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 the rest of the characteristics and properties of this figure, we will consider 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 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 the same line.

Types of quadrilaterals

• A quadrilateral whose opposite sides are pairwise parallel 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.

The 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 always equals 360 degrees.

Special types of quadrilaterals

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

• Parallelogram
• Rectangle
• Square
• Trapeze
• Deltoid
• Counterparallelogram

Quadrilateral and circle

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

The main property of the circumscribed 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 inscribed around a quadrilateral)

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 180 degrees.

Quadrilateral side length properties

Difference modulus of 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 figure is provided solely for ease of understanding.

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 a 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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1 . The sum of the diagonals of a convex quadrilateral is greater than the sum of its two opposite sides.

2 . If the segments connecting the midpoints of opposite sides quadrilateral

a) are equal, then the diagonals of the quadrilateral are perpendicular;

b) are perpendicular, then the diagonals of the quadrilateral are equal.

3 . The bisectors of the angles at the lateral side of the trapezium intersect at its midline.

4 . The sides of the parallelogram are equal and . Then the quadrilateral formed by the intersections of the bisectors of the angles of the parallelogram is a rectangle whose diagonals are equal.

5 . If the sum of the angles at one of the bases of the trapezoid is 90°, then the segment connecting the midpoints of the bases of the trapezoid is equal to their half-difference.

6 . On the sides AB and AD parallelogram ABCD points are taken M and N so that straight MS and NC Divide the parallelogram into three equal parts. Find MN, if BD=d.

7 . A segment of a straight line parallel to the bases of a trapezoid, enclosed inside the trapezoid, is divided by its diagonals into three parts. Then the segments adjacent to the sides are equal to each other.

8 . Through the point of intersection of the diagonals of the trapezoid with the bases and a straight line is drawn, parallel to the bases. The segment of this line, enclosed between the sides of the trapezoid, is equal to.

9 . A trapezoid is divided by a line parallel to its bases equal to and , into two equal trapezoids. Then the segment of this straight line, enclosed between the sides, is equal to .

10 . If one of the following conditions is met, then four points A, B, C and D lie on the same circle.

a) CAD=CBD= 90°.

b) points BUT and AT lie on one side of a straight line CD and angle CAD equal to the angle CBD

c) straight AC and BD intersect at a point O and O A OS=OV OD.

11 . A line connecting a point R intersections of the diagonals of a quadrilateral ABCD with dot Q line intersections AB and CD, divides the side AD in half. Then she bisects and a side Sun.

12 . Each side of a convex quadrilateral is divided into three equal parts. Corresponding division points on opposite sides are connected by segments. Then these segments divide each other into three equal parts.

13 . Two straight lines divide each of the two opposite sides of a convex quadrilateral into three equal parts. Then between these lines lies one third of the area of ​​the quadrilateral.

14 . If a circle can be inscribed in a quadrilateral, then the segment connecting the points at which the inscribed circle touches opposite sides of the quadrilateral passes through the intersection point of the diagonals.

15 . If the sums of opposite sides of a quadrilateral are equal, then a circle can be inscribed in such a quadrilateral.

16. Properties of an inscribed quadrilateral with mutually perpendicular diagonals. Quadrilateral ABCD inscribed in a circle of radius R. Its diagonals AC and BD are mutually perpendicular and intersect at a point R. Then

a) the median of a triangle ARV perpendicular to the side CD;

b) broken line AOC divides the quadrilateral ABCD into two equal figures;

in) AB 2 + CD 2=4R 2 ;

G) AP 2 + BP 2 + SR 2 + DP 2 = 4R 2 and AB 2 + BC 2 + CD 2 + AD 2 = 8R 2;

e) the distance from the center of the circle to the side of the quadrilateral is half the opposite side.

f) if the perpendiculars dropped to the side AD from the peaks AT and WITH, cross diagonals AC and BD at points E and F, then BCFE- rhombus;

g) a quadrilateral whose vertices are projections of a point R on the side of the quadrilateral ABCD,- both inscribed and described;

h) a quadrilateral formed by tangents to the circumscribed circle of the quadrilateral ABCD, drawn at its vertices can be inscribed in a circle.

