Just as in Euclidean geometry, a point and a straight line are the main elements of the theory of planes, so a parallelogram is one of the key figures of convex quadrilaterals. From it, like threads from a ball, flow the concepts of “rectangle”, “square”, “rhombus” and other geometric quantities.
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Definition of parallelogram
convex quadrilateral, consisting of segments, each pair of which is parallel, is known in geometry as a parallelogram.
What a classic parallelogram looks like is depicted by a quadrilateral ABCD. The sides are called bases (AB, BC, CD and AD), the perpendicular drawn from any vertex to the side opposite to this vertex is called height (BE and BF), lines AC and BD are called diagonals.
Attention! Square, rhombus and rectangle are special cases of parallelogram.
Sides and angles: features of the relationship
Key properties, by and large, predetermined by the designation itself, they are proved by the theorem. These characteristics are as follows:
- The sides that are opposite are identical in pairs.
- Angles opposite each other are equal in pairs.
Proof: Consider ∆ABC and ∆ADC, which are obtained by dividing the quadrilateral ABCD with the straight line AC. ∠BCA=∠CAD and ∠BAC=∠ACD, since AC is common for them (vertical angles for BC||AD and AB||CD, respectively). It follows from this: ∆ABC = ∆ADC (the second sign of equality of triangles).
The segments AB and BC in ∆ABC correspond in pairs to the lines CD and AD in ∆ADC, which means that they are identical: AB = CD, BC = AD. Thus, ∠B corresponds to ∠D and they are equal. Since ∠A=∠BAC+∠CAD, ∠C=∠BCA+∠ACD, which are also pairwise identical, then ∠A = ∠C. The property has been proven.
Characteristics of the diagonals of a figure
Main feature of these lines of a parallelogram: the point of intersection divides them in half.
Proof: Let i.e. be the intersection point of diagonals AC and BD of figure ABCD. They form two commensurate triangles - ∆ABE and ∆CDE.
AB=CD since they are opposites. According to the lines and the secant, ∠ABE = ∠CDE and ∠BAE = ∠DCE.
By the second criterion of equality, ∆ABE = ∆CDE. This means that the elements ∆ABE and ∆CDE: AE = CE, BE = DE and at the same time they are proportional parts of AC and BD. The property has been proven.
Features of adjacent corners
Adjacent sides have a sum of angles equal to 180°, since they lie on the same side of parallel lines and a transversal. For quadrilateral ABCD:
∠A+∠B=∠C+∠D=∠A+∠D=∠B+∠C=180º
Properties of the bisector:
- , lowered to one side, are perpendicular;
- opposite vertices have parallel bisectors;
- the triangle obtained by drawing a bisector will be isosceles.
Determination of the characteristic features of a parallelogram using the theorem
The characteristics of this figure follow from its main theorem, which states the following: a quadrilateral is considered a parallelogram in the event that its diagonals intersect, and this point divides them into equal segments.
Proof: let the lines AC and BD of the quadrilateral ABCD intersect at i.e. Since ∠AED = ∠BEC, and AE+CE=AC BE+DE=BD, then ∆AED = ∆BEC (by the first criterion for the equality of triangles). That is, ∠EAD = ∠ECB. They are also the internal cross angles of the secant AC for lines AD and BC. Thus, by definition of parallelism - AD || B.C. A similar property of lines BC and CD is also derived. The theorem has been proven.
Calculating the area of a figure
Area of this figure found by several methods one of the simplest: multiplying the height and the base to which it is drawn.
Proof: draw perpendiculars BE and CF from vertices B and C. ∆ABE and ∆DCF are equal, since AB = CD and BE = CF. ABCD is equal in size to the rectangle EBCF, since they consist of commensurate figures: S ABE and S EBCD, as well as S DCF and S EBCD. It follows from this that the area of this geometric figure is the same as that of a rectangle:
S ABCD = S EBCF = BE×BC=BE×AD.
