General principles: appropriate integration, overall grasp, grasping the outline, gradual infiltration, typical analysis, and key breakthroughs.

First, seriously study the curriculum standards (grasp the outline).

In the curriculum standard, this part of the quadrilateral requires:

(1) Explore and understand the formula of the sum of the inner and outer angles of polygons, and understand the concept of regular polygons.

Master the concepts and properties of parallelogram, rectangle, diamond, square and trapezoid, and understand the relationship between them; Understand the instability of quadrilateral.

(3) Explore and master the related properties of a parallelogram (equal sides, equal diagonal lines and equal diagonal lines) and the conditions that a quadrilateral is a parallelogram (a group of opposite sides are parallel and equal, or two groups of opposite sides are equal, or equal diagonal lines of a quadrilateral).

④ Explore and master the related properties of rectangle, diamond and square (four corners of a rectangle are right angles and diagonal lines are equal; The four sides of a diamond are equal, and the diagonals are perpendicular to each other) and the quadrilateral is a rectangle, a diamond or a square (three corners are right-angled quadrilaterals, or a parallelogram with equal diagonals is a rectangle; A quadrilateral with four equal sides, or a parallelogram with diagonal lines perpendicular to each other is a diamond).

⑤ Explore and understand the related properties of isosceles trapezoid (the two base angles of isosceles trapezoid on the same base are equal and the two diagonals are equal) and the condition that quadrilateral is isosceles trapezoid (the trapezoid with two base angles on the same base is isosceles trapezoid).

⑥ Explore and understand the center of gravity and physical meaning of line segments, rectangles, parallelograms and triangles (such as the center of gravity of a uniform wooden stick and a uniform rectangular board).

⑦ By exploring the mosaic of plane figures, we know that any triangle, quadrilateral or regular hexagon can be used to mosaic planes, and we can use these figures for simple mosaic design.

As can be seen from the above standards, the knowledge points that students need to master are important and difficult, so they should be careful when reviewing, and the general analysis is as follows:

Test center curriculum standards require knowledge and skill objectives.

Understand, understand and master flexible applications.

parallelogram

Parallelogram, rectangle, diamond,

The concept of square

Parallelogram, rectangle, diamond,

The Nature and Judgment Method of Square

trapeziform

The concept of right-angled trapezoid

The concept of isosceles trapezoid

The nature and judgment of isosceles trapezoid

In practical examinations, students are often required to flexibly use the properties and judgment methods of parallelogram, rectangle, diamond and square. )

Second, follow the exam questions (targeted).

Throughout the Guangzhou 06 and 07 senior high school entrance examination papers, this part of the content has different functions:

Examination questions in 2006: (As a mid-level question, if quadrilateral knowledge and other knowledge are put together, students will have a process of analysis, reproduction and logical thinking, instead of examining quadrilateral knowledge alone. )

10. As shown in Figure 3-①, a square board is divided into 36 congruent small squares with dotted lines, and then cut into 7 small pieces with different shapes according to real lines to make a puzzle. When this puzzle is used to make the pattern in Figure 3-②, the area of the shaded part in Figure 3-② is the area of the whole pattern ().

(This problem involves the area calculation of complex graphs) 1

23. (The full score of this short question is 12)

Fig. 8 is a schematic diagram of some streets in a certain district, in which CE vertically bisects AF, AB//DC, BC//DF.

There are only two routes to e station, 1 route is B-D-A-E,

Route 2 is B-C-F-E, please compare the distance between the two routes and give proof.

There are several ways to solve this problem, one of which is to construct a parallelogram, which is a flexible application by using its properties.

Examination questions in 2007: (As a low, medium and high question type, it is mixed with other knowledge, involving calculation and proof)

6. Observe the following four patterns, among which the axis symmetry is (*).

(A) (B) (C) (D)

(indirectly investigate the related properties of special graphics)

16. As shown in Figure 2, point O is the midpoint of AC, and the diamond ABCD with a circumference of 4cm is the diagonal.

If the length of A0 is translated in the AC direction to get a rhombus, the perimeter of the quadrangle OECF is _ * _*__cm. ..

(indirectly investigated the nature of diamonds-all four sides are equal)

25. (The full score of this short question is 12)

It is known that in Rt△ABC, AB = BC is in Rt△ADE, ad = de is connected to EC, and the midpoint of EC is taken as m, which is connected to DM and BM.

