Parallelogram all formulas and properties. Research project "parallelogram and its properties"

The concept of a parallelogram

Definition 1

Parallelogram is a quadrilateral in which opposite sides are parallel to each other (Fig. 1).

Picture 1.

A parallelogram has two main properties. Let's consider them without proof.

Property 1: Opposite sides and angles of a parallelogram are equal, respectively, to each other.

Property 2: Diagonals drawn in a parallelogram are bisected by their intersection point.

Parallelogram features

Consider three features of a parallelogram and present them in the form of theorems.

Theorem 1

If two sides of a quadrilateral are equal to each other and also parallel, then this quadrilateral will be a parallelogram.

Proof.

Let us be given a quadrilateral $ABCD$. In which $AB||CD$ and $AB=CD$ Let us draw a diagonal $AC$ in it (Fig. 2).

Figure 2.

Consider parallel lines $AB$ and $CD$ and their secant $AC$. Then

\[\angle CAB=\angle DCA\]

like crosswise corners.

According to the $I$ criterion for the equality of triangles,

since $AC$ is their common side, and $AB=CD$ by assumption. Means

\[\angle DAC=\angle ACB\]

Consider the lines $AD$ and $CB$ and their secant $AC$; by the last equality of the cross-lying angles, we obtain that $AD||CB$.) Therefore, by the definition of $1$, this quadrilateral is a parallelogram.

The theorem has been proven.

Theorem 2

If opposite sides of a quadrilateral are equal, then it is a parallelogram.

Proof.

Let us be given a quadrilateral $ABCD$. In which $AD=BC$ and $AB=CD$. Let us draw a diagonal $AC$ in it (Fig. 3).

Figure 3

Since $AD=BC$, $AB=CD$, and $AC$ is a common side, then by the $III$ triangle equality test,

\[\triangle DAC=\triangle ACB\]

\[\angle DAC=\angle ACB\]

Consider the lines $AD$ and $CB$ and their secant $AC$, by the last equality of the cross-lying angles we get that $AD||CB$. Therefore, by the definition of $1$, this quadrilateral is a parallelogram.

\[\angle DCA=\angle CAB\]

Consider the lines $AB$ and $CD$ and their secant $AC$, by the last equality of the cross-lying angles we get that $AB||CD$. Therefore, by Definition 1, this quadrilateral is a parallelogram.

The theorem has been proven.

Theorem 3

If the diagonals drawn in a quadrilateral are divided into two equal parts by their intersection point, then this quadrilateral is a parallelogram.

Proof.

Let us be given a quadrilateral $ABCD$. Let us draw the diagonals $AC$ and $BD$ in it. Let them intersect at the point $O$ (Fig. 4).

Figure 4

Since, by the condition $BO=OD,\ AO=OC$, and the angles $\angle COB=\angle DOA$ are vertical, then, by the $I$ triangle equality test,

\[\triangle BOC=\triangle AOD\]

\[\angle DBC=\angle BDA\]

Consider the lines $BC$ and $AD$ and their secant $BD$, by the last equality of the cross-lying angles we get that $BC||AD$. Also $BC=AD$. Therefore, by Theorem $1$, this quadrilateral is a parallelogram.

1. Definition of a parallelogram.

If we intersect a pair of parallel lines with another pair of parallel lines, we get a quadrilateral whose opposite sides are pairwise parallel.

In the quadrilaterals ABDC and EFNM (Fig. 224) BD || AC and AB || CD;

EF || MN and EM || F.N.

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

2. Properties of a parallelogram.

Theorem. The diagonal of a parallelogram divides it into two equal triangle.

Let there be a parallelogram ABDC (Fig. 225) in which AB || CD and AC || BD.

It is required to prove that the diagonal divides it into two equal triangles.

Let's draw a diagonal CB in the parallelogram ABDC. Let us prove that $\Delta$CAB = $\Delta$СDВ.

The NE side is common to these triangles; ∠ABC = ∠BCD, as internal cross lying angles with parallel AB and CD and secant CB; ∠ACB = ∠CBD, same as internal cross lying angles with parallel AC and BD and secant CB.

