One of the most interesting topics in geometry from the school course is "Quadangles" (grade 8). What types of such figures exist, what special properties do they have? What is unique about quadrilaterals with ninety-degree corners? Let's look into all this.

What geometric figure is called a quadrilateral

Polygons, which consist of four sides and, respectively, of four vertices (corners), are called quadrilaterals in Euclidean geometry.

The history of the name of this type of figures is interesting. In the Russian language, the noun "quadrangular" 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 among the ancients this polygon was referred to only as "four-sided".

By the way, such a name (with an emphasis on the presence of four sides, not corners in figures of this type) has been preserved in some modern languages. For example, in English - quadrilateral and in French - quadrilatère.

At the same time, in most Slavic languages, the considered type of figures is still identified by the number of angles, and not sides. For example, in Slovak (štvoruholník), in Bulgarian (“chetirigalnik”), in Belarusian (“chatyrokhkutnik”), in Ukrainian (“chotirikutnik”), in Czech (čtyřúhelník), but in Polish the quadrangle is called by the number of sides - czworoboczny.

What types of quadrangles are studied in the school curriculum

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

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

• Parallelogram. The opposite sides of such a quadrangle are pairwise parallel to each other and, accordingly, are also equal in pairs.
• Trapeze (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 to each other. Such a quadrilateral got its name due to the fact that in appearance it quite strongly 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 subspecies. So, all parallelograms, in turn, are also divided into four groups.

• Classical parallelogram.
• Rhombus (rhombus)- a quadrangular figure with equal sides. Its diagonals intersect at right angles, dividing the rhombus into four equal right 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 (square). Like a rectangle, it is a quadrilateral with right angles, but it has all sides equal to each other. This figure is close to a rhombus. So it can be argued that a square is a cross between a rhombus and a rectangle.

Rectangle Special Properties

Considering figures in which each of the angles between the sides is equal to ninety degrees, it is worth dwelling more closely on the rectangle. So, what special features does it have that distinguish it from other parallelograms?

To assert that the parallelogram under consideration 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, the classic rectangle consists of two right-angled triangles, and in them, as is known, the diagonal of the quadrilateral under consideration acts as the hypotenuse.

The last of the listed signs of this figure is also its special property. Besides this, there are others. For example, the fact that all sides of the studied quadrilateral with right angles are at the same time 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, 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 dealt with 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 just equal to each other, but also intersect at a right angle. Thus, like a rhombus, a square consists of four right-angled 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

Considering the features of Euclidean geometry quadrangles, 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 and other special properties of figures of this type, it is worth knowing what formulas are best used to calculate their perimeter and area.

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

For example, in the KLMN figure, its perimeter can be calculated using the formula: P \u003d 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 \u003d KL x 4. For example: 6 x 4 \u003d 24 (cm).

Area quadrilateral formulas

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: inscribed and circumscribed circles

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 around or inscribe circles inside it:

• If the sums of the opposite angles of the figure are one hundred and eighty degrees each and are pairwise equal to each other, 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 opposite sides of the given figure. Thus, the formula will look like this: KM x LN \u003d KL x MN + LM x KN.
• If you construct a quadrilateral in which the sums of opposite sides are equal to each other, then a circle can be inscribed 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 is worth remembering all this material. In particular, the formulas for finding the perimeter and area of ​​\u200b\u200bthe considered polygons. After all, figures of this form are one of the most common, and this knowledge can be useful for calculations in real life.

Definition. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

Property. In a parallelogram, opposite sides are equal and opposite angles are equal.

Property. The diagonals of a parallelogram are bisected by the intersection point.

1 sign of a parallelogram. If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

2 sign of a parallelogram. If the opposite sides of a quadrilateral are equal in pairs, then the quadrilateral is a parallelogram.

3 sign of a parallelogram. If in a quadrilateral the diagonals intersect and the intersection point is bisected, then this quadrilateral is a parallelogram.

Definition. A trapezoid is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Parallel sides are called grounds.

The trapezoid is called isosceles (isosceles) if its sides are equal. In an isosceles trapezoid, the angles at the bases are equal.

rectangular.

midline of the trapezoid. The middle line is parallel to the bases and equal to their half-sum.