17 . If a a, b, c, d- successive sides of a quadrilateral, S- its area, then, and equality takes place only for an inscribed quadrilateral, the diagonals of which are mutually perpendicular.

18 . Brahmagupta formula. If the sides of the inscribed quadrilateral are equal a, b, c and d, then its area S can be calculated by the formula,

where is the semiperimeter of the quadrilateral.

19 . If a quadrilateral with sides a, b, c, d can be inscribed and a circle can be circumscribed around it, then its area is equal to .

20 . Point P is located inside the square ABCD, and the angle PAB equal to the angle RVA and is equal to 15°. Then the triangle DPC- equilateral.

21 . If for an inscribed quadrilateral ABCD equality CD=AD+BC, then the bisectors of its angles BUT and AT intersect on the side CD.

22 . Continuations of opposite sides AB and CD inscribed quadrilateral ABCD intersect at a point M, and the sides AD and sun- at the point N. Then

a) angle bisectors AMD and DNC mutually perpendicular;

b) straight MQ and NQ intersect the sides of the quadrilateral at the vertices of the rhombus;

c) point of intersection Q of these bisectors lies on the segment connecting the midpoints of the diagonals of the quadrilateral ABCD.

23 . Ptolemy's theorem. The sum of the products of two pairs of opposite sides of an inscribed quadrilateral is equal to the product of its diagonals.

24 . Newton's theorem. In any circumscribed quadrilateral, the midpoints of the diagonals and the center of the inscribed circle lie on the same straight line.

25 . Monge's theorem. Lines drawn through the midpoints of the sides of an inscribed quadrilateral perpendicular to opposite sides intersect at one point.

27 . Four circles, built on the sides of a convex quadrilateral as diameters, cover the entire quadrilateral.

29 . Two opposite corners of a convex quadrilateral are obtuse. Then the diagonal connecting the vertices of these angles is less than the other diagonal.

30. The centers of squares built on the sides of a parallelogram outside it form a square themselves.

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°.

In the school curriculum in geometry lessons, one has to deal with various types of quadrilaterals: rhombuses, parallelograms, rectangles, trapezoids, squares. The very first shapes to study are a rectangle and a square.

So what is a rectangle? The definition for the 2nd grade of a comprehensive school will look like this: this is a quadrilateral, in which all four corners are right. It is easy to imagine what a rectangle looks like: it is a figure with 4 right angles and sides parallel to each other in pairs.

In contact with

How to understand, solving the next geometric problem, what kind of quadrilateral are we dealing with? There are three main features, by which you can accurately determine that we are talking about a rectangle. Let's call them:

• the figure is a quadrilateral with three angles equal to 90°;
• the presented quadrilateral is a parallelogram with equal diagonals;
• a parallelogram that has at least one right angle.

It is interesting to know: what is convex, its features and signs.

Since a rectangle is a parallelogram (that is, a quadrilateral with pairwise parallel opposite sides), then all its properties and features will be fulfilled for it.

Formulas for calculating the length of the sides

in a rectangle opposite sides are equal and mutually parallel. The longer side is usually called the length (denoted by a), the shorter side is called the width (denoted by b). In the rectangle in the image, the lengths are sides AB and CD, and the widths are AC and B.D. They are also perpendicular to the bases (i.e., they are heights).

To find the sides, you can use the formulas below. Conventions are adopted in them: a - the length of the rectangle, b - its width, d - the diagonal (the segment connecting the vertices of two angles lying opposite each other), S - the area of ​​​​the figure, P - the perimeter, α - the angle between the diagonal and the length, β is an acute angle formed by both diagonals. Ways to find the lengths of the sides:

• Using the diagonal and the known side: a \u003d √ (d ² - b ²), b \u003d √ (d ² - a ²).
• By the area of ​​the figure and one of its sides: a = S / b, b = S / a.
• Using the perimeter and the known side: a = (P - 2 b) / 2, b = (P - 2 a) / 2.
• Through the diagonal and the angle between it and the length: a = d sinα, b = d cosα.
• Through the diagonal and the angle β: a = d sin 0.5 β, b = d cos 0.5 β.