(1) If point D is on the AC side and point E is on the AB side, it does not coincide with point B, as shown in Figure 8-①,

Verification: BM=DM and BM ⊥ DM;

(2) If the △ADE in Figure 8-① rotates counterclockwise around point A by an angle less than 45, as shown in Figure 8-②, is the conclusion in (1) still valid? If not, please give a counterexample; If yes, please provide proof.

(second question, the diagonal of parallelogram is used to add auxiliary lines, which is flexible to use. )

For two consecutive years, Guangzhou senior high school entrance examination questions were analyzed, and quadrilateral knowledge points were not examined separately. As an intermediate question, understanding and mastering the content is a part of the final question, which requires students to use it flexibly as a part of pulling students away. This requires us to browse quickly, highlight the key points and break through the difficulties (how to find, discover or construct the graphic expansion problem in calculation and proof quickly). )

Third, lay a solid foundation. Scan the basic knowledge points and understand the recitation (task book) (this part of knowledge includes the understanding of parallelogram in Chapter 8 (1) and Chapter 16); Chapter 8 (2) 20- Determination of parallelogram; Nine (on) the fourth quarter of chapter 24-the center line. Under the guidance of the teacher, students can read books quickly, fill in the blanks and choose by themselves.

1, quadrilateral: the sum of the inner angles of a quadrilateral is equal to the sum of its outer angles, and the quadrilateral is easy to deform, and it has.

2

The properties and judgments of parallelogram, rectangle, diamond, square and trapezoid are all closely developed around edges, angles and diagonals.

2, parallelogram:

(1) Definition: Two sets of opposing quadrangles are called parallelograms.

(2) Properties: ① Opposite sides of parallelogram;

② Diagonal angle of parallelogram;

③ Diagonal line of parallelogram;

(4) parallelogram is a symmetrical figure, and the intersection of two diagonal lines is:

⑤ The two diagonals of the parallelogram divide the parallelogram into four regions. (Fill in "equal" or "unequal")

(3) Judgment: ① From the perspective of sides, the two groups of opposing quadrangles are parallelograms;

(2) Two groups of opposite quadrangles are parallelograms;

③ A set of quadrangles on the opposite side is a parallelogram;

From the angle of internal angle, two groups of diagonal quadrangles are parallelograms;

The quadrilateral of the diagonal is a parallelogram.

3. Rectangular:

(1) Definition: A parallelogram with internal angles is called a rectangle.

(2) Properties: ① Opposite sides of the rectangle;

(2) Four inner corners of a rectangle;

③ Diagonal line of rectangle;

(4) A rectangle is a symmetrical figure with stripes on its axis of symmetry; It is also a symmetrical figure, and the intersection of diagonals is;

⑤ The two diagonal lines of the rectangle divide the rectangle into four area triangles.

three

(3) determine:

The quadrilateral starting from the angle of the inner angle ① is a rectangle;

② Some parallelograms are rectangles;

From the angle of diagonal ③ The quadrilateral of diagonal is a rectangle;

④ Diagonal parallelogram is a rectangle.

4. Diamonds:

(1) Definition: A set of parallelograms with adjacent sides is called a diamond.

(2) Properties: ① Four sides of a diamond;

② Diagonal angle of diamond;

③ diamond diagonal;

④ The diamond is a symmetrical figure with an axis of symmetry; It is also a symmetrical figure, and the intersection of diagonals is;

⑤ The two diagonal lines of the diamond divide the diamond into four triangles with area.

(3) determine:

The quadrangle viewed from the side ① is a diamond;

② Some parallelograms are rhombic;

From the diagonal point of view, the quadrilateral of the diagonal is a diamond;

④ The diagonal parallelogram is a diamond.

5. Square:

(1) Definition: A group of rectangles with adjacent sides or a diamond with internal angles is called a square.

(2) Properties: ① Four sides of a square;

(2) Four inner corners of a square;

③ Diagonal line of the square;

④ A square is a symmetrical figure with an axis of symmetry; It is also a symmetrical figure, and the intersection of diagonals is;

⑤ The two diagonals of a square divide it into four triangles.

(3) determine:

Angle of Side+Angle ① A quadrilateral with four sides and an inner angle is a square;

② A quadrilateral with three internal angles and a group of adjacent sides is a diamond;

From the diagonal angle ③ The quadrilateral of the diagonal is a square. four

6, trapezoidal

(1) defines that the quadrangle of: is called trapezoid.

This trapezoid is called isosceles trapezoid. This trapezoid is called a right-angled trapezoid.