Rectangle

Definition.

Property. The diagonals of a rectangle are equal.

Rectangle sign. If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.

Definition.

Property. The diagonals of a rhombus are mutually perpendicular and bisect its angles.

Definition.

A square is a particular kind of rectangle, and also a particular kind of rhombus. Therefore, it has all their properties.

Properties:
1. All corners of the square are right

Quadrangles all the rules

Keywords:
quadrilateral, convex, sum of angles, area of ​​a quadrilateral

quadrilateral a figure is called, which consists of four points and four segments connecting them in series. In this case, no three of these points should lie on one straight line, and the segments connecting them should not intersect.

• The vertices of the quadrilateral are called neighboring if they are the ends of one of its sides.
• Vertices that are not neighbors , called opposite .
• Line segments connecting opposite vertices of a quadrilateral are called diagonals .
• The sides of a quadrilateral that originate from the same vertex are called neighboring parties.
• Sides that do not have a common end are called opposite parties.
• The quadrilateral is called convex , if it is located in one half-plane relative to the straight line containing any of its sides.

Types of quadrilaterals

1. Parallelogram A quadrilateral with opposite sides parallel
   • Rectangle a parallelogram with all right angles
   • Rhombus - a parallelogram with all sides equal
   • Square - a rectangle with all sides equal
2. Trapeze - a quadrilateral in which two sides are parallel and the other two sides are not parallel
3. Deltoid A quadrilateral whose two pairs of adjacent sides are equal

Quadrangles

quadrilateral a figure is called, which consists of four points and four segments connecting them in series. In this case, no three of these points lie on the same straight line, and the segments connecting them do not intersect.

opposite. opposite.

Types of quadrilaterals

Parallelogram

Parallelogram is called a quadrilateral whose opposite sides are pairwise parallel.

Parallelogram properties

• opposite sides are equal;
• opposite angles are equal;
• the sum of the squares of the diagonals is equal to the sum of the squares of all the sides:

Parallelogram features

Trapeze A quadrilateral is called, in which two opposite sides are parallel, and the other two are not parallel.

The parallel sides of a trapezoid are called its grounds and the non-parallel sides sides. The segment connecting the midpoints of the sides is called middle line.

The trapezoid is called isosceles(or isosceles) if its sides are equal.

A trapezoid with one right angle is called rectangular.

Trapezoid Properties

Signs of a trapezoid

Rectangle

Rectangle A parallelogram is called if all angles are right angles.

Rectangle properties

Rectangle Features

A parallelogram is a rectangle if:

1. One of its corners is right.
2. Its diagonals are equal.

Rhombus A parallelogram is called if all sides are equal.

Rhombus Properties

• all the properties of a parallelogram;
• the diagonals are perpendicular;

Signs of a rhombus

Square A rectangle is called in which all sides are equal.

Square properties

• all corners of the square are right;
• the diagonals of the square are equal, mutually perpendicular, the intersection point is divided in half and the corners of the square are divided in half.

Square signs

Basic formulas

S=d 1 d 2 sin

Parallelogram
a and b- adjacent parties; - the angle between them; h a - height to side a.

S = ab sin

S=d 1 d 2 sin

Trapeze
a and b- grounds; h- the distance between them; l- middle line .

Rectangle

S=d 1 d 2 sin

S = a 2 sin

S=d 1 d 2

Square
d- diagonal.

www.univer.omsk.su

Properties of quadrilaterals. Types of quadrilaterals. Properties of arbitrary quadrilaterals. Parallelogram properties. Rhombus properties. Rectangle properties. Square properties. trapezoid properties. Approximately 7-9 grade (13-15 years old)

Properties of quadrilaterals. Types of quadrilaterals. Properties of arbitrary quadrilaterals.
Parallelogram properties. Rhombus properties. Rectangle properties. Square properties. trapezoid properties.

Types of quadrilaterals:

• Parallelogram is a quadrilateral whose opposite sides are parallel

• Rhombus is a parallelogram with all sides equal.

• Rectangle is a parallelogram with all right angles.

