The definition of a quadrilateral is the sum of the interior angles of a quadrilateral. Quadrilaterals

INSCRIBED AND CIRCULAR POLYGONS,

§ 106. PROPERTIES OF INSCRIBED AND DESCRIBED QUADRIGONS.

Theorem 1. The sum of the opposite angles of a cyclic quadrilateral is 180°.

Let a quadrilateral ABCD be inscribed in a circle with center O (Fig. 412). It is required to prove that / A+ / C = 180° and / B + / D = 180°.

/ A, as inscribed in circle O, measures 1/2 BCD.
/ C, as inscribed in the same circle, measures 1/2 BAD.

Consequently, the sum of angles A and C is measured by the half-sum of arcs BCD and BAD; in sum, these arcs make up a circle, i.e. they have 360°.
From here / A+ / C = 360°: 2 = 180°.

Similarly, it is proved that / B + / D = 180°. However, this can be deduced in another way. We know that the sum of the interior angles of a convex quadrilateral is 360°. The sum of angles A and C is equal to 180°, which means that the sum of the other two angles of the quadrilateral also remains 180°.

Theorem 2(reverse). If in a quadrilateral the sum of two opposite angles is equal 180° , then a circle can be described around such a quadrilateral.

Let the sum of the opposite angles of the quadrilateral ABCD be equal to 180°, namely
/ A+ / C = 180° and / B + / D = 180° (drawing 412).

Let us prove that a circle can be described around such a quadrilateral.

Proof. Through any 3 vertices of this quadrilateral you can draw a circle, for example through points A, B and C. Where will point D be located?

Point D can only take one of the following three positions: be inside the circle, be outside the circle, be on the circumference of the circle.

Let's assume that the vertex is inside the circle and takes position D" (Fig. 413). Then in the quadrilateral ABCD" we will have:

/ B + / D" = 2 d.

Continuing side AD" to the intersection with the circle at point E and connecting points E and C, we obtain the cyclic quadrilateral ABCE, in which, by the direct theorem

/ B+ / E = 2 d.

From these two equalities it follows:

/ D" = 2 d - / B;
/ E=2 d - / B;

/ D" = / E,

but this cannot be, because / D", being external relative to the triangle CD"E, must be greater than angle E. Therefore, point D cannot be inside the circle.

It is also proved that vertex D cannot take position D" outside the circle (Fig. 414).

It remains to recognize that vertex D must lie on the circumference of the circle, i.e., coincide with point E, which means that a circle can be described around the quadrilateral ABCD.

Consequences. 1. A circle can be described around any rectangle.

2. A circle can be described around an isosceles trapezoid.

In both cases, the sum of opposite angles is 180°.

Theorem 3. In a circumscribed quadrilateral, the sums of opposite sides are equal. Let the quadrilateral ABCD be described about a circle (Fig. 415), i.e., its sides AB, BC, CD and DA are tangent to this circle.

It is required to prove that AB + CD = AD + BC. Let us denote the points of tangency by the letters M, N, K, P. Based on the properties of tangents drawn to a circle from one point (§ 75), we have:

AR = AK;
VR = VM;
DN = DK;
CN = CM.

Let us add these equalities term by term. We get:

AR + BP + DN + CN = AK + VM + DK + SM,

i.e. AB + CD = AD + BC, which is what needed to be proven.

Exercises.

1. In an inscribed quadrilateral, two opposite angles are in the ratio 3:5,
and the other two are in the ratio 4:5. Determine the magnitude of these angles.

2. In the described quadrilateral, the sum of two opposite sides is 45 cm. The remaining two sides are in the ratio 0.2: 0.3. Find the length of these sides.

Polygon concept

Definition 1

Polygon is a geometric figure in a plane that consists of segments connected in pairs, the adjacent ones do not lie on the same straight line.

In this case, the segments are called sides of the polygon, and their ends - vertices of the polygon.

Definition 2

An $n$-gon is a polygon with $n$ vertices.

Types of polygons

Definition 3

If a polygon always lies on the same side of any line passing through its sides, then the polygon is called convex(Fig. 1).

Figure 1. Convex polygon

Definition 4

If a polygon lies on opposite sides of at least one straight line passing through its sides, then the polygon is called non-convex (Fig. 2).

Figure 2. Non-convex polygon

Sum of angles of a polygon

Let us introduce a theorem on the sum of the angles of a triangle.

Theorem 1

The sum of the angles of a convex triangle is determined as follows

\[(n-2)\cdot (180)^0\]

Proof.

Let us be given a convex polygon $A_1A_2A_3A_4A_5\dots A_n$. Let's connect its vertex $A_1$ with all other vertices of this polygon (Fig. 3).

Figure 3.

With this connection we get $n-2$ triangles. By summing their angles we get the sum of the angles of a given -gon. Since the sum of the angles of a triangle is equal to $(180)^0,$ we obtain that the sum of the angles of a convex -gon is determined by the formula

\[(n-2)\cdot (180)^0\]

The theorem is proven.

Concept of a quadrilateral

Using the definition of $2$, it is easy to introduce the definition of a quadrilateral.

