9.(20 1 1 Yangzhou) As shown in the figure, Island C is 60 east-northeast of Island A and 45 north-northwest of Island B, so the perspective relationship between Island A and Island B and Island C is ∠ ACB = _ _ _ _ _ _
The answer is 105.
The analysis is as follows: ∫ (60+∠ cab)+(45+∠ ABC) =180, ∴ cab+∠ ABC = 75, in △ABC, ∞ is obtained.
12. As shown in the figure, in △ABC, ∠ A = 80, ∠ B = 30, CD ∠ACB, DE∨AC.
(1) Number of times to find ∠DEB;
(2) Find the degree of ∠ EDC.
The solution in △ABC (1), ∠ A = 80, ∠ B = 30,
∴∠ACB= 180 -∠A-∠B=70。
∫DE∑AC，
∴∠DEB=∠ACB=70。
(2) CD sharing ∵ ∠ACB,
∴∠DCE=∠ACB=35。
∠∠DEB =∠DCE+∠EDC，
∴∠EDC=70 -35 =35。
13. as shown in the figure, ∠ 1 = ∠ 2, CF⊥AB in f, DE⊥AB in e, verification: fg∨BC. (Please complete the certificate).
Prove that ∵CF⊥AB, DE⊥AB (known),
∴ED∥FC()。
∴∠ 1=∠BCF()。
And ≈ 1 =∠2 (known),
∴∠ 2 =∠ BCF (equivalent substitution),
∴FG∥BC()。
The solution is in the same plane, and two straight lines perpendicular to the same straight line are parallel to each other; Two straight lines are parallel and the same angle is equal; Internal dislocation angles are equal and two straight lines are parallel.
14. As shown in the figure, the triangle ABC is known. Verification: ∠ A+∠ B+∠ C = 180.
Analysis: By drawing parallel lines, replace ∠A, ∠B and ∠C with equal angles, so that the sum of all angles is just a flat angle, and various proofs are obtained according to different auxiliary lines, as follows:
Proof method 1: As shown in Figure A, extend BC to D, and draw CE∨BA after C. 。
∵BA∨CE (drawing known),
∴∠ B = ∠ 1, ∠ a =∠∠ a =∠ 2 (two straight lines are parallel, with the same complementary angle and internal dislocation angle).
∠∠ BCD =∠ BCA +∠ 2 +∠1=180 (definition of right angle)
∴∠ A+∠ B+∠ ACB = 180 (equivalent substitution).
As shown in Figure B, draw FH∨AC and FG∨AB at any point F passing through BC. Can this method of adding auxiliary lines prove that ∠ A+∠ B+∠ C = 180? Please have a try.
Solve ∵FH∑AC,
∴∠BHF=∠A,∠ 1=∠C.
∫FG∨AB，
∴∠BHF=∠2,∠3=∠B，
∴∠2=∠A.
∫∠BFC = 180，
∴∠ 1+∠2+∠3= 180 ,
That is, ∠ A+∠ B+∠ C = 180.
15.(20 10 Yuxi) Two straight lines in the plane have intersecting and parallel positional relationships.
(1) As shown in Figure A, if AB∨CD and point P are outside AB and CD, there is ∠ B = ∠BOD. Since ∠ BOD is the external angle of △POD, ∠BOD =∠BPD+∞. If yes, explain the reasons; If not, what is the quantitative relationship between ∠BPD, ∠B and ∠D? Please prove your conclusion;
(2) In Figure B, rotate the straight line AB counterclockwise around point B for a certain angle and intersect with the straight line CD at point Q, as shown in Figure C, then what is the quantitative relationship between ∠BPD, ∠B, ∠D and ∠BQD? (No proof required)
(3) According to the conclusion of (2), find the degree of ∠ A+∠ B+∠ C+∠ D+∠ E+∠ F in Figure D.
The solution (1) does not hold, and the conclusion is ∠ bpd = ∠ b+∠ d. 。
Extend the BP intersection CD to point e,
∵AB∥CD,∴∠B=∠BED.
And < bpd = < bed+< d,
∴∠BPD=∠B+∠D.
