1.AD is parallel to BC, the angle A = 90 E is a point on AB, and the triangle DEC is an isosceles triangle (that is, the angle DEC = 90 and DE=CD).

(1) Try to explain why AD=BE.

(2) If AB = 7 and BC = 4, find the area of quadrilateral ABCD.

2. in rt △ ABC, ∠ c = 90, CD is the height AB on the hypotenuse, AC=20, BC= 15.

(1). Take E and F on AC and BC respectively (E and F at least coincide with A or B or C) to make △DEF and △ABC similar. (Write down the main points of painting and proof)

(2) Take e and f on AC and BC respectively (which cannot coincide with A, B and C) to make △DEF and △ABC similar. (Write down the main points of painting and proof)

3. In △abc,1/(a+b)+1/(b+c) = 3/(a+b+c) find ∠ b Note that abc is the corresponding side of the angle.

4. It is known that there is a point P in the equilateral triangle ABC.

AP=3，BP=4，CP=5

Find the degree of ∠APB

fill (up) a vacancy

1, (1) congruent triangles's _ _ _ _ _ and _ _ _ _ are equal;

(2) The methods for judging the congruence of two triangles are: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _;

The other two methods to judge the congruence of right triangle can also be used: _ _ _ _ _ _ _;

(3) As shown in the right figure, it is known that AB=DE, ∠B=∠E,

If you want to make △ ABC△ def, then you need another condition.

This condition can be: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

This condition can also be: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

(4) As shown in the right figure, it is known that ∠ b = ∠ d = 90, and another condition is needed to make △ ABC △ Abd.

This condition can be: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

This condition can also be: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

This situation can also be _ _ _ _ _ _ _ _ _ _ _ for the following reasons: _ _ _ _ _ _ _ _ _ _ _;

2. As shown in Figure 5, ⊿ABC≌⊿ADE, if ∠ B = 40, ∠ EAB = 80,

∠ C = 45, then ∠EAC=, ∠D=, ∠DAC=.

3. As shown in Figure 6, if AB=CD and AD=BC are known, then.

4. As shown in Figure 7, it is known that ∠ 1=∠2, AB⊥AC, BD⊥CD, then congruent triangles in the figure has _ _ _ _ _ _ _ _ _ _;

5. As shown in Figure 8, if AO=OB, ∠ 1=∠2, plus conditions, there is Δ AOC ≌ΔBOC.

6. As shown in Figure 9, AE=BF, AD‖BC and AD=BC, then there are δ ADF ≌ and DF=.

7. As shown in figure 10, in ABC and DEF, if AB=DE and BE=CF, just add ∞ =∞.

Or ‖, Δ ABC≌Δdef can prove it.

8. As shown in figure ∠B=∠DEF and AB=DE, it is necessary to explain △ ABC △ def.

(1) If it is based on "ASA", it still lacks conditions.

(2) If it is based on "AAS", it still lacks conditions.

(3) If it is based on "SAS", the conditions are still lacking.

Second, multiple choice questions

1. The correct one in the following propositions is ().

① The corresponding sides of congruent triangles are equal; ② Three angles correspond to the coincidence of two equal triangles;

(3) Three sides correspond to the coincidence of two equal triangles; (4) There are two triangles with equal sides.

A.4 B，3 C，2 D， 1。

2. As shown in the figure, it is known that AB=CD and AD=BC, then congruent triangles * * * in the figure has ().

A.2 to b, 3 to c, 4 to d, 5 to.

3. Of the two triangles that meet the following conditions, the one that is not necessarily congruent is ().

(a) Two sides and an angle are equal; (b) tripartite equality.

(c) One side of the two corners is equal; (d) There are two right triangles with equal sides.

3. Conditions that can make two right triangles congruent ()

(a) Two right-angled sides are equal; (b) An acute angle is equal to.

(c) The two acute angles are equal; (d) The hypotenuse is equal.

4. Given △ ABC △ def, ∠ A = 70, ∠ E = 30, the degree of ∠F is ().

80 (B) 70 (C) 30 (D) 100

5. For the following groups of conditions, the group that cannot determine △△ is ().

(A)A =∠A′，B =∠B′，AB = A′B′

(B)A =∠A′，AB = A′B′，AC = A′C′

(C)A =∠A′，AB = A′B′，BC = B′C′

(D)AB = A′B′，AC = A′C′，BC = B′C′

6. As shown in the figure, △ ABC △ CDA, and AB=CD, the following conclusion is wrong ().

(A)DAC =∠BCA(B)AC = CA D

(C)D =∠B(D)AC = BC

7. As shown in the figure, D is on AB, E is on AC, ∠B=∠C,

In the following conditions, it cannot be determined that △ Abe△ ACD is ().

(A)AD=AE (B)AB=AC

(C)BE=CD (D)∠AEB=∠ADC

Three. Drawing: 1. Copy the triangle below with compasses and straightedge (drawing traces must be kept).