(2) Trapezoids are often divided into parallelogram (rectangle) and triangle to explore. Commonly used auxiliary lines are as follows:

(3) The nature of isosceles trapezoid: ① the opposite side of isosceles trapezoid;

② the internal angle of isosceles trapezoid;

③ Diagonal isosceles trapezoid.

④ An isosceles trapezoid is a symmetrical figure, and its straight line is its axis of symmetry.

⑤ area formula: or

(4) Determination of isosceles trapezoid: ① The trapezoid at the two waists is isosceles trapezoid.

② Diagonal trapezoid is isosceles trapezoid.

(3) The two internal angle trapeziums on the same base are isosceles trapeziums.

(5) Definition of trapezoid midline: The line segment of is the midline of trapezoid.

Trapezoidal midline attribute: Trapezoidal midline.

7. Basic standard practices:

(1)AB‖CD, to make the quadrilateral ABCD a parallelogram,

One more condition needs to be added:

(2) As shown in the figure, in □ABCD, EF‖BC, GH‖AB, EF and GH intersect at point O,

Then there is a parallelogram in the picture.

(3) The circumference of □ ABCD is 28cm, AC and BD meet in O, and the circumference of △OAB is 4cm longer than that of △OBC.

So AB =, BC =

(4) Given that the circumference of □ABCD is 50 and AB: BC = 4: 1, then AB =, BC =.

(5) One diagonal AC of a given rectangle ABCD is 24, and the other diagonal BD is equal to.

(6) The included angle between two diagonals of a rectangle is 60, and the sum of one diagonal and the short side is 2 1cm, so the length of this diagonal is.

(7) If the diagonal lines of the diamond are 7 and 16, the area of the diamond is .5.

(8) When the side length of a square is 5cm, its perimeter is and its area is.

(9) If the diagonal of a square is 8, then the area of this square is.

(10) As shown in the figure, CD is the middle vertical line of AB. If AC = 2 and BD = 3,

Then the perimeter of the quadrilateral ABCD is.

(1 1) As shown in the figure, trapezoidal ABCD, AD‖BC, diagonal AC, BD intersect at O, which is in the figure.

There are several pairs of triangles with equal areas.

In (12) trapezoidal ABCD, AD//BC, AB = CD = 2cm, BC = 4cm, BD bisects ∠ABC,

Then ad = and the circumference of the trapezoid is.

(13) As shown in the figure, in the trapezoidal ABCD, AD//BC, AB//DE, ∠ B = 800, ∠ C = 500,

The equal line segments in the figure are:.

(14) as shown in the figure, in the trapezoidal ABCD, AD//BC, ∠ A = 900,

Ad = ab, BD = CD, then ∠ c =.

(15) If the area of the trapezoid is 12cm 2 and the height is 3cm, the length of the midline of the trapezoid is cm.

(16) If the difference between the two bottoms of an isosceles trapezoid is equal to the length of a waist, the acute angle of this isosceles trapezoid is.

(17) (Changchun City, Jilin Province, 2006) As shown in the figure, the midpoint of each side of an arbitrary quadrilateral ABCD is e, f, g and h respectively. If the length of diagonal AC and BD are both 20cm, the perimeter of quadrilateral EFGH is ().

A.80cm B.40cm C.20cm D.10cm

(18) (Yangzhou City, Jiangsu Province, 2006) □ The diagonal of ABCD passes through point O, and the following conclusion is wrong ().

A.□ABCD is a centrosymmetric figure B. △ AOB △ COD.

C. △ AOB △BOC D. △ AOB and △ BOC have the same area.

(19) (Heilongjiang Province, 2006) As shown in the figure, in the rectangular ABCD, the intersection point P of EF‖AB, GH‖BC, EF and GH is on BD, and the quadrilateral with equal area in the figure is ().

A.3 to B.4 to C.5 to D.6.

(20) (Hainan Province in 2006) As shown in the figure, in the diamond-shaped ABCD, E, F, G and H are the midpoints of the four sides of the diamond, and EG and FH are connected at point O, then the diamond in the figure has ().

A.4 B.5 C.6 D.7

(2 1) (Tianjin, 2006) as shown in the figure, in the trapezoidal ABCD, AB//CD, the central line EF and the pair.

Angle lines AC and BD intersect at two points M and N. If EF = 18cm and Mn = 8cm, then AB.