Hence $\Delta$CAB = $\Delta$СDВ.

In the same way, one can prove that the diagonal AD divides the parallelogram into two equal triangles ACD and ABD.

Consequences:

1 . Opposite angles of a parallelogram are equal.

∠A = ∠D, this follows from the equality of triangles CAB and CDB.

Similarly, ∠C = ∠B.

2. Opposite sides of a parallelogram are equal.

AB \u003d CD and AC \u003d BD, since these are sides of equal triangles and lie opposite equal angles.

Theorem 2. The diagonals of a parallelogram are bisected at the point of their intersection.

Let BC and AD be the diagonals of the parallelogram ABDC (Fig. 226). Let us prove that AO = OD and CO = OB.

To do this, let's compare some pair of opposite triangles, for example $\Delta$AOB and $\Delta$COD.

In these triangles AB = CD, as opposite sides of a parallelogram;

∠1 = ∠2, as interior angles crosswise lying at parallel AB and CD and secant AD;

∠3 = ∠4 for the same reason, since AB || CD and CB are their secant.

It follows that $\Delta$AOB = $\Delta$COD. And in equal triangles, opposite equal angles are equal sides. Therefore, AO = OD and CO = OB.

Theorem 3. The sum of the angles adjacent to one side of the parallelogram is equal to 180°.

Draw a diagonal AC in parallelogram ABCD and get two triangles ABC and ADC.

The triangles are congruent because ∠1 = ∠4, ∠2 = ∠3 (cross-lying angles at parallel lines), and side AC is common.
The equality $\Delta$ABC = $\Delta$ADC implies that AB = CD, BC = AD, ∠B = ∠D.

The sum of the angles adjacent to one side, for example, angles A and D, is equal to 180 ° as one-sided with parallel lines.

A parallelogram is a quadrilateral whose opposite sides are pairwise parallel. The following figure shows parallelogram ABCD. It has side AB parallel to side CD and side BC parallel to side AD.

As you may have guessed, a parallelogram is a convex quadrilateral. Consider the basic properties of a parallelogram.

Parallelogram Properties

1. In a parallelogram opposite corners and opposite sides are equal. Let's prove this property - consider the parallelogram shown in the following figure.

Diagonal BD divides it into two equal triangles: ABD and CBD. They are equal in side BD and two angles adjacent to it, since the angles lying at the secant of BD are parallel lines BC and AD and AB and CD, respectively. Therefore, AB = CD and
BC=AD. And from the equality of angles 1, 2,3 and 4 it follows that angle A = angle1 + angle3 = angle2 + angle4 = angle C.

2. The diagonals of the parallelogram are bisected by the intersection point. Let the point O be the point of intersection of the diagonals AC and BD of the parallelogram ABCD.

Then the triangle AOB and the triangle COD are equal to each other, along the side and two angles adjacent to it. (AB=CD since they are opposite sides of the parallelogram. And angle1 = angle2 and angle3 = angle4 as cross-lying angles at the intersection of lines AB and CD by secants AC and BD, respectively.) It follows that AO = OC and OB = OD, which and needed to be proven.

All main properties are illustrated in the following three figures.

A parallelogram is a quadrilateral whose opposite sides are pairwise parallel. This definition is already sufficient, since the remaining properties of a parallelogram follow from it and are proved in the form of theorems.

The main properties of a parallelogram are:

a parallelogram is a convex quadrilateral;
a parallelogram has opposite sides equal in pairs;
a parallelogram has opposite angles that are equal in pairs;
the diagonals of a parallelogram are bisected by the point of intersection.

Parallelogram - a convex quadrilateral

Let us first prove the theorem that a parallelogram is a convex quadrilateral. A polygon is convex when whatever side of it is extended to a straight line, all other sides of the polygon will be on the same side of this straight line.

Let a parallelogram ABCD be given, in which AB is the opposite side for CD, and BC is the opposite side for AD. Then it follows from the definition of a parallelogram that AB || CD, BC || AD.

Parallel segments do not have common points, they do not intersect. This means that CD lies on one side of AB. Since segment BC connects point B of segment AB with point C of segment CD, and segment AD connects other points AB and CD, segments BC and AD also lie on the same side of line AB, where CD lies. Thus, all three sides - CD, BC, AD - lie on the same side of AB.

