Key points of knowledge

1, the sum of the internal angles of the quadrilateral is equal to 1800, and the sum of the internal angles of the n polygon is equal to (n-2)? 6? 1 1800, arbitrary polygon

The sum of the external angles of is equal to 3600, and the diagonal number of n polygons is n(n-3)/2.

2. Parallelogram

Properties: (1) The opposite sides of the parallelogram are parallel and equal, the diagonal lines are equal, and the diagonal lines are equally divided;

(2) The parallelogram is a figure with central symmetry.

Judgment: (1) defines judgment;

(2) Two groups of quadrangles with equal opposite sides are parallelograms;

(3) Two groups of quadrangles with equal diagonal are parallelograms;

(4) Quadrilaterals whose diagonals bisect each other are parallelograms;

(5) A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

3. rectangular

Properties: (1) has all the properties of a parallelogram;

(2) All four corners are right angles;

(3) The diagonal lines are equal (inference: the median line on the hypotenuse of the right triangle is equal to half of the hypotenuse);

(4) It is both a central symmetrical figure and an axisymmetric figure;

(5) Its area is equal to the product of two adjacent sides.

Judgment: (1) defines judgment;

(2) A quadrilateral with three right angles;

(3) Parallelogram with equal diagonals.

4. Diamond shape

Properties: (1) has all the properties of a parallelogram;

(2) Four sides are equal;

(3) diagonal lines are divided vertically, and each diagonal line divides a set of diagonal lines equally;

(4) It is both a central symmetrical figure and an axisymmetric figure;

(5) Its area is equal to half of the product of two diagonals (applicable to all quadrangles whose diagonals are perpendicular to each other).

Judgment: (1) defines judgment;

(2) A quadrilateral with four equilateral sides;

(3) Parallelogram with diagonal lines perpendicular to each other.

5. Square

Attribute: It has all the attributes of rectangle and diamond.

Judgment: (1) defines judgment;

(2) First, determine that the quadrilateral is a rectangle, and then determine that it is also a diamond;

(3) Make sure that the quadrilateral is a diamond, and then make sure that it is also a rectangle.

6, isosceles trapezoid

Nature: (1) Two waists are equal;

(2) The two diagonals are equal;

(3) The two bottom angles on the same bottom are equal;

(4) It is an axisymmetric figure.

Judgment: (1) A trapezoid with two equal angles on the same base is an isosceles trapezoid;

(2) A trapezoid with equal diagonals is an isosceles trapezoid.

7. Theorem of bisection of parallel lines: If a group of parallel lines have equal line segments on a straight line, then the line segments on other straight lines are also equal.

Inference 1: A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.

Inference 2: A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.

8. Two midline theorems

The midline theorem of triangle: the midline of triangle is parallel to the third side and equal to half of it.

The midline theorem of trapezium: the midline of trapezium is parallel to the two bottoms and equal to half of the sum of the two bottoms (inference: the area of trapezium is equal to the product of the length and height of the midline).

9. Central symmetry

Definition: It is emphasized that 180 needs to be rotated to coincide.

Theorem: (1) Two graphs that are symmetric about the center are congruent.

(2) For two graphs with symmetrical centers, the connecting lines of symmetrical points pass through the symmetrical center and are divided into two by the symmetrical center (there is an inverse theorem).

10, the relationship of various quadrangles.

Method summary

1. Problems related to the angle, number of sides and diagonal of a polygon are generally solved by formula column equations.

2. Distinguish the relations and differences of various quadrangles, and understand the definition, nature and correct use of judgment methods (which can be determined according to the order of conditions and conclusions).

3. Diagonal line is a commonly used auxiliary line to study quadrilateral, which can not only transform quadrilateral into triangle, but also fully embody all the characteristics of quadrilateral.

4. Auxiliary lines are usually added to the trapezoid to convert it into a parallelogram or triangle:

(1) The vertex passing through the shorter base is the height of a trapezoid;

(2) Parallel lines with the vertex as the waist;

(3) Parallel lines crossing a vertex as diagonal lines;

(4) extend the intersection of the two waists;

(5) Connect one vertex of the upper sole with the midpoint of the other waist, and extend the intersection with the extension line of the lower sole.

Auxiliary lines commonly used in trapezoid are as follows:

5. When we encounter problems about the midpoint, we often consider constructing the midline or using the "double long midline method".

6. Solve the problem of folding, and grasp the key of "the overlapping figure before and after folding is symmetrical about the straight line where the crease is located".

7. The beauty of judging "bisymmetric figure": an axisymmetric figure, after drawing an axis of symmetry, is also a central symmetrical figure if it can draw another axis of symmetry perpendicular to it, and the vertical foot is the center of symmetry; If you can't draw another axis of symmetry perpendicular to it, then this symmetrical figure must not be a central symmetrical figure.

8. To find the area of a special figure, it is usually necessary to add auxiliary lines to convert it into a standard figure. There are two main conversion methods: "cut" and "fill".

9. Among many theorems, it is necessary to strictly distinguish whether there is an inverse theorem. For example, there is no inverse theorem for the theorem that parallel lines bisect line segments.

Typical case analysis

Example 1 If the sum of the inner angles of a convex polygon is equal to the sum of its outer angles, then the number of its sides is _ _ _ _ _.

Analysis: Let the number of sides of this convex polygon be n, and list the equations and get the results according to the formula of the sum of the inner angles of the polygon and the inference that the sum of the outer angles is equal to 3600.

(n - 2)？ 6? 1 1800 =3600.

The solution is n=4.

Example 2 The following patterns are both central and axial symmetry ()

A.B. C. D。

Analysis: according to article 7 of "method summary", it is easy to choose a.