• Square is a rectangle with all sides equal.

Properties of arbitrary quadrilaterals:

Parallelogram properties:

Rhombus properties:

Rectangle properties:

Square properties:

Trapeze properties:

Consulting and technical
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Quadrangles all the rules

Non-Euclidean geometry, geometry similar to geometry Euclid in that it defines the movement of figures, but differs from Euclidean geometry in that one of its five postulates (second or fifth) is replaced by its negation. The denial of one of the Euclidean postulates (1825) was a significant event in the history of thought, for it served as the first step towards theory of relativity.

Euclid's second postulate states that any line segment can be extended indefinitely. Euclid apparently believed that this postulate also contained the statement that the straight line has infinite length. However in "elliptic" geometry any straight line is finite and, like a circle, is closed.

The fifth postulate states that if a line intersects two given lines in such a way that the two interior angles on one side of it are less than two right angles in sum, then these two lines, if extended indefinitely, will intersect on the side where the sum of these angles is less than the sum two straight lines. But in "hyperbolic" geometry, there may exist a line CB (see Fig.), Perpendicular at point C to a given line r and intersecting another line s at an acute angle at point B, but, nevertheless, the infinite lines r and s will never intersect .

From these revised postulates it followed that the sum of the angles of a triangle, equal to 180° in Euclidean geometry, is greater than 180° in elliptic geometry and less than 180° in hyperbolic geometry.

Quadrilateral

Quadrilateral is a polygon containing four vertices and four sides.

Quadrilateral, a geometric figure - a polygon with four corners, as well as any object, a device of this form.

Two non-adjacent sides of a quadrilateral are called opposite. Two vertices that are not adjacent are also called opposite.

Quadrangles are convex (like ABCD) and
non-convex (A 1 B 1 C 1 D 1).

Types of quadrilaterals

• Parallelogram- a quadrilateral in which all opposite sides are parallel;
• Rectangle- a quadrilateral with all right angles;
• Rhombus- a quadrilateral in which all sides are equal;
• Square- a quadrilateral in which all angles are right and all sides are equal;
• Trapeze- a quadrilateral with two opposite sides parallel;
• Deltoid A quadrilateral whose two pairs of adjacent sides are equal.

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

Parallelogram (from the Greek parallelos - parallel and gramme - line) i.e. lie on parallel lines. Special cases of a parallelogram are a rectangle, a square and a rhombus.

• opposite sides are equal;
• opposite angles are equal;
• the diagonals of the intersection point are divided in half;
• the sum of the angles adjacent to one side is 180°;
• the sum of the squares of the diagonals is equal to the sum of the squares of all the sides.

A quadrilateral is a parallelogram if:

1. Its two opposite sides are equal and parallel.
2. Opposite sides are equal in pairs.
3. Opposite angles are equal in pairs.
4. The diagonals of the intersection point are divided in half.

Rectangle

A rectangle is a parallelogram with all right angles.

• opposite sides are equal;
• opposite angles are equal;
• the diagonals of the intersection point are divided in half;
• the sum of the angles adjacent to one side is 180°;
• the diagonals are equal.

A parallelogram is a rectangle if:

1. One of its corners is right.
2. Its diagonals are equal.

A rhombus is a parallelogram in which all sides are equal.

• opposite sides are equal;
• opposite angles are equal;
• the diagonals of the intersection point are divided in half;
• the sum of the angles adjacent to one side is 180°;
• the sum of the squares of the diagonals is equal to the sum of the squares of all the sides;
• the diagonals are perpendicular;
• the diagonals are the bisectors of its angles.

A parallelogram is a rhombus if:

1. Its two adjacent sides are equal.
2. Its diagonals are perpendicular.
3. One of the diagonals is the bisector of its angle.

A square is a rectangle in which all sides are equal.

• all corners of the square are right;
• the diagonals of the square are equal, mutually perpendicular, the intersection point is divided in half and the corners of the square are divided in half.

1. A rectangle is a square if it has some characteristic of a rhombus.

A trapezoid is a quadrilateral in which two opposite sides are parallel and the other two are not parallel.

