A closed plane figure surrounded by four line segments that are not on the same line is called a quadrilateral. It consists of convex quadrilateral and concave quadrilateral.

Convex quadrilateral:

Make a straight line on the same side as the other sides.

Parallelogram (including: ordinary parallelogram, rectangle, diamond and square)

Trapezoid (including: common trapezoid, right-angled trapezoid, isosceles trapezoid)

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

Concave quadrilateral:

Make a straight line between one side and the other.

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The quadrilateral obtained by connecting the midpoints of the sides of the quadrilateral in turn is called the midpoint quadrilateral. No matter how the shape of the original quadrangle changes, the shape of the midpoint quadrangle is always a parallelogram. The midpoint quadrangle of a diamond is a rectangle, the midpoint quadrangle of a rectangle is a diamond, and the midpoint quadrangle of a square is a square. [Edit this paragraph] The nature and judgment of parallelogram 1. Definition: Two groups of parallelograms with opposite sides are called parallelograms.

2. Nature:

(1) If a quadrilateral is a parallelogram, then two opposite sides of the quadrilateral are equal.

(abbreviation for "the opposite sides of a parallelogram are equal")

(2) If the quadrilateral is a parallelogram, then the two diagonal corners of the quadrilateral are equal.

(Simply described as "parallelogram diagonal is equal")

(3) If a quadrilateral is a parallelogram, then the adjacent angles of this quadrilateral are complementary.

(simply described as "complementary adjacent angles of parallelogram")

(4) The parallel segments sandwiched between two parallel lines are equal.

(5) If a quadrilateral is a parallelogram, then the two diagonals of this quadrilateral are equally divided.

(Simply described as "the two diagonals of a parallelogram are equally divided")

(6) A parallelogram is a central symmetric figure, and the center of symmetry is the intersection of two diagonals.

3. Judges:

(1) If two opposite sides of a quadrilateral are equal, then it is a parallelogram.

(Simply stated as "two sets of quadrangles with equal opposite sides are parallelograms")

(2) If a set of opposite sides of a quadrilateral are parallel and equal, the quadrilateral is a parallelogram.

(briefly stated as "a set of quadrilaterals with parallel and equal opposite sides is a parallelogram")

(3) If two diagonal lines of a quadrilateral are equally divided, then the quadrilateral is a parallelogram.

(In short, it means that "quadrilaterals with diagonal lines bisecting each other are parallelograms")

(4) A quadrilateral is a parallelogram if its two opposite corners are equal.

(Simply stated as "two groups of diagonally equal quadrilaterals are parallelograms"

(5) If two opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

(Simply described as "two groups of parallelograms with parallel opposite sides are parallelograms") [Edit this paragraph] The nature and judgment definition of a rectangle: a parallelogram with right angles is called a rectangle.

Property ① All four corners are right angles.

Diagonal lines of rectangles are equal.

Note: A rectangle has all the attributes of a parallelogram.

Judgment: ① A parallelogram with right angles is a rectangle;

② A quadrilateral with three right angles is a rectangle;

③ Parallelograms with equal diagonals are rectangles. [Edit this paragraph] The nature and judgment of diamonds

Definition: A set of parallelograms with equal adjacent sides is called a diamond.

Properties: ① All four sides of the diamond are equal;

Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

Note: A diamond also has all the attributes of a parallelogram.

Judgment: ① A group of parallelograms with equal adjacent sides are rhombic;

② A quadrilateral with four equal sides is a diamond;

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

(4). A parallelogram whose diagonal bisects a group of diagonals is a diamond [edit this paragraph] The nature and judgment definition of a square: a group of parallelograms with equal adjacent sides and a right angle is called a square.

Properties: ① All four corners of a square are right angles and all four sides are equal;

The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.

Judgment: Because a square has all the properties of parallelogram, rectangle and diamond, there are three ways to judge a square.

① A set of rectangles with equal adjacent sides is a square.

Diamonds with right angles are squares.

(3) A quadrilateral with two equal diagonals and bisected vertically.

(4) the definition of parallelogram [edit this paragraph] trapezoid and special trapezoid with equal diagonal lines and perpendicular to each other.