1+12), …, 180° (nk+12). Plus or minus is taken depending on the direction of bypassing broken lines. The sum of the angles of the polygon M0, remaining from the polygon M after the removal of the polygons M1, ..., Mk, is equal to 180° (n-n1- ...-nk+k2). The sums of the angles of the polygons M0, M1, …, Mk give the sum of the angles of the polygon M, and at each vertex A1, …, Ak we additionally obtain 360°. Therefore, we have the equality

180° (n1+12)+…+180° (nk+12)+180° (n-n1-…-nk+k2)=+360°k.

180° (n2…2) = 180° (n+2m),

where m is the degree of the polygon M.

As an example, consider the calculation of the sum of the angles of a five-pointed asterisk (Fig. 17, a). The degree of the corresponding closed polyline is -2. Therefore, the desired sum of the angles is 180.

broken line

Definition

broken line, or shorter, broken line, is called a finite sequence of segments, such that one of the ends of the first segment serves as the end of the second, the other end of the second segment serves as the end of the third, and so on. In this case, adjacent segments do not lie on the same straight line. These segments are called polyline links.

Types of broken line

The broken line is called closed if the beginning of the first segment coincides with the end of the last one.

The broken line can cross itself, touch itself, lean on itself. If there are no such singularities, then such a broken line is called simple.

Polygons

Definition

A simple closed polyline, together with a part of the plane bounded by it, is called polygon.

Comment

At each vertex of a polygon, its sides define some angle of the polygon. It can be either less than deployed, or more than deployed.

Property

Each polygon has an angle less than $180^\circ$.

Proof

Let a polygon $P$ be given.

Let's draw some straight line that does not intersect it. We will move it parallel to the side of the polygon. At some point, for the first time we obtain a line $a$ that has at least one common point with the polygon $P$. The polygon lies on one side of this line (moreover, some of its points lie on the line $a$).

The line $a$ contains at least one vertex of the polygon. Its two sides converge in it, located on the same side of the line $a$ (including the case when one of them lies on this line). So, at this vertex, the angle is less than the developed one.

Definition

The polygon is called convex if it lies on one side of each line containing its side. If the polygon is not convex, it is called non-convex.

Comment

A convex polygon is the intersection of half-planes bounded by lines that contain the sides of the polygon.

Properties of a convex polygon

A convex polygon has all angles less than $180^\circ$.

A line segment connecting any two points of a convex polygon (in particular, any of its diagonals) is contained in this polygon.

Proof

Let's prove the first property

Take any corner $A$ of a convex polygon $P$ and its side $a$ coming from the vertex $A$. Let $l$ be a line containing side $a$. Since the polygon $P$ is convex, it lies on one side of the line $l$. Therefore, its angle $A$ also lies on the same side of this line. Hence the angle $A$ is less than the straightened angle, that is, less than $180^\circ$.

Let's prove the second property

Take any two points $A$ and $B$ of a convex polygon $P$. The polygon $P$ is the intersection of several half-planes. The segment $AB$ is contained in each of these half-planes. Therefore, it is also contained in the polygon $P$.

Definition

Diagonal polygon is called a segment connecting its non-neighboring vertices.

Theorem (on the number of diagonals of an n-gon)

The number of diagonals of a convex $n$-gon is calculated by the formula $\dfrac(n(n-3))(2)$.

Proof

From each vertex of an n-gon one can draw $n-3$ diagonals (one cannot draw a diagonal to neighboring vertices and to this vertex itself). If we count all such possible segments, then there will be $n\cdot(n-3)$, since there are $n$ vertices. But each diagonal will be counted twice. Thus, the number of diagonals of an n-gon is $\dfrac(n(n-3))(2)$.

Theorem (on the sum of the angles of an n-gon)

The sum of the angles of a convex $n$-gon is $180^\circ(n-2)$.

Proof

Consider the $n$-gon $A_1A_2A_3\ldots A_n$.

Take an arbitrary point $O$ inside this polygon.

The sum of the angles of all triangles $A_1OA_2$, $A_2OA_3$, $A_3OA_4$, \ldots, $A_(n-1)OA_n$ is $180^\circ\cdot n$.

On the other hand, this sum is the sum of all interior angles of the polygon and the total angle $\angle O=\angle 1+\angle 2+\angle 3+\ldots=30^\circ$.

Then the sum of the angles of the considered $n$-gon is equal to $180^\circ\cdot n-360^\circ=180^\circ\cdot(n-2)$.

Consequence

The sum of the angles of a non-convex $n$-gon is $180^\circ(n-2)$.

Proof

Consider a polygon $A_1A_2\ldots A_n$ whose only angle $\angle A_2$ is non-convex, that is, $\angle A_2>180^\circ$.

Let's denote the sum of his catch $S$.

Connect the points $A_1A_3$ and consider the polygon $A_1A_3\ldots A_n$.

The sum of the angles of this polygon is:

$180^\circ\cdot(n-1-2)=S-\angle A_2+\angle 1+\angle 2=S-\angle A_2+180^\circ-\angle A_1A_2A_3=S+180^\circ-( \angle A_1A_2A_3+\angle A_2)=S+180^\circ-360^\circ$.

Therefore, $S=180^\circ\cdot(n-1-2)+180^\circ=180^\circ\cdot(n-2)$.

If the original polygon has more than one non-convex corner, then the operation described above can be done with each such corner, which will lead to the assertion being proved.

Theorem (on the sum of the exterior angles of a convex n-gon)

The sum of the exterior angles of a convex $n$-gon is $360^\circ$.

Proof

The exterior angle at vertex $A_1$ is $180^\circ-\angle A_1$.

The sum of all external angles is:

$\sum\limits_(n)(180^\circ-\angle A_n)=n\cdot180^\circ - \sum\limits_(n)A_n=n\cdot180^\circ - 180^\circ\cdot(n -2)=360^\circ$.