The parallel sides of a trapezoid are called its bases, and the non-parallel sides are called its sides. The segment connecting the midpoints of the sides is called the midline.

A trapezoid is called isosceles (or isosceles) if its sides are equal.

A trapezoid with one right angle is called a right angled trapezoid.

• its middle line is parallel to the bases and equal to their half-sum;
• if the trapezoid is isosceles, then its diagonals are equal and the angles at the base are equal;
• if the trapezoid is isosceles, then a circle can be described around it;
• if the sum of the bases is equal to the sum of the sides, then a circle can be inscribed in it.

1. A quadrilateral is a trapezoid if its parallel sides are not equal

Deltoid A quadrilateral with two pairs of sides of the same length. Unlike a parallelogram, two pairs of adjacent sides are not equal, but two pairs of adjacent sides. The deltoid is shaped like a kite.

• The angles between sides of unequal length are equal.
• The diagonals of the deltoid (or their extensions) intersect at right angles.
• A circle can be inscribed in any convex deltoid, besides this, if the deltoid is not a rhombus, then there is another circle that touches the extensions of all four sides. For a non-convex deltoid, one can construct a circle tangent to two larger sides and extensions of two smaller sides, and a circle tangent to two smaller sides and extensions of two larger sides.

• If the angle between the unequal sides of the deltoid is a straight line, then a circle can be inscribed in it (the described deltoid).
• If a pair of opposite sides of a deltoid are equal, then such a deltoid is a rhombus.
• If a pair of opposite sides and both diagonals of a deltoid are equal, then the deltoid is a square. An inscribed deltoid with equal diagonals is also a square.

The emergence of geometry dates back to ancient times and was due to the practical needs of human activity (the need to measure land, measure the volumes of various bodies, etc.).

The simplest geometric information and concepts were known in ancient Egypt. During this period, geometric statements were formulated in the form of rules given without proof.

From the 7th century BC e. to the 1st century AD e. geometry as a science developed rapidly in ancient Greece. During this period, not only the accumulation of various geometric information took place, but also the methodology for proving geometric statements was worked out, and the first attempts were made to formulate the basic primary provisions (axioms) of geometry, from which many different geometric statements are derived by purely logical reasoning. The level of development of geometry in ancient Greece is reflected in the work of Euclid's "Beginnings".

In this book, for the first time, an attempt was made to give a systematic construction of planimetry on the basis of basic undefined geometric concepts and axioms (postulates).

A special place in the history of mathematics is occupied by the fifth postulate of Euclid (the axiom of parallel lines). For a long time, mathematicians unsuccessfully tried to derive the fifth postulate from the rest of Euclid's postulates, and only in the middle of the 19th century, thanks to the studies of N. I. Lobachevsky, B. Riemann and J. Boyai, it became clear that the fifth postulate cannot be derived from the rest, and the system of axioms, proposed by Euclid is not the only possible one.

Euclid's "Elements" had a huge impact on the development of mathematics. For more than two thousand years this book was not only a textbook on geometry, but also served as a starting point for many mathematical studies, as a result of which new independent branches of mathematics arose.

The systematic construction of geometry is usually carried out according to the following plan:

I. The main geometric concepts are listed, which are introduced without definitions.

II. A formulation of the axioms of geometry is given.

III. On the basis of axioms and basic geometric concepts, other geometric concepts and theorems are formulated.

1. Origin of the name Non-Euclidean geometry?
2. What shapes are called quadrilaterals?
3. Properties of a parallelogram?
4. Types of quadrilaterals?

List of sources used

1. A.G. Tsypkin. Handbook of Mathematics
2. “Unified state exam 2006. Mathematics. Educational and training materials for the preparation of students / Rosobrnadzor, ISOP - M .: Intellect-Center, 2006 "
3. Mazur K. I. "Solving the main competitive problems in mathematics of the collection edited by M. I. Scanavi"

Working on the lesson

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With four corners and four sides. A quadrilateral is formed by a closed polyline, consisting of four links, and that part of the plane that is inside the polyline.

