parallelogram

A kind of mathematical plane geometric figure

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This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Parallelogram is a closed figure composed of two groups of parallel line segments in the same two-dimensional plane. A parallelogram is generally named by the shape name plus four vertices in turn. Note: When using letters to represent quadrangles, be sure to press Clockwise or anti-clockwise Each vertex shall be indicated in the direction.

stay Euclidean geometry In, a parallelogram is a simple (non self intersecting) quadrilateral with two pairs of parallel edges. The opposite or opposite sides of a parallelogram have the same length, and the opposite angles of the parallelogram are equal.

In contrast, a quadrilateral with only one pair of parallel sides is a trapezoid. The three-dimensional correspondence of parallelogram is Parallelepiped 。

Chinese name: parallelogram
Foreign name: Parallelogram
Features: The opposite sides are parallel and equal, easy to deform
Category: Plane figure
Nature 1: Two sets of opposite sides are equal

Nature 2: The two groups are equal diagonally
Nature 3: Diagonal lines are divided equally
Nature 4: Two sets of opposite sides are parallel to each other
Internal angle and: 360°
Number of sides: 4

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definition

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Two sets of paralleled quadrilaterals with opposite sides are called parallels quadrilateral 。

1. The parallelogram belongs to Plane figure 。

2. The parallelogram belongs to quadrilateral 。

3. The parallelogram belongs to Centrosymmetric figure 。

nature

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rectangle

（ rectangle 、 diamond 、 square Are special parallelograms.)

(1) If a quadrilateral is a parallelogram, the two opposite sides of the quadrilateral are equal.

(Briefly described as "two sets of opposite sides of parallelogram are equal respectively"^[1] ）

(2) If a quadrilateral is a parallelogram, then the two sets of diagonals of the quadrilateral are equal.

(briefly described as "two sets of diagonals of parallelogram are equal respectively"^[1] ）

(3) If a quadrilateral is a parallelogram, then the Adjacent angle complementary 。

(Briefly described as "the complementation of neighboring angles of parallelogram")

(4) Clamp in two strips Parallel line Parallelogram between Height of equal. (Briefly described as "the high distance between parallel lines is equal everywhere")

(5) If a quadrilateral is a parallelogram, the two diagonals of the quadrilateral are equally divided with each other.

(Briefly described as "the diagonals of parallelogram are equally divided with each other"^[1] ）

(6) The graph obtained by connecting the midpoints of the sides of an arbitrary quadrilateral is a parallelogram. (corollary)

(7) The area of a parallelogram is equal to the product of the base and the height. (It can be regarded as a rectangle.)

(8) A straight line passing through the intersection of the diagonals of a parallelogram divides the parallelogram into two congruent parts.

(9) The parallelogram is Centrosymmetric The center of symmetry is the intersection of two diagonals

(10) A parallelogram is not an axisymmetric figure, but a parallelogram is a centrosymmetric figure. Rectangles and diamonds are axial symmetry graphical. notes : Square, rectangle and diamond are also special parallelograms, which have the nature of parallelograms.

(11) In the parallelogram ABCD, E is the midpoint of AB, then AC and DE are mutually three Bisection In general, if E is the n bisection point near A on AB, then AC and DE are equally divided into each other (n+1).

(12) In the parallelogram ABCD, AC and BD are the diagonals of the parallelogram ABCD, so the sum of the squares of the four sides is equal to the sum of the squares of the diagonals.

(13) The parallelogram diagonal divides the parallelogram area into four equal parts.

(14) In a parallelogram, the included angle formed by two heights on different opposite sides. The smaller angle is equal to the smaller angle in the parallelogram, and the larger angle is equal to the larger angle in the parallelogram.

(15) Parallelogram the measure of area It is equal to the product of the sine of adjacent two sides and their included angle^[2]

Other properties

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The opposite sides of a parallelogram are parallel (by definition) and therefore never intersect.

The area of a parallelogram is twice that of a triangle created by one of its diagonals.

The area of a parallelogram is also equal to the cross product of vectors of two adjacent sides.

Any line passing through the midpoint of the parallelogram bisects the area.

Any nondegenerate affine transformation adopts parallelogram of parallelogram.

The parallelogram has rotational symmetry of order 2 (to 180 °) (order 4 if it is a square). If it also has two lines of reflection symmetry, it must be a diamond or rectangle (non rectangular rectangle). If it has four lines of reflection symmetry, it is a square.

The perimeter of a parallelogram is 2 (a+b), where a and b are the lengths of adjacent sides.

Unlike any other convex polygon, a parallelogram cannot be engraved on any triangle less than twice its area.

The center of the four squares constructed on the inside or outside of the parallelogram is the vertex of the square.

If two lines parallel to the parallelogram are formed in parallel with the diagonal, the area of the parallelogram formed on the opposite side of the diagonal is equal [7]

The diagonal of a parallelogram divides it into four triangles of equal area.

determine

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1. Two sets of parallelograms whose opposite sides are parallel are parallelograms (definition decision method);

2. A group Opposite side Parallel and equal quadrilaterals are parallelograms;

3. Two groups Opposite side A quadrilateral that is equal to each other is a parallelogram;

4. Two sets of quadrilaterals with equal diagonals are parallelograms (two sets of opposite sides are judged to be parallel);

5. A quadrilateral whose diagonals are bisected is a parallelogram.

Supplement: Condition 3 is only true when it is a planar quadrilateral. If it is not a planar quadrilateral, even if it is two groups of quadrilateral whose opposite sides are equal, it is not a parallelogram.

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1、 Connect or translate diagonals.

2、 Having opposite edges across vertices vertical constitute right triangle 。

3、 Connect Diagonal intersection Forming one side at the midpoint of one side or crossing the intersection of diagonal lines Parallel line , consisting of Line segments parallel or Median line 。

4、 Connects a vertex to a point on the opposite edge line segment Or extend this line segment to form a similar triangle or Equal product triangle 。

5、 Over vertex A vertical line used as a diagonal line to form a parallel or triangular line segment Congruent 。

Correlation calculation

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1. (1) Parallelogram the measure of area Formula: bottom × height (cutting and compensation method can be used, and the derivation method is shown in Figure 1); If "h" represents height, "a" represents bottom, and "S" represents area of parallelogram, then S parallelogram=a * h.

(2) The area of a parallelogram is equal to the product of two sets of adjacent edges times the included angle sine ； If "a" and "b" are used to indicate the length of two adjacent sides, α is the included angle of both sides, and "S" is the area of the parallelogram, then S parallelogram=ab * sin α.

Fig. 1 Parallelogram

2. Perimeter of parallelogram: sum of four sides. It can be doubled (bottom 1+bottom 2); If "a" represents bottom 1, "b" represents bottom 2, and "c flat" represents the perimeter of the parallelogram, then the perimeter of the parallelogram c=2 (a+b).^[3]

Special parallelogram

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rectangle

Definition: A parallelogram with a right angle is rectangle 。

Judgment:

1. A parallelogram with a right angle is a rectangle;

2. A parallelogram with equal diagonals is a rectangle;

3. A quadrilateral with three right angles is a rectangle;

4. Quadrilateral shapes with equal diagonals and bisecting each other are rectangles.

nature:

1. A rectangle has all the properties of a parallelogram;

2. Diagonal lines of rectangles are equal;

3. The four corners of the rectangle are 90 degrees;

4. A rectangle is an axisymmetric figure as well as a central symmetric figure. It has two symmetry axes, which are the straight lines where the lines connecting the midpoints of each group of opposite sides are located; The center of symmetry is the intersection of two diagonal lines.