(2) Conclusion: ∠ BPD = ∠ BQD+∠ B+∠ D.
(3) Let AC and BF meet at G point.
From the conclusion of (2): ∠ AGB = ∠ A+∠ B+∠ E.
Question 14
∠∠agb =∠CGF，∠ CGF+∠ C+∠ D+∠ F = 360，∴∠a+∠b+∠c+∞。
14. Put a pair of commonly used triangles together as shown in the figure, then ∠ADE in the figure is the degree.
2. As shown in the figure, in △ABC and △ABD, the following three conclusions are given: ① AD = BC; ②∠C =∠D； ③∠ 1=∠2。 Please choose two of them as conditions, one as a conclusion and one as a proposition.
(1) Write all the correct propositions (in the form of "",denoted by serial number):.
(2) Please choose a correct proposition to explain. The correct proposition you chose is:
Description:
3. As shown in the figure, the straight line AD and BC intersect at O, AB∥CD, ∠AOC = 95, ∠ B = 50, and find ∠ A and ∠ D. 。
4. As shown in the figure, in △ABC, the angular bisectors AD, BE and CF intersect at point H, point H is HG⊥AB, and the vertical foot is G, so ∠ ahe = ∠ CHG? Why?
5. As shown in figure 17, in △ABC, AD is the bisector of ∠BAC, DE⊥AB is in E, DF⊥AC is in F, and the area of △ABC is, AB = 20cm, AC = 8cm, so find the length of DE.
Question 5
6. As shown in the figure, it is known that the vertical foot of AB⊥CD is B, AB=DB and AC = DE. Please judge the relationship between ∠D and ∠A and explain the reasons.
Question 6
7. As shown in the figure, AD=BC, DC=AB, AE = CF Find a pair of congruent triangles in the figure and explain your reasons.
Question 7
8. As shown in the figure, it is known that M is on AB, BC=BD, MC = MD, please specify: AC = AD.
Question 8
9. As shown in the figure, in △ABC, AB=AC, and the middle line BD on the side of AC divides the circumference of △ABC into 2 1 cm.
12 cm, and find the length of each side of △ABC.
D
A
B
C
10. Given AE⊥BD, CF⊥BD and AD=BC, BE=DF, try to judge the positional relationship between AD and BC. Explain your conclusion.
1 1. As shown in the figure, ∠ ACB = ∠ BDA = 90, AD=BC, AB//CD. Trial description: ∠ 1 = ∠ 2.
12. As shown in Figure 3, AC⊥BD, AC=DC, CB=CE, please explain: De ⊥ AB.
13. As shown in the figure, AB//DE, AB=DE and BE=CF are known. Try to explain the reason of △ ABC △ def.
Xiao Ming's reasoning process is as follows:
Because AB//DE, ∠ 1=∠2,
In △ABC and △DEF,
Because BE=CF, ∠ 1=∠2, AB=DE, so △ ABC △ def (SAS).
Is Xiaoming's reasoning correct? If it is not correct, please point out the mistakes and help Xiao Ming get out of the misunderstanding of reasoning.
14. As shown in Figure 2, AC and BD intersect at point E, with AD=BC and ∠ D = ∠ C. Try to explain the reason why AC and BD are congruent.
Xiaohua's reasoning process is as follows:
At △ABD and △BAC,
Because AD=BC, AB=BA, ∠C=∠D,
so△ABD?△BAC(SSA)
So AC=BD.
3.( 10) As shown in figure 15, in △ABC, point D is on AB, BD=BE,
(1) Please add another condition to make △ bea △ BDC,
And explain the reason, you add the condition is
The reason is:
(2) Write a pair of congruent triangles in the diagram according to the conditions you added.
Only one pair of congruent triangles is needed, no other line segments are added, no other letters are marked or used, and no reason is needed.
4.( 10) Known: as shown in figure 16, Rt△ABC≌Rt△ADE, ∠ ABC =
∠ ade = 90, try to connect two lines with the point marked with letters as the endpoint.
Line segments, as shown in the figure, the two line segments you connect satisfy the equal, vertical or parallel relationship.
