Modern Mathematics: Square is one of the special parallelograms. That is, a group of parallelograms with equal adjacent sides and a right angle is called a square.

Primary school mathematics: the primary school mathematics textbook does not give a clear definition of a square. Generally, we know a square through two stages. For example, in 2003, the first stage of the People's Education Edition textbook was on page 34 of grade one. On the basis of understanding the cube, realize that the faces are on the object, and draw one of the faces on the paper to form a square, so that students can understand it intuitively and learn to recognize it. The second stage is on page 34 of the first volume of the third grade. By measuring, folding and comparing, find out the characteristics of the sides and corners of a square: the four sides are equal and the four corners are right angles, so that students can judge whether the figure is a square according to the characteristics.

Two. Conceptual explanation

What is the prototype of a square in life? People first knew squares by dividing fields. In China's ancient calculations, a square field was called a square field, also known as a square figure. In real life, when people observe an object, they often see a shape with length, width and height, and then draw a certain face of the shape on paper or take pictures to abstract it into their minds, thus forming a plane figure. For example, the surfaces of dining table, floor tile, glass and Rubik's cube are often square.

What are the characteristics of a square? As shown in the figure below:

In the parallelogram ABCD, if AB= BC, and ∠A = Rt∞, the parallelogram ABCD is a square, which is marked as the port ABCD and pronounced as the square ABCD. The length of each side of a square is called the side length of a square.

A square has the following characteristics.

① Side: Two groups of opposite sides are parallel, four sides are equal, and adjacent sides are perpendicular to each other.

② Internal angle: All four angles are 90. .

③ Diagonal lines: Diagonal lines are perpendicular to each other, equal and equally divided, and each diagonal line is equally divided into a set of diagonal lines.

④ Symmetry: It is both a central symmetrical figure and an axisymmetric figure (with four axes of symmetry).

⑤ Shape: Square is a kind of rectangle and a kind of diamond.

⑥ Square has all the properties of parallelogram, rhombus and rectangle.

⑦ Special properties: A diagonal line of a square divides the square into two isosceles right triangles, and the included angle between the diagonal line and the side is 45 degrees; The two diagonal lines of a square divide it into four congruent isosceles right-angled triangles.

How to judge whether a quadrilateral is a square? There are several sufficient conditions for judging a square. A quadrilateral that satisfies one of the following conditions is a square: