A quadrilateral whose opposite sides are equal is called. Everything you need to know about the properties of quadrilaterals
A parallelogram is a quadrilateral whose opposite sides are pairwise parallel. B A C D AVIIDC, ADIIBC
How many parallelograms can be seen in the drawing? a d e c a II c, d II e II f II b II g f b g
Properties of a parallelogram 10. In a parallelogram, opposite sides are equal and opposite angles are equal. В 3 2 1 С , AD=BC B= D A= C
Properties of a parallelogram 20. The diagonals of a parallelogram are bisected by the point of intersection. Proof: B 2 4 A C 1 \u003d 2, as NLU at 3 D O AVIIDC and secant BD 3 \u003d 4, like NLU at ABIIDC and secant AC AB \u003d CD, as opposite sides of the parallelogram 1 ABO \u003d CDO along the side and two adjacent to her angles AO=OC, BO=OD
These figures illustrate all the considered properties B C B A D A B C O A C D D
Additional properties. The sum of adjacent angles of a parallelogram is 1800. B C D A ABIIDC, ADIIBC Justify ...
The perimeter of a parallelogram is 20 cm. Can one of the diagonals be 11 cm? cm 11 Semiperimeter B Ten centimeters C A D What is the largest integer value that the length of one of the diagonals of this parallelogram can take?
Training tasks on ready-made drawings. Find the sides of the parallelogram ABCD, knowing that its perimeter is 24 cm. AD - AB \u003d 3 cm B C The side AD is 3 cm larger than the side AB x A x + 3 D P \u003d 24 cm p \u003d 12 cm x + x + 3 \u003d 12
Find the sides of the parallelogram ABCD, knowing that its perimeter is 24 cm. 12
Find the sides of the parallelogram ABCD, knowing that its perimeter is 24 cm. MS - MB \u003d 3 cm B x M x + 3 450 A P \u003d 24 cm 2 (x + x + x + 3) = 24 segment MV C D p \u003d 12 cm x + x + x + 3 \u003d 12
The length of one side of a parallelogram is 80% of the length of the other side. Find the length of the smaller side of this parallelogram if its half-perimeter is 18 cm. B x C 0.8 x A D p \u003d 18 cm x + 0.8 x \u003d 18
The length of one side of a parallelogram is 15% longer than the length of the other side. Find the length of the longest side of this parallelogram if its semi-perimeter is 8.6 cm B 1.15 x C x A D p \u003d 8.6 cm x + 1.15 x \u003d 8.6
Find the angles of the parallelogram ABCD. B– B C x + 30 A x D A = 300 Angle B is 300 more than angle A
The sum of the degree measures of the three angles of the parallelogram is 3000. Find the value of the obtuse angle of this parallelogram. B C x A 180's D
Find the angles of the parallelogram ABCD (3600 - 400 2): 2 C B 1800 -400 140 A 400 D
No. 376 (c) Find the angles of the parallelogram ABCD if B 1090 A 710 C 710 1090 D
No. 376 (c) Find the angles of the parallelogram ABCD if B C x 2 x A A \u003d 2 B Angle A is 2 times the angle B D
In this article, we will cover all the main properties and signs of quadrilaterals.
To begin with, I will arrange all types of quadrilaterals in the form of such a summary diagram:
The scheme is remarkable in that the quadrangles in each row have ALL THE PROPERTIES OF THE QUADRANGLES LOCATED ABOVE THEM. So there is very little to remember.
Trapeze is a quadrilateral, two sides of which are parallel and the other two are not parallel. Parallel sides are called bases of a trapezoid, not parallel sides.
1
. in a trapeze the sum of the angles adjacent to the side equals 180°: A+B=180°, C+D=180°
2
. Bisector of any angle of a trapezoid cuts off on its base a segment equal to the lateral side: 
3. Bisectors of adjacent angles of a trapezoid intersect at right angles.
4 .trapezium is called isosceles if its sides are equal:
In an isosceles trapezoid
5. Area of a trapezoid is equal to the product of half the sum of the bases and the height:
parallelogram
is a quadrilateral whose opposite sides are pairwise parallel: 
In a parallelogram:
- opposite sides and opposite angles are equal
- the diagonals of a parallelogram are bisected by the point of intersection:
Accordingly, if a quadrilateral has these properties, then it is a parallelogram.
Parallelogram area is equal to the product of the base and the height:
or the product of the sides by the sine of the angle between them:
:
Rhombus is a parallelogram with all sides equal:
- opposite angles are equal
- the diagonals of the intersection point are bisected
- diagonals are mutually perpendicular
- the diagonals of a rhombus are the bisectors of the angles
Rhombus area is equal to half the product of the diagonals:
or the product of the square of a side and the sine of the angle between the sides:
A quadrilateral is a polygon consisting of four points (vertices) and four segments (sides) connecting these points in pairs.
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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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 a 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
- Historical reference. Non-Euclidean geometry.
- Quadrilateral.
- 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 so 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.