The length of is equal to ()

A.10cm B.13cm C. 20cm D. 26cm

six

(22) (Changsha City, Hunan Province, 2006) As shown in the figure, it can be seen that in the isosceles trapezoid,

Then the circumference of this isosceles trapezoid is ()

A. 19

(23) (Fuzhou, Fujian, 2007) Among the following propositions, the wrong one is ()

A the diagonal lines of a rectangle are divided into two and equal to each other. A quadrilateral with diagonal lines perpendicular to each other is a diamond.

C. The two diagonals of the isosceles trapezoid are equal. D the distance between the midpoint on the bottom of the isosceles triangle and the waist is equal.

(24)(2007 Zhejiang Volunteer Bird) Among the following propositions, the correct one is ()

A. A set of quadrilaterals with parallel opposite sides is a parallelogram. B. A quadrilateral with right angles is a rectangle.

C. A set of parallelograms with equal adjacent sides is a diamond D. Quadrilaterals whose diagonals bisect each other vertically are squares.

(25) (Meishan, Sichuan, 2007) The false proposition in the following propositions is (). D.

A. A set of parallelograms with equal adjacent sides is a diamond B, and a set of rectangles with equal adjacent sides is a square.

C. A set of quadrangles with parallel and equal opposite sides is a parallelogram D. A set of quadrangles with equal opposite sides and a right angle is a rectangle.

(26) (Longnan, Gansu, 2007) The quadrilateral obtained by connecting the midpoints of all sides of any quadrilateral in turn must be ().

A. parallelogram b rhombus c rectangle d square

(27) (Jiaxing, Zhejiang, 2007) As shown in the figure, () is not necessarily true in the diamond-shaped ABCD.

(a) quadrilateral ABCD is a parallelogram (B)AC⊥BD (C)△ABD is an equilateral triangle (d) ∠ cab = ∠ CAD.

(28) (Chengdu, Sichuan, 2007) Among the following propositions, the correct proposition is () D.

A. Two quadrangles with equal diagonals are rectangles. Two quadrangles with perpendicular diagonal lines are diamonds.

C. A quadrilateral with two diagonal lines perpendicular to each other and equal is a square D. A quadrilateral with two diagonal lines bisecting each other is a parallelogram.

(29) (Tianjin, 2007) If ABCD, AD//BC, diagonal AC⊥BD, BD= 12c m in the trapezoid, the length of the midline in the trapezoid is equal to ().

A.7.5 cm wide 7 cm high 6.5 cm deep 6 cm

(30) (Jinhua, Zhejiang, 2007) Jinhua, a national historical and cultural city, has beautiful scenery and flowers. A flower bed in the shape of a parallelogram (pictured) in the square is planted with six colors of flowers: red, yellow, blue, green, orange and purple. If so, then the following statement is wrong () C.

A. The planting area of safflower and green flower must be equal.

B. The planting areas of purple flowers and orange flowers must be equal.

C. the planting area of safflower and blue flower must be equal.

D. the planting areas of blue flowers and yellow flowers must be equal.

(3 1) (Shaoxing, Zhejiang, 2007) As shown in the figure, in the rhombic ABCD, the diagonal AC and BD phase points O and E are the midpoint of BC, then the following formula must be established ().

A.AC=2OE B.BC=2OE

C.AD=OE D.OB=OE

(32). (Foshan, Guangdong, 2007) As shown in the figure, two cylindrical water cups with the same height, Cup A is filled with liquid and Cup B is empty. If all the liquid in Cup A is poured into Cup B, the distance between the liquid level in Cup B and the midpoint of the graph is ().

A.B.

C.D.

seven

Four, typical example analysis (breakthrough)

Selected exercises of textbooks

1. As shown in the figure, in the mouth ABCD, the line segments EF and GH drawn by point P are parallel to AB and BC respectively.

Then there are four parallelograms in the diagram. (Test students' ability to try)

2. As shown in the figure, in the rectangular ABCD, two adjacent sides AB and BC are 15cm and 25cm respectively.

The length of the bisector between the inner angle ∠BAD and the side BC at the point e.ce is \ \.

(Examining students' basic computing ability)

3. As shown in the figure, in the regular triangle ABC, D, E and F are the midpoints of three sides BC, CA and AB respectively.

In the whole figure, there are parallelogram, rhombus and isosceles trapezoid.

