About parallelograms, diamond properties and how to determine them

1. There are two groups of quadrilaterals with parallel sides are parallelograms 2. Two groups of quadrilaterals with equal opposite sides are parallelograms.

3. There are two groups of quadrilaterals with equal diagonal diagonals that are parallelograms4. The quadrilaterals with two diagonal lines bisected by each other are parallelograms, and they have all the properties of parallelograms, and the other four sides are equal, and the two diagonals are bisected perpendicular to each other.

Rhombic judgment 1, there is a set of parallelograms with equal adjacent sides is a rhombus 2, and parallelograms with diagonal perpendicular to each other are rhombuses.

3. A quadrilateral with all four sides equal is a parallelogram.

Parallelogram nature.

1. The two groups of opposite sides of the parallelogram are equal respectively.

2. Two groups of parallelograms are parallel to each other.

3. The parallelograms are equal diagonally.

4. The parallelograms are bisected by diagonals.

Parallelogram determination.

1. There are two groups of quadrilaterals with opposite sides parallel to each other are parallelograms.

2. Two sets of quadrilaterals with equal opposite sides are parallelograms.

3. There are two groups of quadrilaterals with equal diagonal angles that are parallelograms.

4. A quadrilateral with two diagonals bisecting each other is a parallelogram.

5. A group of quadrilaterals with parallel sides and equal diagonal sides is a horizontal quadrilateral (this is not a judgment that cannot be used directly, but it can be obtained by proof).

Rhomboid-shaped nature. It has all the properties of a parallelogram, and the other four sides are equal.

The two diagonals are bisected perpendicular to each other.

Rhombus determination. 1. There is a group of parallelograms with equal adjacent sides that are diamond-shaped.

2. A parallelogram with diagonal perpendicular to each other is a diamond.

3. A quadrilateral with all four sides equal is a parallelogram.

Determination of rectangles.

1) There is a parallelogram with an angle of right angles and a rectangle.

2) A parallelogram with an equal dime is a rectangle.

3) There are three corners that are right angles, and the quadrilateral is rectangular.

Determination of the diamond.

1) A group of parallelograms with equal adjacent sides is a rhombus

2) A parallelogram perpendicular to the diagonal is a diamond

3) A quadrilateral with all four sides equal is a diamond

Determination of parallelograms.

Decision: Two sets of quadrilaterals with opposite sides parallel to each other are parallelograms;

Two sets of quadrilaterals with equal opposite sides are parallelograms;

Two sets of quadrilaterals with equal diagonal angles are parallelograms;

A quadrilateral with diagonals bisecting each other is a parallelogram;

A set of quadrilaterals with opposite sides parallel and equal is a parallelogram

The properties and judgments of parallelograms, rectangles, diamonds, squares are as follows:

Parallelogram nature.

1. The opposite sides of the parallelogram are equal.

2. The opposite diagonal of the parallelogram is equal.

3. The diagonal lines of the parallelogram are bisected from each other.

Parallelogram determination.

1. Two sets of quadrilaterals with equal opposite sides are parallelograms.

2. A quadrilateral with diagonals bisecting each other is a parallelogram.

3. A group of quadrilaterals that are parallel and equal to the opposite sides is a parallelogram.

4. Two sets of quadrilaterals with equal diagonal angles are parallelograms.

5. Two sets of quadrilaterals with opposite sides parallel to each other are parallelograms.

Rectangle properties: (1) It has all the properties of a parallelogram.

2) Unique properties: All four corners are right angles, and the diagonals are equal.

Rectangle judgment: 1. A parallelogram with a right angle is called a rectangle.

2. A quadrilateral with three corners that are right angles is a rectangle.

3. A parallelogram with equal diagonal lines is a rectangle.

Rhombic properties: 1. It has all the properties of a parallelogram.

2. The four sides of the diamond are equal.

3. The diagonals of the diamond are perpendicular to each other, and each diagonal is divided into a set of diagonals.

4. Diamond area = base height = half of the diagonal product.

Rhombus determination: 1. There is a group of parallelograms with equal adjacent sides called rhombus.

2. A quadrilateral with all four sides equal is a diamond.

3. A parallelogram with diagonal perpendicular to each other is a diamond.

Square properties: A square has all the properties of a quadrilateral, parallelogram, rectangle, and diamond.

How to determine the square:

1. First, it is proved that it is rectangular, and then it is proved that there is a group of adjacent sides equal or diagonally perpendicular.

2. First prove that it is diamond-shaped, and then prove that it has an angle that is right or equal diagonal.

Brief description of the subject of mathematics:

Mathematics: English: mathematics, from the ancient Greek máthēma); Often abbreviated as math or maths], it is a discipline that studies concepts such as quantity, structure, change, space, and information.

Mathematics is a general means for humans to strictly describe the abstract structure and pattern of things, and can be applied to any problem in the real world, and all mathematical objects are inherently artificially defined.

In this sense, mathematics belongs to the formal sciences, not the natural sciences. Different mathematicians and philosophers have a range of opinions on the exact scope and definition of mathematics. In the historical development and social life of mankind, mathematics plays an irreplaceable role, and it is also an indispensable basic tool for learning and researching modern science and technology.

Definition of parallelogram: A group of quadrilaterals where two sets of opposing sides are parallel to each other is a parallelogram.

Properties of parallelograms:

1): The parallelogram is equal on opposite sides.

2): The parallelogram is equal diagonally.

3): The parallelogram is parallel to the opposite side.