The designation of a quadrilateral is made up of the letters at its vertices, naming them in order. For example, they say or write: quadrilateral ABCD :

In a quadrilateral ABCD points A, B, C and D- This quadrilateral vertices, segments AB, BC, CD and DA - sides.

Vertices that belong to the same side are called neighboring, vertices that are not adjacent are called opposite:

In a quadrilateral ABCD peaks A and B, B and C, C and D, D and A are adjacent, and the vertices A and C, B and D- opposite. Angles lying at adjacent vertices are also called neighboring, and at opposite vertices - opposite.

The sides of a quadrilateral can also be divided in pairs into adjacent and opposite sides: sides that have a common vertex are called neighboring(or related), sides that do not have common vertices - opposite:

Parties AB and BC, BC and CD, CD and DA, DA and AB are adjacent, and the sides AB and DC, AD and BC- opposite.

If opposite vertices are connected by a segment, then such a segment will be called the diagonal of the quadrilateral. Considering that there are only two pairs of opposite vertices in the quadrilateral, then there can be only two diagonals:

Segments AC and BD- diagonals.

Consider the main types of convex quadrilaterals:

• Trapeze- a quadrilateral in which one pair of opposite sides are parallel to each other, and the other pair is not parallel.
  • Isosceles trapezoid- a trapezoid whose sides are equal.
  • Rectangular trapezoid A trapezoid with one of the right angles.
• Parallelogram A quadrilateral in which both pairs of opposite sides are parallel to each other.
  • Rectangle A parallelogram in which all angles are equal.
  • Rhombus A parallelogram with all sides equal.
  • Square A parallelogram with equal sides and angles. Both a rectangle and a rhombus can be a square.

Corner properties of convex quadrilaterals

All convex quadrilaterals have the following two properties:

1. Any internal angle less than 180°.
2. The sum of the interior angles is 360°.

Lesson topic

• Definition of a quadrilateral.

Lesson Objectives

• Educational - repetition, generalization and testing of knowledge on the topic: “Quadrangles”; development of basic skills.
• Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
• Educational - through the lesson to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.

Lesson objectives

• To form skills in building a quadrilateral using a scale bar and a drawing triangle.
• Check students' ability to solve problems.

Lesson plan

1. History reference. Non-Euclidean geometry.
2. Quadrilateral.
3. Types of quadrilaterals.

Non-Euclidean geometry

Non-Euclidean geometry, geometry similar to geometry Euclid in that it defines the movement of figures, but differs from Euclidean geometry in that one of its five postulates (second or fifth) is replaced by its negation. The denial of one of the Euclidean postulates (1825) was a significant event in the history of thought, for it served as the first step towards theory of relativity.

Euclid's second postulate states that any line segment can be extended indefinitely. Euclid apparently believed that this postulate also contained the statement that the straight line has infinite length. However in "elliptic" geometry any straight line is finite and, like a circle, is closed.

The fifth postulate states that if a line intersects two given lines in such a way that the two interior angles on one side of it are less than two right angles in sum, then these two lines, if extended indefinitely, will intersect on the side where the sum of these angles is less than the sum two straight lines. But in "hyperbolic" geometry, there may exist a line CB (see Fig.), Perpendicular at point C to a given line r and intersecting another line s at an acute angle at point B, but, nevertheless, the infinite lines r and s will never intersect .

From these revised postulates it followed that the sum of the angles of a triangle, equal to 180° in Euclidean geometry, is greater than 180° in elliptic geometry and less than 180° in hyperbolic geometry.

Quadrilateral

Subjects > Mathematics > Mathematics Grade 8

A convex quadrilateral is a figure consisting of four sides connected to each other at the vertices, forming four angles together with the sides, while the quadrangle itself is always in the same plane relative to the straight line on which one of its sides lies. In other words, the entire figure is on one side of any of its sides.

In contact with

As you can see, the definition is quite easy to remember.

Basic properties and types

Almost all figures known to us, consisting of four corners and sides, can be attributed to convex quadrilaterals. The following can be distinguished:

1. parallelogram;
2. square;
3. rectangle;
4. trapezoid;
5. rhombus.

