I. Parallelogram
Definition: Two groups of parallelograms with parallel opposite sides are called parallelograms.
Nature:
The opposite sides of a parallelogram are equal.
The diagonals of parallelogram are equal.
Diagonal bisection of parallelogram.
Judge:
Two groups of parallelograms with parallel opposite sides are parallelograms.
Two sets of quadrilaterals with equal opposite sides are parallelograms.
Two sets of diagonally equal quadrilaterals are parallelograms.
A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
Quadrilaterals whose diagonals bisect each other are parallelograms.
Second, the rectangle:
Definition: A parallelogram with a right angle is called a rectangle.
Properties of 1. rectangle
(1) has all the properties of a parallelogram.
(2) Unique property: all four corners are right angles and diagonal lines are equal. A rectangle is an axisymmetric figure.
2. Determination of rectangle
Definition: A parallelogram with a right angle is called a rectangle.
(2) Theorem 1: A quadrilateral with three right angles is a rectangle.
Theorem 2: Parallelograms with equal diagonals are rectangles.
Third, diamonds.
1. Definition:
A set of parallelograms with equal adjacent sides is called a diamond.
2. The nature of diamonds
(1) has all the properties of a parallelogram.
(2) All four sides of a diamond are equal.
(3) The diagonals of the rhombus are perpendicular to each other, and each diagonal bisects a set of diagonals.
(4) The rhombus is an axisymmetric figure.
(5) Diamond area = bottom × height = half of diagonal product.
3. Determination of diamond shape
(1) Definition: A group of parallelograms with equal adjacent sides is called a rhombus.
(2) Theorem 1: A quadrilateral with four equilateral sides is a diamond.
Theorem 2: Parallelograms with diagonal lines perpendicular to each other are diamonds.
Fourth, the square.
1. Definition:
The definition of a square can be understood from two aspects:
(1) A diamond with right angles is called a square.
(2) A group of rectangles with equal adjacent sides is called a square.
2. Square attribute
A square has all the attributes of quadrilateral, parallelogram, rectangle and diamond.
(1) edge-Four edges are equal and adjacent edges are vertical.
(2) Angle-all four angles are right angles.
(3) Diagonal lines-① Equal; ② Divided vertically; ③ Each diagonal line bisects a set of diagonal lines.
(4) It is an axisymmetric figure with four axes of symmetry.
3, the method of judging the square:
(1) According to the definition, there are two ways to judge whether a quadrilateral is a square:
First, it is proved that it is a rectangle, and then a group of adjacent sides are equal or diagonally perpendicular.
Prove that it is a diamond, and then prove that it has a right angle or diagonal equal.
Five, the relationship between square and rectangle, diamond, parallelogram:
Rectangle, diamond and square are all special parallelograms, in which square is both a special rectangle and a special diamond. Rectangular, rhombic and square are all special parallelograms, and their inclusion relations are shown in the figure.
Sixth, the relationship between the midpoint quadrangle and the original quadrangle:
Connect the midpoints of quadrilateral sides with equal diagonals in turn to obtain a diamond;
Connect the midpoints of each side of the quadrilateral in turn with diagonal lines perpendicular to each other to obtain a rectangle;
Connect the midpoints of each side of the quadrilateral with equal diagonal lines and vertical lines in turn to obtain a quadrilateral, which is square;
Seven, isosceles trapezoid
1, the nature of isosceles trapezoid: the two waists of isosceles trapezoid are equal; Two angles on the same base of isosceles trapezoid are equal; The diagonal lines of the isosceles trapezoid are equal.
2, isosceles trapezoid judgment:
Two trapezoid with equal waist are isosceles trapezoid; A trapezoid with two equal angles on the same base is an isosceles trapezoid.
I hope it helps you.