4. The parallelograms are bisected by diagonals.

5): The adjacent angles of parallelograms are complementary.

How to determine a parallelogram.

Two sets of quadrilaterals with opposite sides parallel to each other are parallelograms.

A set of quadrilaterals with opposite sides parallel and equal is a parallelogram

Two sets of quadrilaterals with equal opposite sides are parallelograms;

Two sets of quadrilaterals with equal diagonal angles are parallelograms;

A quadrilateral with diagonals bisecting each other is a parallelogram;

Rhomboid-shaped nature. 1. The diagonals are perpendicular to each other and bisected, and each diagonal is bisected by a group of diagonals;

2. All four sides are equal;

3. The diagonals are equal, and the adjacent angles are complementary;

4. The rhombus is both an axisymmetric figure, the symmetry axis is the straight line where the two diagonals are located, and it is also a central symmetrical figure, 5. In a 60° rhombus, the short diagonal is equal to the side length, and the long diagonal is 3 times that of the short diagonal.

6. The rhombus is a special parallelogram, which has all the properties of a parallelogram.

Determine 1. A set of parallelograms with equal adjacent sides is a diamond.

2. A quadrilateral with equal quadrilaterals is a diamond.

3. A quadrilateral with diagonals perpendicular to each other and bisected is a diamond.

The quadrilateral obtained by connecting the midpoints of each side of the quadrilateral in turn is called a midpoint quadrilateral. No matter how the shape of the original quadrilateral changes, the shape of the midpoint quadrilateral is always parallelogram. The midpoint quadrilateral of a rhombus is a rectangle (the midpoint quadrilateral of a quadrilateral whose diagonals are perpendicular to each other is defined as a rhombus.)

The midpoint quadrilateral of a quadrilateral with equal diagonals is defined as a rectangle. ）

The rhombus is defined on the premise of a parallelogram, first of all it is a parallelogram, but it is a special parallelogram, and the special feature is that "there is a group of adjacent sides equal", so it adds some special properties and different judgment methods from parallelograms.

Rectangle Properties: 1 The 4 corners of a rectangle are all right angles.

Rectangle 2 The diagonal lines of the rectangle are equal and bisected from each other.

3 The sum of squares of the distance from any point in the plane of the rectangle to the ends of its two diagonals is equal.

4 A rectangle is both axisymmetric and center-symmetric (the axis of symmetry is a line connecting the midpoints of any set of opposite sides), and it has two axes of symmetry.

5 Rectangles have all the properties of a parallelogram.

Verdict: 1A parallelogram with an angle of right angles is a rectangle.

2.A parallelogram with equal diagonal lines is a rectangle.

3.A quadrilateral with three corners that are right angles is a rectangle.

Parallelogram: A quadrilateral with two pairs of sides of equal length. (Feature: Parallel to each other).

Rhombus: A parallelogram with diagonal lines perpendicular to each other. (Feature: four sides of equal length).

Rectangle: A parallelogram with all four corners at right angles.

Properties: Parallelogram: Opposite sides are parallel and equal, diagonally equal, two diagonals bisect each other, and the center is symmetrical.

Rectangle: The opposite sides are parallel and equal, the four corners are right angles, the two diagonals are bisected and equal to each other, the axis is symmetrical, and the center is symmetrical.

Rhombus: the opposite sides are parallel, the four sides are equal, the diagonals are equal, the two diagonals are bisected perpendicular to each other, each diagonal is bisected by a set of diagonals, axisymmetric, and center-symmetrical.

Square: The opposite sides are parallel and all four sides are equal, the four corners are all right angles, the two diagonals are bisected and equal to each other, and each diagonal is bisected by a set of diagonals, symmetrical in axis and symmetrical in the center.

Judgment method: parallelogram: (1) Two sets of quadrilaterals with equal sides are parallelograms.

2) Two sets of quadrilaterals with opposite sides parallel to each other are parallelograms.

3. A group of quadrilaterals that are parallel and equal to the opposite sides is a parallelogram.

4) Two sets of quadrilaterals with equal diagonals.

5) A quadrilateral with two diagonals bisecting each other is a parallelogram.

Rectangle: (1) A quadrilateral with three corners that are right angles is a rectangle.

2) There is a parallelogram with an angle that is a right angle and is a rectangle.

3. A parallelogram with equal diagonal lines is a rectangle.

Rhombus: (1) A quadrilateral with all four sides equal is a rhombus.

2) There is a group of parallelograms with equal adjacent sides that are rhombuses.

3. A parallelogram with diagonal perpendicular to each other is a diamond.

Square: (1) There is an angle that is a right angle, and a group of parallelograms with equal adjacent sides is a square.

2) There is a set of rectangles with equal adjacent sides that are squares.

3) There is a diamond with a right angle that is a square.

Absolutely accurate, you can ask me again if you have any questions in the future, never get tired of asking?

The rhombus is one of the special parallelograms. There is a group of parallelograms with equal adjacent sides called diamonds. As shown in the figure on the right, in the parallelogram ABCD, if AB=BC, then the parallelogram ABCD is said to be a diamond, denoted as ABCD, and read as a diamond ABCD.

Properties: 1. The rhombus has all the properties of a parallelogram;

2. The four sides of the diamond are equal.

3. The diagonals of the diamond are bisected perpendicular to each other and each group of diagonals is equally divided;

4. The diamond is an axisymmetric figure, and there are 2 axes of symmetry, that is, the straight line where the two diagonals are located;

5. The diamond shape is a central symmetrical figure;