Definition 5

A quadrilateral is a polygon with $4$ vertices (Fig. 4).

Figure 4. Quadrangle

For a quadrilateral, the concepts of a convex quadrilateral and a non-convex quadrilateral are similarly defined. Classic examples of convex quadrilaterals are square, rectangle, trapezoid, rhombus, parallelogram (Fig. 5).

Figure 5. Convex quadrilaterals

Theorem 2

The sum of the angles of a convex quadrilateral is $(360)^0$

Proof.

By Theorem $1$, we know that the sum of the angles of a convex -gon is determined by the formula

\[(n-2)\cdot (180)^0\]

Therefore, the sum of the angles of a convex quadrilateral is equal to

\[\left(4-2\right)\cdot (180)^0=(360)^0\]

The theorem is proven.

One of the most interesting topics in geometry from the school course is “Quadrilaterals” (8th grade). What types of such figures exist, what special properties do they have? What is unique about quadrilaterals with ninety-degree angles? Let's figure it all out.

What geometric figure is called a quadrilateral?

Polygons that consist of four sides and, accordingly, four vertices (angles) are called quadrilaterals in Euclidean geometry.

The history of the name of this type of figure is interesting. In the Russian language, the noun “quadrangle” is formed from the phrase “four corners” (just like “triangle” - three corners, “pentagon” - five corners, etc.).

However, in Latin (through which many geometric terms came to most languages ​​of the world) it is called quadrilateral. This word is formed from the numeral quadri (four) and the noun latus (side). So we can conclude that the ancients called this polygon nothing more than a “quadrilateral”.

By the way, this name (with an emphasis on the presence of four sides, rather than corners, in figures of this type) has been preserved in some modern languages. For example, in English - quadrilateral and in French - quadrilatère.

Moreover, in most Slavic languages, the type of figure in question is still identified by the number of angles, not sides. For example, in Slovak (štvoruholník), in Bulgarian ("chetirigalnik"), in Belarusian ("chatyrokhkutnik"), in Ukrainian ("chotirikutnik"), in Czech (čtyřúhelník), but in Polish the quadrilateral is called by the number of sides - czworoboczny.

What types of quadrilaterals are studied in the school curriculum?

In modern geometry, there are 4 types of polygons with four sides.

However, due to the overly complex properties of some of them, schoolchildren are introduced to only two types in geometry lessons.

• Parallelogram. The opposite sides of such a quadrilateral are parallel to each other in pairs and, accordingly, are also equal in pairs.
• Trapezium (trapezium or trapezoid). This quadrilateral consists of two opposite sides parallel to each other. However, the other pair of sides does not have this feature.

Types of quadrilaterals not studied in the school geometry course

In addition to the above, there are two more types of quadrilaterals that schoolchildren are not introduced to in geometry lessons because of their particular complexity.

• Deltoid (kite)- a figure in which each of two pairs of adjacent sides is equal in length. This quadrilateral got its name due to the fact that in appearance it quite closely resembles the letter of the Greek alphabet - “delta”.
• Antiparallelogram- this figure is as complex as its name. In it, two opposite sides are equal, but at the same time they are not parallel to each other. In addition, the long opposite sides of this quadrilateral intersect each other, as do the extensions of the other two, shorter sides.

Types of parallelogram

Having dealt with the main types of quadrangles, it is worth paying attention to its subtypes. So, all parallelograms, in turn, are also divided into four groups.

• Classic parallelogram.
• Rhombus- a quadrangular figure with equal sides. Its diagonals intersect at right angles, dividing the rhombus into four equal right-angled triangles.
• Rectangle. The name speaks for itself. Since it is a quadrilateral with right angles (each of them is equal to ninety degrees). Its opposite sides are not only parallel to each other, but also equal.
• Square. Like a rectangle, it is a quadrilateral with right angles, but all its sides are equal. In this way, this figure is close to a rhombus. So we can say that a square is a cross between a rhombus and a rectangle.

Special properties of a rectangle

When considering figures in which each of the angles between the sides is equal to ninety degrees, it is worth taking a closer look at the rectangle. So, what special features does it have that distinguish it from other parallelograms?

To claim that the parallelogram in question is a rectangle, its diagonals must be equal to each other, and each of the angles must be right. In addition, the square of its diagonals must correspond to the sum of the squares of two adjacent sides of this figure. In other words, a classic rectangle consists of two right triangles, and in them, as is known, the diagonal of the quadrilateral in question acts as the hypotenuse.

The last of the listed features of this figure is also its special property. Besides this, there are others. For example, the fact that all sides of the quadrilateral being studied with right angles are also its heights.

In addition, if a circle is drawn around any rectangle, its diameter will be equal to the diagonal of the inscribed figure.

Among other properties of this quadrilateral is that it is flat and does not exist in non-Euclidean geometry. This is due to the fact that in such a system there are no quadrangular figures, the sum of the angles of which is equal to three hundred and sixty degrees.

Square and its features

Having understood the signs and properties of a rectangle, it is worth paying attention to the second quadrilateral known to science with right angles (this is a square).

Being in fact the same rectangle, but with equal sides, this figure has all its properties. But unlike it, the square is present in non-Euclidean geometry.