Perimeter and area

The perimeter of a quadrilateral is called the sum of the lengths of all its sides. To calculate the perimeter, the following formulas can be used:

• Through both sides: P = 2 (a + b).
• Through the area and one of the sides: P \u003d (2S + 2a ²) / a, P \u003d (2S + 2b ²) / b.

An area is a space bounded by a perimeter. Three main ways to calculate the area:

• Through the lengths of both sides: S = a*b.
• Using the perimeter and any one known side: S \u003d (Pa - 2 a ²) / 2; S = (Pb - 2b²) / 2.
• Diagonally and angle β: S = 0.5 d² sinβ.

In the tasks of a school mathematics course, it is often required to have a good command of properties of the diagonals of a rectangle. We list the main ones:

1. The diagonals are equal to each other and are divided into two equal segments at the point of their intersection.
2. The diagonal is defined as the root of the sum of both sides squared (follows from the Pythagorean theorem).
3. The diagonal divides the rectangle into two triangles with a right angle.
4. The point of intersection coincides with the center of the circumscribed circle, and the diagonals themselves coincide with its diameter.

The following formulas are used to calculate the length of the diagonal:

• Using the length and width of the figure: d = √ (a ² + b ²).
• Using the radius of a circle circumscribed around a quadrilateral: d = 2 R.

Definition and properties of a square

A square is a special case of a rhombus, parallelogram, or rectangle. Its difference from these figures is that all its angles are right, and all four sides are equal. A square is a regular quadrilateral.

A quadrilateral is called a square in the following cases:

1. If it is a rectangle whose length a and width b are equal.
2. If it is a rhombus with equal length diagonals and four right angles.

The properties of a square include all the previously discussed properties related to a rectangle, as well as the following:

1. Diagonals are perpendicular to each other (property of a rhombus).
2. The point of intersection coincides with the center of the inscribed circle.
3. Both diagonals divide the quadrilateral into four identical right-angled and isosceles triangles.

Here are some frequently used formulas for calculating the perimeter, area and elements of a square:

• Diagonal d = a √2.
• Perimeter P = 4 a.
• Area S = a².
• The radius of the circumscribed circle is half the diagonal: R = 0.5 a √2.
• The radius of an inscribed circle is defined as half the side length: r = a / 2.

Sample questions and tasks

Let's analyze some of the questions that you may encounter when studying mathematics at school, and solve a few simple problems.

Task 1. How will the area of ​​a rectangle change if the length of its sides is tripled?

Decision : Let's denote the area of ​​the original figure as S0, and the area of ​​the quadrilateral with triple the length of the sides - S1. According to the formula considered earlier, we obtain: S0 = ab. Now let's increase the length and width by 3 times and write: S1= 3 a 3 b = 9 ab. Comparing S0 and S1, it becomes obvious that the second area is 9 times larger than the first.

Question 1. Is a quadrilateral with right angles a square?

Decision : It follows from the definition that a figure with right angles is a square only if the lengths of all its sides are equal. Otherwise, the figure is a rectangle.

Task 2. The diagonals of a rectangle form an angle of 60 degrees. The width of the rectangle is 8. Calculate what the diagonal is.

Decision: Recall that the diagonals are bisected by the intersection point. Thus, we are dealing with an isosceles triangle with an angle at the vertex equal to 60°. Since the triangle is isosceles, the angles at the base will also be the same. By simple calculations, we get that each of them is equal to 60 °. It follows that the triangle is equilateral. The width we know is the base of the triangle, so half of the diagonal is also 8, and the length of the whole diagonal is twice that and equal to 16.

Question 2. Does a rectangle have all sides equal or not?

Decision : Suffice it to recall that all sides must be equal for a square, which is a special case of a rectangle. In all other cases, a sufficient condition is the presence of at least 3 right angles. The equality of the parties is not a mandatory feature.

Task 3. The area of ​​the square is known and equal to 289. Find the radii of the inscribed and circumscribed circles.

Decision : According to the formulas for the square, we will carry out the following calculations:

• Let's determine what the main elements of the square are equal to: a = √ S = √289 = 17; d = a √2 =1 7√2.
• Let's calculate what the radius of the circle described around the quadrilateral is equal to: R = 0.5 d = 8.5√2.
• Let's find the radius of the inscribed circle: r = a / 2 = 17 / 2 = 8.5.

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.

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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:

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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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