For determining general formula The area of the parallelogram is denoted by the height as hb, and the side - b. Respectively:
Other ways to find area
Area calculations through the sides of the parallelogram and the angle, which they form, is the second known method.
,
Spr-ma - area;
a and b are its sides
α is the angle between segments a and b.
This method is practically based on the first, but in case it is unknown. always cuts off right triangle, whose parameters are trigonometric identities, that is . Transforming the relation, we get . In the equation of the first method, we replace the height with this product and obtain a proof of the validity of this formula.
Through the diagonals of a parallelogram and the angle, which they create when they intersect, you can also find the area.
Proof: AC and BD intersect to form four triangles: ABE, BEC, CDE and AED. Their sum is equal to the area of this quadrilateral.
The area of each of these ∆ can be found by the expression , where a=BE, b=AE, ∠γ =∠AEB. Since , then in the calculations it is used single meaning sinus That is . Since AE+CE=AC= d 1 and BE+DE=BD= d 2, the area formula reduces to:
.
Application in vector algebra
The features of the constituent parts of this quadrilateral have found application in vector algebra, namely the addition of two vectors. The parallelogram rule states that if given vectorsAndNotare collinear, then their sum will be equal to the diagonal of this figure, the bases of which correspond to these vectors.
Proof: from an arbitrarily chosen beginning - i.e. - construct vectors and . Next, we construct a parallelogram OASV, where the segments OA and OB are sides. Thus, the OS lies on the vector or sum.
Formulas for calculating the parameters of a parallelogram
The identities are given under the following conditions:
- a and b, α - sides and the angle between them;
- d 1 and d 2, γ - diagonals and at the point of their intersection;
- h a and h b - heights lowered to sides a and b;
| Parameter | Formula |
| Finding the sides | |
| along the diagonals and the cosine of the angle between them | 
|
| along diagonals and sides | 
|
| through the height and the opposite vertex | |
| Finding the length of diagonals | |
| on the sides and the size of the apex between them | |
A parallelogram is a quadrangular figure whose opposite sides are parallel and equal in pairs. Its opposite angles are also equal, and the point of intersection of the diagonals of the parallelogram divides them in half, being the center of symmetry of the figure. Special cases of a parallelogram are: geometric figures like square, rectangle and rhombus. The area of a parallelogram can be found different ways, depending on what initial data accompanies the problem statement.
The key characteristic of a parallelogram, very often used when finding its area, is its height. The height of a parallelogram is usually called a perpendicular drawn from an arbitrary point on the opposite side to a straight segment forming that side.
- In the simplest case, the area of a parallelogram is defined as the product of its base and its height.
S = DC ∙ h
where S is the area of the parallelogram;
a - base;
h is the height drawn to the given base.This formula is very easy to understand and remember if you look at the following figure.
As you can see from this image, if we cut off an imaginary triangle to the left of the parallelogram and attach it to the right, the result will be a rectangle. As you know, the area of a rectangle is found by multiplying its length by its height. Only in the case of a parallelogram will the length be the base, and the height of the rectangle will be the height of the parallelogram lowered to a given side.
- The area of a parallelogram can also be found by multiplying the lengths of two adjacent bases and the sine of the angle between them:
S = AD∙AB∙sinα
where AD, AB are adjacent bases forming an intersection point and an angle a between themselves;
α is the angle between the bases AD and AB.
- You can also find the area of a parallelogram by dividing in half the product of the lengths of the diagonals of the parallelogram by the sine of the angle between them.
S = ½∙AC∙BD∙sinβ
where AC, BD are the diagonals of the parallelogram;
β is the angle between the diagonals.
- There is also a formula for finding the area of a parallelogram through the radius of the circle inscribed in it. It is written as follows:
Parallelogram is a quadrilateral whose sides are parallel in pairs.
