1 First of all, we need to know the definition of rectangle and square. A rectangle is a quadrilateral with right angles and equal sides. A square is a quadrilateral, in which all corners are right angles and all sides are equal.
Now let's prove that a rectangle with equal adjacent sides is a square. Suppose there is a rectangular ABCD, where AB=BC. According to the definition of rectangle, we know that ∠ A = ∠ B = 90, ∠ C = ∠ D = 90. Since AB=BC, we can get AC=BD. This means that the diagonal lines of rectangular ABCD are equal.
3. Next, let's prove that AC⊥BD. Because ∠ A = ∠ B = 90, ∠ C = ∠ D = 90, we can get ∠ACB=∠DBC. And because AB=BC, so △ ABC△ CBD (SAS). Therefore ∠ACB=∠CBD. Since ∠ ACB+∠ CBD = 180, we can get ∠ ACB = ∠ CBD = 90. This means AC⊥BD.
Finally, let's prove that AC=BD. Because of △ ABC △ CBD, we can get AC=BD. This means that all sides of the rectangular ABCD are equal. Therefore, a rectangle with equal adjacent sides is a square.
The relationship between rectangle and square
1, rectangle and square are two common quadrangles, and they are closely related. First of all, rectangle and square are both a kind of rectangle. A rectangle refers to a quadrilateral with four internal angles at right angles, and both rectangles and squares satisfy this condition. Therefore, we can say that both rectangle and square are special forms of rectangle.
Secondly, the diagonals of rectangle and square are equal. For a rectangle, it is obvious that its diagonal lines are equal, because the diagonal lines of the rectangle divide the rectangle into two equal isosceles right triangles. For a square, its diagonal lines are equal, because the diagonal lines of the square divide the square into two equal isosceles right triangles.
In addition, the area and perimeter of rectangle and square are also related. For a rectangle, its area is equal to length times width, and its circumference is equal to twice the length plus twice the width. For a square, its area is equal to the square of its side length, and its circumference is equal to four times its side length.
It can be seen that the area of a square is the square of the side length, while the area of a rectangle is the length times the width. Therefore, if the length and width of a rectangle are equal, it is a square.