All these figures are united not only by the fact that they are quadrangular, but also by the fact that they are also convex. Just look at the diagram:

The figure shows a convex trapezoid. Here you can see that the trapezoid is on the same plane or on one side of the segment. If you carry out similar actions, you can find out that in the case of all other sides, the trapezoid is convex.

Is a parallelogram a convex quadrilateral?

Above is an image of a parallelogram. As can be seen from the figure, parallelogram is also convex. If you look at the figure with respect to the lines on which the segments AB, BC, CD and AD lie, it becomes clear that it is always on the same plane from these lines. The main features of a parallelogram are that its sides are pairwise parallel and equal in the same way as opposite angles are equal to each other.

Now, imagine a square or a rectangle. According to their main properties, they are also parallelograms, that is, all their sides are arranged in pairs in parallel. Only in the case of a rectangle, the length of the sides can be different, and the angles are right (equal to 90 degrees), a square is a rectangle in which all sides are equal and the angles are also right, while the lengths of the sides and angles of a parallelogram can be different.

As a result, the sum of all four corners of the quadrilateral must be equal to 360 degrees. The easiest way to determine this is by a rectangle: all four corners of the rectangle are right, that is, equal to 90 degrees. The sum of these 90-degree angles gives 360 degrees, in other words, if you add 90 degrees 4 times, you get the desired result.

Property of the diagonals of a convex quadrilateral

The diagonals of a convex quadrilateral intersect. Indeed, this phenomenon can be observed visually, just look at the figure:

The figure on the left shows a non-convex quadrilateral or quadrilateral. As you wish. As you can see, the diagonals do not intersect, at least not all of them. On the right is a convex quadrilateral. Here the property of diagonals to intersect is already observed. The same property can be considered a sign of the convexity of the quadrilateral.

Other properties and signs of convexity of a quadrilateral

Specifically, according to this term, it is very difficult to name any specific properties and features. It is easier to isolate according to different kinds of quadrilaterals of this type. You can start with a parallelogram. We already know that this is a quadrangular figure, the sides of which are pairwise parallel and equal. At the same time, the property of the diagonals of the parallelogram to intersect with each other, as well as the sign of the convexity of the figure itself, is also included here: the parallelogram is always in the same plane and on one side relative to any of its sides.

So, the main features and properties are known:

1. the sum of the angles of a quadrilateral is 360 degrees;
2. the diagonals of the figures intersect at one point.

Rectangle. This figure has all the same properties and features as a parallelogram, but all its angles are equal to 90 degrees. Hence the name, rectangle.

Square, the same parallelogram, but its corners are right, like a rectangle. Because of this, a square is rarely called a rectangle. But the main distinguishing feature of a square, in addition to those already listed above, is that all four of its sides are equal.

The trapezoid is a very interesting figure.. This is also a quadrilateral and also convex. In this article, the trapezoid has already been considered using the example of a drawing. It is clear that she is also convex. The main difference, and, accordingly, a sign of a trapezoid is that its sides can be absolutely not equal to each other in length, as well as its angles in value. In this case, the figure always remains on the same plane with respect to any of the straight lines that connect any two of its vertices along the segments forming the figure.

Rhombus is an equally interesting figure. Partly a rhombus can be considered a square. A sign of a rhombus is the fact that its diagonals not only intersect, but also divide the corners of the rhombus in half, and the diagonals themselves intersect at right angles, that is, they are perpendicular. If the lengths of the sides of the rhombus are equal, then the diagonals are also divided in half at the intersection.

Deltoids or convex rhomboids (rhombuses) may have different side lengths. But at the same time, both the main properties and features of the rhombus itself and the features and properties of convexity are still preserved. That is, we can observe that the diagonals bisect the corners and intersect at right angles.

Today's task was to consider and understand what convex quadrilaterals are, what they are and their main features and properties. Attention! It is worth recalling once again that the sum of the angles of a convex quadrilateral is 360 degrees. The perimeter of figures, for example, is equal to the sum of the lengths of all segments forming the figure. The formulas for calculating the perimeter and area of ​​quadrilaterals will be discussed in the following articles.

Types of convex quadrilaterals