In this figure, opposite sides and angles are equal to each other. The diagonals of a parallelogram intersect at one point and bisect it. Formulas for the area of a parallelogram allow you to find the value using the sides, height and diagonals. A parallelogram can also be presented in special cases. They are considered a rectangle, square and rhombus.
First, let's look at an example of calculating the area of a parallelogram by height and the side to which it is lowered.
This case is considered classic and does not require additional investigation. It’s better to consider the formula for calculating the area through two sides and the angle between them. The same method is used in calculations. If the sides and the angle between them are given, then the area is calculated as follows: 
Suppose we are given a parallelogram with sides a = 4 cm, b = 6 cm. The angle between them is α = 30°. Let's find the area: 
Area of a parallelogram through diagonals
The formula for the area of a parallelogram using the diagonals allows you to quickly find the value.
For calculations, you will need the size of the angle located between the diagonals.
Let's consider an example of calculating the area of a parallelogram using diagonals. Let a parallelogram be given with diagonals D = 7 cm, d = 5 cm. The angle between them is α = 30°. Let's substitute the data into the formula: 
An example of calculating the area of a parallelogram through the diagonal gave us an excellent result - 8.75.
Knowing the formula for the area of a parallelogram through the diagonal, you can solve the set interesting tasks. Let's look at one of them.
Task: Given a parallelogram with an area of 92 square meters. see Point F is located in the middle of its side BC. Let's find the area of the trapezoid ADFB, which will lie in our parallelogram. First, let's draw everything we received according to the conditions.
Let's get to the solution: 
According to our conditions, ah =92, and accordingly, the area of our trapezoid will be equal to 
Parallelogram called a quadrilateral whose opposite sides are parallel to each other. The main tasks in school on this topic are to calculate the area of a parallelogram, its perimeter, height, and diagonals. The indicated values and formulas for their calculation will be given below.
Properties of a parallelogram
The opposite sides of a parallelogram, as well as the opposite angles, are equal to each other:
AB=CD, BC=AD,
The diagonals of a parallelogram at the point of intersection are divided into two equal parts:
AO=OC, OB=OD.
Angles adjacent to any side (adjacent angles) add up to 180 degrees.
Each of the diagonals of a parallelogram divides it into two triangles of equal area and geometric dimensions.
Another remarkable property that is often used when solving problems is that the sum of the squares of the diagonals in a parallelogram is equal to the sum of the squares of all sides:
AC^2+BD^2=2*(AB^2+BC^2) . 
The main features of parallelograms:
1. A quadrilateral whose opposite sides are parallel in pairs is a parallelogram.
2. A quadrilateral with equal opposite sides is a parallelogram.
3. A quadrilateral with equal and parallel opposite sides is a parallelogram.
4. If the diagonals of a quadrilateral at the intersection point are divided in half, then it is a parallelogram.
5. A quadrilateral whose opposite angles are equal in pairs is a parallelogram
Bisectors of a parallelogram
The bisectors of opposite angles in a parallelogram can be parallel or coincident.
Bisectors of adjacent angles (adjacent to one side) intersect at right angles (perpendicular).
Parallelogram height
Parallelogram height- this is a segment drawn from an angle perpendicular to the base. It follows from this that two heights can be drawn from each angle.
Parallelogram area formula
Area of a parallelogram is equal to the product of the side and the height drawn to it. The area formula is as follows 
The second formula is no less popular in calculations and is defined as follows: the area of a parallelogram is equal to the product of adjacent sides and the sine of the angle between them
Based on the above formulas, you will know how to calculate the area of a parallelogram.
Perimeter of a parallelogram
The formula for calculating the perimeter of a parallelogram is
that is, the perimeter is equal to twice the sum of the sides. Problems involving parallelograms will be discussed in adjacent materials, but for now, study the formulas. Most problems in calculating the sides and diagonals of a parallelogram are quite simple and boil down to knowledge of the theorem of sines and the Pythagorean theorem.
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