Parallelogram quadrilateral then opposite sides are equal. Parallelogram and its properties. The area of a parallelogram. Angle bisectors of a parallelogram
It is a quadrilateral whose opposite sides are pairwise parallel.
Property 1 . Any diagonal of a parallelogram divides it into two equal triangles.
Proof . According to the II sign (cross-lying corners and a common side).
Theorem proven.
Property 2 . In a parallelogram, opposite sides are equal and opposite angles are equal.
Proof .
Likewise,
Theorem proven.
Property 3. In a diagonal parallelogram, the intersection point is divided in half.
Proof .
Theorem proven.
Property 4 . The angle bisector of a parallelogram intersecting the opposite side divides it into isosceles triangle and a trapezoid. (Ch. word - top - two isosceles? -ka).
Proof .
Theorem proven.
Property 5 . In a parallelogram, a segment with ends on opposite sides, passing through the point of intersection of the diagonals, is bisected by this point.
Proof .
Theorem proven.
Property 6 . The angle between the heights dropped from the vertex of the obtuse angle of the parallelogram is equal to the acute angle of the parallelogram.
Proof .
Theorem proven.
Property 7 . The sum of the angles of a parallelogram adjacent to one side is 180°.
Proof .
Theorem proven.
Construction of the bisector of an angle. Properties of the angle bisector of a triangle.
1) Construct an arbitrary ray DE.
2) On a given ray, construct an arbitrary circle with a center at the vertex and the same
centered at the beginning of the constructed ray.
3) F and G - points of intersection of the circle with the sides of the given angle, H - point of intersection of the circle with the constructed ray
Construct a circle with center at point H and radius equal to FG.
5) I - the point of intersection of the circles of the constructed beam.
6) Draw a line through the vertex and I.
IDH - required angle.
)
Property 1 . The angle bisector of a triangle divides the opposite side in proportion to the adjacent sides.
Proof . Let x, y be segments of the side c. We continue the ray BC. On the ray BC, we plot a segment CK from C equal to AC.
A parallelogram is a quadrilateral whose opposite sides are pairwise parallel. The following figure shows the 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 angles 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 parallel in pairs. 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.
No parallel lines 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.
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs (Fig. 233).
An arbitrary parallelogram has the following properties:
1. Opposite sides of a parallelogram are equal.
Proof. Draw a diagonal AC in parallelogram ABCD. Triangles ACD and AC B are equal as having a common side AC and two pairs of equal angles adjacent to it:
(as cross-lying angles with parallel lines AD and BC). Hence, and as sides of equal triangles lying opposite equal angles, which was required to be proved.
2. Opposite angles of a parallelogram are:
3. Neighboring angles of a parallelogram, that is, angles adjacent to one side, add up, etc.
The proof of properties 2 and 3 immediately follows from the properties of angles at parallel lines.
4. The diagonals of a parallelogram bisect each other at the point of their intersection. In other words,
Proof. Triangles AOD and BOC are equal, since their sides AD and BC are equal (property 1) and the angles adjacent to them (as cross-lying angles with parallel lines). This implies the equality of the corresponding sides of these triangles: AO which was required to be proved.
Each of these four properties characterizes a parallelogram, or, as they say, is its characteristic property, i.e., any quadrangle that has at least one of these properties is a parallelogram (and, therefore, has all the other three properties).
We carry out the proof for each property separately.
1". If the opposite sides of a quadrilateral are pairwise equal, then it is a parallelogram.
Proof. Let the quadrilateral ABCD have sides AD and BC, AB and CD, respectively, equal (Fig. 233). Let's draw the diagonal AC. Triangles ABC and CDA will be congruent as having three pairs of equal sides.
But then the angles BAC and DCA are equal and . The parallelism of the sides BC and AD follows from the equality of the angles CAD and DIA.
2. If a quadrilateral has two pairs opposite corners are equal, then it is a parallelogram.
Proof. Let . Since both the sides AD and BC are parallel (on the basis of parallel lines).
3. We leave the formulation and proof to the reader.
4. If the diagonals of a quadrilateral are mutually divided at the point of intersection in half, then the quadrilateral is a parallelogram.
Proof. If AO \u003d OS, BO \u003d OD (Fig. 233), then the triangles AOD and BOC are equal, as having equal angles (vertical!) At the vertex O, enclosed between pairs of equal sides AO and CO, BO and DO. From the equality of triangles we conclude that the sides AD and BC are equal. The sides AB and CD are also equal, and the quadrilateral turns out to be a parallelogram according to the characteristic property Г.
Thus, in order to prove that a given quadrilateral is a parallelogram, it suffices to verify the validity of any of the four properties. The reader is invited to independently prove one more characteristic property of a parallelogram.
5. If a quadrilateral has a pair of equal, parallel sides, then it is a parallelogram.
Sometimes any pair of parallel sides of a parallelogram is called its bases, while the other two are called lateral sides. The segment of a straight line perpendicular to two sides of a parallelogram, enclosed between them, is called the height of the parallelogram. The parallelogram in fig. 234 has a height h drawn to the sides AD and BC, its second height is represented by a segment .
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, you must 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. 
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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.
From the equality of triangles it follows that on the basis of the parallelism of lines at the intersection of their secant. We get:
parallelogram by definition. Q.E.D.
Proven.
Let's consider an example of applying the features of a parallelogram.
Example 1. In a convex quadrilateral Find: a) the corners of the quadrilateral; b) side.
Solution. Let's depict Fig. 4.
Rice. 4
parallelogram according to the first attribute of a parallelogram.