2. The following figure shows three equilateral triangles. Please divide them into two, three and four congruent triangles:

Fourth, the proof questions

1. As shown in the figure on the right, it is known that AB=AD and AC share ∠BAD equally. Verification: BC =DC.

2. It is known that points A, C, B and D are on the same straight line, AC=BD, ∠ M = ∠ N = 90, AM=CN.

Verification: MB‖ND

3. As shown in the right figure, AB=AD, ∠ AB=AD = ∠ CAE, AC=AE, and verification: AB=AD.

4. Known: As shown in the figure, AB = CD, AB ‖ DC. Verification: AD‖BC, AD = BC.

5. AB=AC and DB = DC are known. F is a point on the extension line of AD.

Proof: (1) ∠ Abd = ∠ ACD (2) BF = CF.

6. It is known that AO divides ∠EAD and ∠EOD equally, as shown in the figure.

Verification: ①△AOE?△aod2eb = DC

7. As shown in the figure, there are two points A and B in a small reservoir, and the distance between A and B cannot be measured directly. The method is as follows: take a point that can reach points C of A and B at the same time, connect AC and extend it to D, so that AC = DC；; In the same way, connect BC and extend it to E, so BC = EC. In this way, as long as the length of CD is measured, the distance between A and B can be obtained. Why? According to the above description, please draw a picture and write what is known, verified and proved.

There is the following statement about congruent triangles:

① congruent triangles has the same shape and size.

② The corresponding sides of congruent triangles are equal.

③ The angles corresponding to congruent triangles are equal.

④ The circumference and area of congruent triangles are equal.

The correct statement is ()

A.①③④

B.①②④

C.②③④

D.①②③④

, △ABC and △DEF are congruent triangles, if AB=DE, ∠ B = 50, ∠ C = 70, ∠ E = 50, then ∠D is _ _ _.

As shown in the figure, △ ABC △ bad, A and B, C and D are corresponding vertices, AB=6, BD=5, AD=4, then BC=____

3. It is known that when △ABC and ∠ C = 90, AD bisects ∠A and intersects with BC at point D. If BC=8 and BD=5, the distance from D to AB is _ _ _ _ _.

4. As shown in figure ∠ 1=∠2, in order to make △ ABC △ ADC, a condition _ _ needs to be added.

5. As shown in the figure, in the right triangle ABC, the distance from the O point to the three sides of the triangle is equal, then ∠AOB=___.

Second, multiple-choice questions (6 points for each question, ***30 points)

6. In △ ABC, d and e are points on AC and BC respectively. If △ ADB △ EDB △ EDC, the degree of △ C is ().

A. 15

7. As shown in the figure, AB=A 1B 1, ∠A=∠A 1, and it is necessary to () make △ ABC△ a1b1.

A.∠ b = ∠ b 1b。 ∠ c = ∠ c 1c。 AC = a 1c 1d。 All the above answers are acceptable.

8. As shown in the figure, it is known that in △ABC, DF=FE, BD=CE, AF⊥BC, and the vertical foot is F, then congruent triangles * * * in the figure has () pairs.

A.5 to B.4 to C.3 to d.2.

9. If two triangles have two sides and an angle corresponding to each other, then two triangles ().

A. it must be congruent. B. it must be congruent. C. it may or may not be congruent. D. none of the above.

10. As shown in the figure, given AD‖BC and AD=BC, the following conclusion is correct ().

( 1)AB=CD

(2)B =∠D

(3)∠ 1=∠2

(4)∠B+∠DCB= 180

A.4 B.3 C.2 D. 1

Three. Problem solving (each question 10, ***40)

1 1. As shown in the figure, BD=CD, BF⊥AC, CE⊥AB are known, which proves that D is on the bisector of ∠BAC.

12. As shown in the figure, points D and E are known to be on BC, AB=AC, AH⊥BC is in H, ∠DAH=∠EAH, and verification: BD=CE.

13. As shown in the figure, AB=CD, DE⊥AC, BF⊥AC, E, F are vertical scales, DE=BF, and verification: AB‖CD.

14. In quadrilateral ABCD, if the bisector AE of AD‖BC and ∠DAB intersects with CD and E, then BE connected, and BE happens to be the bisector ∠ABC. Find the relationship between the length of AB and the length of AD+BC and prove your conclusion.

15. Known: As shown in the figure, in △ABC, ∠ ACB = 90, AC=BC, AE is the center line on the side of BC, CD⊥AE is in F, and CD=AE.

(1) If BD is connected, find ∠ DBC;

(2) If AC= 12cm, find the length of BD.

16. It is known that in equilateral △ABC, D and E are points on AC and BC respectively, BD and AE intersect at N point, BM⊥AE is in M, if AD=CE.

(1) verification: △ Abd △ AEC

(2) Verification:

That's all. Some of them may be out of line. Pick something to do.