Parallelogram properties:

1, and the opposite sides of the parallelogram are equal.

2. The diagonals of parallelograms are equal.

3. The diagonal of parallelogram is equally divided.

Parallelogram judgment:

1, two sets of quadrilaterals with equal opposite sides are parallelograms.

2. Quadrilaterals whose diagonals bisect each other are parallelograms.

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

4. Two groups of diagonally equal quadrilaterals are parallelograms.

5. Two groups of parallelograms with parallel opposite sides are parallelograms.

Rectangular properties:

(1) has all the properties of a parallelogram.

(2) Uniqueness: All four corners are right angles and diagonal lines are equal.

Rectangular decision:

1, a parallelogram with a right angle is called a rectangle.

A quadrilateral with three right angles is a rectangle.

3. A parallelogram with equal diagonals is a rectangle.

Diamond properties:

1 has all the properties of a parallelogram.

All four sides of the diamond are equal.

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

4. Diamond area = bottom × height = half of diagonal product.

Diamond decision:

1, a set of parallelograms with equal adjacent sides is called a diamond.

2. A quadrilateral with four equilateral sides is a diamond.

3. Parallelograms with diagonal lines perpendicular to each other are rhombic.

Properties of Square: Square has all the properties of quadrilateral, parallelogram, rectangle and diamond.

How to judge the square:

1, it is proved to be a rectangle first, and then it is proved that a group of adjacent sides are equal or diagonal is vertical.

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

Introduction to Mathematics:

Mathematics: English: Mathematics, which originated from the ancient Greek μ θ η μ α (má th ē ma); Often abbreviated as math or maths], it is a discipline that studies concepts such as quantity, structure, change, space and information. Mathematics is a universal means for human beings to strictly describe the abstract structure and mode of things, and can be applied to any problem in the real world. All mathematical objects are artificially defined in essence.

In this sense, mathematics belongs to formal science, not natural science. Different mathematicians and philosophers have a series of views on the exact scope and definition of mathematics. Mathematics plays an irreplaceable role in the development of human history and social life, and it is also an indispensable basic tool for studying and studying modern science and technology.

Detailed definition:

Aristotle defined mathematics as "quantitative mathematics", which lasted until18th century. /kloc-since the 0/9th century, mathematical research has become more and more rigorous, and it has begun to involve abstract topics such as group theory and projection geometry that have no clear relationship with quantity and measurement. Mathematicians and philosophers have begun to put forward various new definitions.

Some of these definitions emphasize the deductive nature of a lot of mathematics, some emphasize its abstraction, and some emphasize some themes in mathematics. Even among professionals, the definition of mathematics has not been reached. Whether mathematics is an art or a science has not even been decided. Many professional mathematicians are not interested in the definition of mathematics or think it is undefined. Some just said, "Mathematics is done by mathematicians."

The three main mathematical definitions are called logicians, intuitionists and formalists, each of which reflects a different school of philosophical thought. Everyone has serious problems, no one generally accepts it, and no reconciliation seems feasible.

The early definition of mathematical logic is Benjamin Peirce's Science of Drawing Inevitable Conclusions (1870). In Principles of Mathematics, Bertrand Russell and alfred north whitehead put forward a philosophical program called logicism, trying to prove that all mathematical concepts, statements and principles can be defined and proved by symbolic logic. The logical definition of mathematics is Russell's "All mathematics is symbolic logic" (1903).

The definition of intuitionism comes from mathematician L. E. J. Brouwer, who equates mathematics with some psychological phenomena. An example of the definition of intuitionism is that "mathematics is a psychological activity constructed one after another". Intuitionism is characterized by rejecting some mathematical ideas that are considered effective according to other definitions. In particular, although other mathematical philosophies allow objects that can be proved to exist, even if they cannot be constructed, intuitionism only allows mathematical objects that can actually be constructed.

Formalism defines mathematics through mathematical symbols and operational rules. Haskell Curry simply defined mathematics as "formal system science". A formal system is a set of symbols, or symbols, and there are some rules that tell how the symbols are combined into formulas. In the formal system, the word axiom has a special meaning, which is different from the ordinary meaning of "self-evident truth" in the formal system. Axiom is a combination of symbols contained in a given formal system, without using the rules of the system to deduce it.