What shape is special for rectangles and squares

A closed plane figure or three-dimensional figure enclosed by four line segments that are not on the same straight line and do not cross each other end to end is called a quadrilateral, which is composed of a convex quadrilateral and a concave quadrilateral. The quadrilateral obtained by sequentially connecting the midpoints on any quadrilateral is called a midpoint quadrilateral, and the midpoint quadrilateral is a parallelogram. The midpoint quadrilateral of a rhombus is a rectangle, the midpoint quadrilateral of a rectangle is a rhombus, the midpoint quadrilateral of an isosceles trapezoid is a rhombus, and the midpoint quadrilateral of a square is a square.

Convex quadrilateral. The four vertices are in the same plane, the opposite edges do not intersect and make a straight line on one side, and the other edges are on the same side.

Parallelograms (including: normal parallelograms, rectangles, diamonds, squares).

Trapezoidal (including: ordinary trapezoidal, right-angled trapezoid, isosceles trapezoid).

The sum of the inner and outer angles of a convex quadrilateral is 360 degrees.

Concave quadrilateral. The four vertices of a concave quadrilateral are in the same plane, the opposite sides do not intersect and make a straight line on one side, and some of the other sides are on the opposite side. Do not focus on research.

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 shape of the midpoint quadrilateral depends on the diagonal of the original quadrilateral.

If the diagonal of the original quadrilateral is perpendicular, the midpoint quadrilateral is rectangular; If the diagonals of the original quadrilateral are equal, then the midpoint quadrilateral is a diamond; If the diagonals of the original quadrilateral are both perpendicular and equal, then the midpoint quadrilateral is a square.

First, to put it simply: rectangles and squares are special quadrilaterals.

The second subdivision: rectangles and squares are all special parallelograms.

Rectangles are special parallelograms.

Rectangles are special parallelograms. Because quadrilaterals with parallel and equal opposite sides are called parallelograms. So a rectangle is a parallelogram with right angles, and a rectangular is a special form of a parallelogram.

Difference between parallelogram and rectangle, diamond, square:

For parallelograms, the rectangle has unique properties: all four corners are right angles; The two diagonals are equal and bisected (the basis for determining that the middle line on the hypotenuse of a right triangle is equal to half of the hypotenuse). Unique properties of the rhombus:

All four sides are equal; The two diagonals are perpendicular to each other, and each diagonal is bisected by a set of diagonals. The sum of the properties unique to the rectangle and the rhombus is the unique property of the square to the parallelogram.

In general, if we are to prove that a quadrilateral is a rectangle or a rhombus, we should first prove that the quadrilateral is a parallelogram and then prove whether the parallelogram is a rectangle or a rhombus. When proving whether it is a square, we can start from two ways, the same as proving a rectangle and a rhombus, first proving that it is a parallelogram, then proving that it is a rectangle or a rhombus, and finally proving that it is a square through known conditions or verification.

Both squares and rectangles are special parallelograms.

A parallelogram is a closed figure composed of two sets of parallel line segments in the same two-dimensional plane. Parallelograms are generally named with the name of the figure plus four vertices. Note: When using letters to represent quadrilaterals, be sure to indicate each vertex in a clockwise or counterclockwise direction.

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

In contrast, a quadrilateral with only a pair of parallel sides is trapezoidal. The three-dimensional counterpart of a parallelogram is a parallelepiped.

Auxiliary lines: 1. Connect diagonals or translate diagonal lines.

2. The perpendicular line that passes over the vertex as the opposite side forms a right-angled triangle.

3. Connect the intersection of the diagonal line with the midpoint of one side, or cross the intersection of the diagonal line as a parallel line on one side, forming a parallel or median line of the line segment.

4. Connect the line segment of the vertex with the point on the opposite side or extend the line segment to construct a similar triangle or equal area triangle.

5. The perpendicular line that crosses the vertex as a diagonal line constitutes a parallel or triangular congruence of line segments.

A group of quadrilaterals with two opposing sides parallel to each other is called a parallelogram. Rectangles, diamonds, squares are all special parallelograms. If a quadrilateral is a parallelogram, then the two sets of opposite sides of the quadrilateral are equal.

Special parallelograms. Because the opposite sides are parallel and equal; But each of them has its own characteristics. So, both rectangles and squares are special parallelograms.

The square has the characteristics of a rectangle: the opposite sides are parallel and equal, and the four corners are right angles; But the square has its own characteristics: all four sides are equal.

So the square is a special rectangle.

Definition of a square

Square, is one of the special parallelograms. That is, a group of parallelograms with equal adjacent sides and one angle is at right angles is called a square, also known as a regular quadrilateral. Square, with all the characteristics of a rectangle and a rhombus.

Square

Determination theorem. 1. A diamond with equal diagonals is a square.

2. A diamond with a right angle is a square.

3. Rectangles with diagonal lines perpendicular to each other are squares.

4. A group of rectangles with equal adjacent sides is a square.

5. A group of parallelograms with equal adjacent sides and one angle is a right angle.

Definition of a rectangle

A rectangle is a parallelogram with a corner that is at a right angle. A square is a special rectangle with four sides of equal length.

RectangularDetermination theorem.

1. There is a parallelogram with a right angle that is a rectangle.

2. A parallelogram with equal diagonals is a rectangle.

3. A parallelogram with adjacent sides perpendicular to each other is a rectangle.

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

5. Quadrilaterals with equal diagonals and bisected from each other are rectangular.

A group of parallelograms with equal adjacent sides and one angle that is at right angles is called a square, also known as a regular quadrilateral. Square, with all the characteristics of a rectangle and a rhombus. A square is a special rectangle and is one of the special parallelograms.

The same point between a rectangle and a square:

1. The opposite sides are parallel and equal.

2. Both rectangles and squares have 4 right angles.

3. Rectangles and squares are special parallelograms.

4. All are axisymmetric and center-symmetrical figures.

The difference between a rectangle and a square:

1. The opposite sides of the rectangle are equal, and the 4 sides of the square are equal.

2. The rectangle has two axes of symmetry, and the square has four axes of symmetry.

The circumference of the square is the sum of the lengths of the four sides of the square. Because the sides of the square are all of the same length, four times the length of the sides is the circumference of the square. The side length is set to a, the perimeter is set to l, and the perimeter l=4a.

The area of the square is equal to the square of the side length: area s = a a, which is the side length multiplied by the side length. According to the area formula, the side length a = root number s is obtained.

Mistake. The analysis process is as follows:

According to the definition of a square, a quadrilateral with all four sides equal and all four corners at right angles is a square. A rectangle, on the other hand, does not satisfy the condition that all four sides are equal.

So the rectangle is not a square, so the rectangle is not a special square. Squares are special rectangular shapes.

According to the characteristics of squares and rectangles, it can be seen that a square is a special rectangle, and a rectangle is not only a special square. Definitions:

A quadrilateral with all four sides equal and one corner at a right angle is a square. A quadrilateral with equal sides and three angles that refers to the right angle of the rock is called a square.

Determination of squares1. A diamond with equal diagonals is a square.

2. A diamond with a right angle is a square.

3. Rectangles with diagonal lines perpendicular to each other are squares. Tease.

4. A group of rectangles with equal adjacent sides is a square.

5. A group of parallelograms with equal adjacent sides and one angle is a right angle.

6. A parallelogram with diagonal lines perpendicular to each other and equal to each other is a square.