Similarly, it is proved that with respect to the other sides of the parallelogram, the other three sides lie on the same side.

Opposite sides and angles are equal

One of the properties of a parallelogram is that in a parallelogram opposite sides and opposite angles are equal. For example, if a parallelogram ABCD is given, then it has AB = CD, AD = BC, ∠A = ∠C, ∠B = ∠D. This theorem is proved as follows.

A parallelogram is a quadrilateral. So it has two diagonals. Since a parallelogram is a convex quadrilateral, any of them divides it into two triangles. Consider the triangles ABC and ADC in the parallelogram ABCD obtained by drawing the diagonal AC.

These triangles have one side in common - AC. The angle BCA is equal to the angle CAD, as are the verticals with parallel BC and AD. Angles BAC and ACD are also equal, as are the vertical angles when AB and CD are parallel. Therefore, ∆ABC = ∆ADC over two angles and the side between them.

In these triangles, side AB corresponds to side CD, and side BC corresponds to AD. Therefore, AB = CD and BC = AD.

Angle B corresponds to angle D, i.e. ∠B = ∠D. Angle A of a parallelogram is the sum of two angles - ∠BAC and ∠CAD. The angle C equals consists of ∠BCA and ∠ACD. Since the pairs of angles are equal to each other, then ∠A = ∠C.

Thus, it is proved that in a parallelogram opposite sides and angles are equal.

Diagonals cut in half

Since a parallelogram is a convex quadrilateral, it has two two diagonals, and they intersect. Let a parallelogram ABCD be given, its diagonals AC and BD intersect at a point E. Consider the triangles ABE and CDE formed by them.

These triangles have sides AB and CD equal as opposite sides of a parallelogram. The angle ABE is equal to the angle CDE as they lie across parallel lines AB and CD. For the same reason, ∠BAE = ∠DCE. Hence, ∆ABE = ∆CDE over two angles and the side between them.

You can also notice that the angles AEB and CED are vertical, and therefore also equal to each other.

Since triangles ABE and CDE are equal to each other, so are all their corresponding elements. Side AE of the first triangle corresponds to side CE of the second, so AE = CE. Similarly, BE = DE. Each pair of equal segments makes up the diagonal of the parallelogram. Thus, it is proved that the diagonals of a parallelogram are bisected by the point of intersection.

In today's lesson, we will repeat the main properties of a parallelogram, and then we will pay attention to the consideration of the first two features of a parallelogram and prove them. In the course of the proof, let us recall the application of the signs of equality of triangles, which we studied last year and repeated in the first lesson. At the end, an example will be given on the application of the studied features of a parallelogram.

Theme: Quadrangles

Lesson: Signs of a parallelogram

Let's start by recalling the definition of a parallelogram.

Definition. Parallelogram- a quadrilateral in which every two opposite sides are parallel (see Fig. 1).

Rice. 1. Parallelogram

Let's remember basic properties of a parallelogram:

In order to be able to use all these properties, it is necessary to be sure that the figure about which in question, is a parallelogram. To do this, you need to know such facts as the signs of a parallelogram. We will consider the first two of them today.

Theorem. The first feature of a parallelogram. If in a quadrilateral two opposite sides are equal and parallel, then this quadrilateral is parallelogram. .

Rice. 2. The first sign of a parallelogram

Proof. Let's draw a diagonal in the quadrilateral (see Fig. 2), she divided it into two triangles. Let's write down what we know about these triangles:

according to the first sign of equality of triangles.

From the equality of these triangles it follows that, on the basis of the parallelism of the lines at the intersection of their secant. We have that:

Proven.

Theorem. The second sign of a parallelogram. If in a quadrilateral every two opposite sides are equal, then this quadrilateral is parallelogram. .

Rice. 3. The second sign of a parallelogram

Proof. Let's draw a diagonal in the quadrilateral (see Fig. 3), it divides it into two triangles. Let's write down what we know about these triangles, based on the formulation of the theorem:

according to the third criterion for the equality of triangles.