In addition, this figure has other distinctive features of its own. For example, the fact that the diagonals of a square are not only equal to each other, but also intersect at right angles. Thus, like a rhombus, a square consists of four right triangles into which it is divided by diagonals.

In addition, this figure is the most symmetrical among all quadrilaterals.

What is the sum of the angles of a quadrilateral?

When considering the features of quadrangles of Euclidean geometry, it is worth paying attention to their angles.

So, in each of the above figures, regardless of whether it has right angles or not, their total sum is always the same - three hundred and sixty degrees. This is a unique distinguishing feature of this type of figure.

Perimeter of quadrilaterals

Having figured out what the sum of the angles of a quadrilateral is equal to and other special properties of figures of this type, it is worth finding out what formulas are best used to calculate their perimeter and area.

To determine the perimeter of any quadrilateral, you just need to add the lengths of all its sides together.

For example, in the KLMN figure, its perimeter can be calculated using the formula: P = KL + LM + MN + KN. If you substitute the numbers here, you get: 6 + 8 + 6 + 8 = 28 (cm).

In the case when the figure in question is a rhombus or a square, to find the perimeter you can simplify the formula by simply multiplying the length of one of its sides by four: P = KL x 4. For example: 6 x 4 = 24 (cm).

Formulas for quadrilaterals with area

Having figured out how to find the perimeter of any figure with four corners and sides, it is worth considering the most popular and simple ways to find its area.

Other properties of quadrilaterals: incircles and circumcircles

Having considered the features and properties of a quadrilateral as a figure of Euclidean geometry, it is worth paying attention to the ability to describe circles around or inscribe circles inside it:

• If the sums of the opposite angles of a figure are one hundred and eighty degrees and are equal in pairs, then a circle can be freely described around such a quadrilateral.
• According to Ptolemy's theorem, if a circle is circumscribed outside a polygon with four sides, then the product of its diagonals is equal to the sum of the products of the opposite sides of the given figure. Thus, the formula will look like this: KM x LN = KL x MN + LM x KN.
• If you build a quadrilateral in which the sums of the opposite sides are equal to each other, then you can inscribe a circle in it.

Having figured out what a quadrilateral is, what types of it exist, which of them have only right angles between the sides and what properties they have, it’s worth remembering all this material. In particular, formulas for finding the perimeter and area of ​​the polygons considered. After all, figures of this shape are among the most common, and this knowledge can be useful for calculations in real life.

Today we will consider a geometric figure - a quadrilateral. From the name of this figure it already becomes clear that this figure has four corners. But we will consider the remaining characteristics and properties of this figure below.

What is a quadrilateral

A quadrilateral is a polygon consisting of four points (vertices) and four segments (sides) connecting these points in pairs. The area of ​​a quadrilateral is equal to half the product of its diagonals and the angle between them.

A quadrilateral is a polygon with four vertices, three of which do not lie on a straight line.

Types of quadrilaterals

• A quadrilateral whose opposite sides are parallel in pairs is called a parallelogram.
• A quadrilateral in which two opposite sides are parallel and the other two are not is called a trapezoid.
• A quadrilateral with all right angles is a rectangle.
• A quadrilateral with all sides equal is a rhombus.
• A quadrilateral in which all sides are equal and all angles are right is called a square.

A quadrilateral can be:

Self-intersecting

Non-convex

Convex

Self-intersecting quadrilateral is a quadrilateral in which any of its sides have an intersection point (in blue in the figure).

Non-convex quadrilateral is a quadrilateral in which one of the internal angles is more than 180 degrees (indicated in orange in the figure).

Sum of angles any quadrilateral that is not self-intersecting is always equal to 360 degrees.

Special types of quadrilaterals

Quadrilaterals can have additional properties, forming special types of geometric shapes:

• Parallelogram
• Rectangle
• Square
• Trapezoid
• Deltoid
• Counterparallelogram

Quadrangle and circle

A quadrilateral circumscribed around a circle (a circle inscribed in a quadrilateral).

The main property of the described quadrilateral:

A quadrilateral can be circumscribed around a circle if and only if the sums of the lengths of opposite sides are equal.

Quadrilateral inscribed in a circle (circle circumscribed around a quadrilateral)

The main property of an inscribed quadrilateral:

A quadrilateral can be inscribed in a circle if and only if the sum of the opposite angles is equal to 180 degrees.

Properties of the lengths of the sides of a quadrilateral

Modulus of the difference between any two sides of a quadrilateral does not exceed the sum of its other two sides.

|a - b| ≤ c + d

|a - c| ≤ b + d

|a - d| ≤ b + c

|b - c| ≤ a + d

|b - d| ≤ a + b

|c - d| ≤ a + b

Important. The inequality is true for any combination of sides of a quadrilateral. The drawing is provided solely for ease of perception.

In any quadrilateral the sum of the lengths of its three sides is not less than the length of the fourth side.

Important. When solving problems within the school curriculum, you can use strict inequality (<). Равенство достигается только в случае, если четырехугольник является "вырожденным", то есть три его точки лежат на одной прямой. То есть эта ситуация не попадает под классическое определение четырехугольника.

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