(Important basic drawings, often used)

4. It is known that a diagonal AC of a square ABCD is 4cm long, so its side length is and its area is. (Common formula test)

5. As shown in the figure, in the trapezoidal ABCD, AD‖BC, AD = AB, BC = BD, A =120,

Then the degrees of other internal angles of the trapezoid are respectively. (More complicated logical reasoning)

6. As shown in the figure, the diagonal of the square ABCD passes through point O, and point O is another one.

Vertex of a square a' b' c' O. If both sides of two squares are A,

Then the area of the overlapping part of two squares is. (transformed mathematical thinking, starting from special circumstances)

Please divide a rectangle into two parts with equal area in different ways.

(1) Observe the positional relationship between two divided graphs;

(2) If straight lines are used, how many such straight lines are there? What's the connection between them?

(3) If you divide a rectangle into four parts with equal area, what will you find?

(according to the nature of graphics, investigate graphics segmentation)

Review of classic test questions, skills and skills

1. As shown in the figure, point H is any point on the CD on the side of rectangular ABCD, and HM⊥AC is in M.

If HN⊥BD is in n and the lengths of two adjacent sides of the rectangle are 3 and 4 respectively, then the value of HM+HN.

Equal to (with the help of rectangle to examine the ability of mathematical transformation, including the application of area method)

2. Vertex D of diamond ABCD is DEAB in E, DFBC in F, and point E is the midpoint of AB and BC respectively.

If the length of DE is, the area of diamond ABCD is.

The problem is set in a diamond, which is actually divided into special triangles. Let's examine the judgment of equilateral triangle and special acute angle trigonometric function. ) 8

3. Find a point P on the plane where the square ABCD is located, so that the four triangles △PAB, △PBC, △PCD and △PAD are isosceles triangles, then there are () such points P ... (indirectly investigate the symmetry of the square, the nature of the vertical line in the line segment, and comprehensively consider the mathematical idea of the problem)

A. 1

4. It is known that the two diagonals of an isosceles trapezoid are perpendicular to each other, and the height of this isosceles trapezoid is 5, so the area of this isosceles trapezoid is.

(Translate the diagonal of isosceles trapezoid to convert the area of isosceles trapezoid into the area of isosceles right triangle)

Selected teaching materials for solving problems:

1. It is known that when △ABC, ∠ c = 90, the quadrilateral ABDE and AGFC are both squares.

Verification: BG = EC.

2. It is known that in the parallelogram ABCD, AD and BC are taken as positive △ADE and positive △BFC respectively, and the connecting line DB and EF intersect at point O,

It is proved that the quadrilateral DEBF is a parallelogram.

3. It is known that in equilateral △ABC, point D is the midpoint of AC, f is the midpoint of BC, and BD is equilateral △BDE.

Verification: AB=EF, quadrilateral AEBF is a rectangle.

4. Known parallelogram ABCD, AB = 2bc, point E is on the extension line of DA, AE = AD, and point F is on the extension line of AD.

Df = ad, CE crosses AB at g point, BF crosses CD at m point and CE crosses BF at h point.

It is proved that the quadrilateral GBCM is a diamond.

nine

Improvement questions: (selected from textbook review exploration questions)

1. In the parallelogram ABCD, the midpoint O of diagonal AC is a straight line EF that intersects with AD and BC at points E and F, connects BE and AF at points G, and connects EC and FD at points H. How many parallelograms are there in the figure? Why?

2. It is known that △ABC and △ADE are equilateral triangles, and CD = BF.

It is proved that the quadrilateral CDEF is a parallelogram.

3.( 1) As shown in the figure, take three sides of △ABC as equilateral △ACD, △ABE and △BCF to judge the shape of quadrilateral ADFE;

(2) Is there a parallelogram ADFE in (1)? If so, write down the conditions that △ABC should meet; If it does not exist, please explain the reason;

(3) When △ ABC meets what conditions, the quadrilateral ADFE is a rectangle?

(4) When △ ABC meets what conditions, the quadrilateral ADFE is a diamond?

(5) When △ ABC meets what conditions, the quadrilateral ADFE is a square?

4.( 1) As shown in Figure A, MN is a straight line outside ABCD, AA ′, BB ′, CC ′ and DD ′ are all perpendicular to MN, and A ′, B ′, C ′ and D ′ are vertical feet. Verification: aa ′+cc ′ = bb ′+DD ′.

(2) If the straight line MN moves upward so that the point C is on one side of the straight line and the points A, B and D are on the other side of the straight line (as shown in Figure B), what is the relationship between the vertical lines AA ′, BB ′, CC ′ and DD ′? Guess the conclusion first and then prove it.

10