One, then please write it down to prove it.
1. The existing two sticks are 3 cm and 5 cm in length respectively. If you want to choose the third stick to form a triangle with the first two sticks, its length can be ().
A. 1 cm b. 2 cm c. 5 cm d. 10 cm
E
D
C
B
A
Figure 1 Figure 2
2. As shown in figure 1, AD is the height of △ABC. If BC is extended to E, CE = BC, the area of △ ABC is S 1, and the area of △ ACE is S2, then ().
a . s 1 > s2b . s 1 = s2c . s 1 < s2d。 Not sure.
2. If the three sides of a triangle are 5 and 8 respectively, the value range of is _.
3.( 10) As shown in figure 16, △ABC, the angular bisectors AD, BE and CF intersect at point H, the point passing through H is HG⊥AB, and the vertical foot is G, so ∠ ahe = ∠ CHG? Why?
Figure 17
E
D
C
B
A
G
H
F
Figure 16
4.( 10) as shown in figure 17, in △ABC, AD is the bisector of ∠BAC, DE⊥AB is in E, DF⊥AC is in F, and the area of △ABC is, AB = 20cm, AC = 8cm, and.
Fourth, broaden the exploration! (This big topic ***22 points)
1.( 10) As shown in Figure 18, in △ABC, point D is on AB, and BD=BE.
(1) Please add another condition to make △ bea △ BDC,
And explain the reason, you add the condition is
The reason is:
(2) Write a pair of congruent triangles in the diagram according to the conditions you added.
(Only one pair of congruent triangles is needed, no other line segments are added, no other letters are marked or used, and there is no need to explain why. )
2.( 12 point) (1) As shown in figure19 (1), if a right triangle XYZ is placed on △ABC, it happens that two right-angled sides XY and XZ of the triangle XYZ pass through △ABC at point B and point C respectively, and ∠ A = 30, then ∠ ABC.
(2) As shown in Figure 19②, if the position of the right triangle XYZ is changed so that the two right-angled sides XY and xz of the triangle XYZ still pass through B and C respectively, will the size of ∠ABX+∠ACX change? If yes, please give examples; If not, request the size of ∠ABX+∠ACX.
②
①
Third, answer questions.
2 1, draw two known line segments A and B (A > B) first, and then draw line segment B (A > B = A-B. 。
22, as shown in the figure, AE∨BD, ∠ 1 = 3 ∠ 2, ∠ 2 = 28. Looking for ∠ C.
(Figure 22)
23. As shown in the figure, l∨m is known, and the number of times to find ∠x and ∠y is known.
24. As shown in the figure, straight lines l 1 and l2 intersect with straight lines l3 and l4 respectively, and ∠ 1 is complementary to ∠3, ∠2 is complementary to ∠3, and ∠ 4 = 165438+.
25. As shown in the figure, it is known that ∠ C = ∠C=∠D, DB∨EC. AC and DF are parallel? Try to explain your reasons.
(Figure 25)
26. As shown in the figure, AB and AE are two rays, ∠ 2+∠ 3+∠ 4 = ∠1+∠ 2+∠ 5 =180, and find ∠65438+.
27. As shown in the figure, it is known that DB∨FG∨EC, ∠ Abd = 60, ∠ ACE = 60, and AP is the bisector of ∠BAC. Find the degree of ∠PAG
28. As shown in the figure, CD∨AB, ∠ DCB = 70, ∠ CBF = 20, ∠ EFB = 130, what is the positional relationship between straight line EF and AB, and why?
C
F
A
B
E
D
29, as shown in the figure: AB⊥BF, CD⊥BF, ∠ BAF = ∠ AFE. Please explain why ∠ DCE+∠ E = 180.
7. As shown in the figure, AB∥CD and straight line EF intersect AB and CD at points E and F respectively, and ED divides ∠BEF equally. If ∠ 1 = 72, ∠ 2 = _ _ _ _ _ _.
8. As shown in the figure, if DE∥BC, ∠ DBE = 40, ∠ EBC = 25, ∠ Bed = _ _ _ _ _ _, ∠ BDE = _ _ _ _ _ _.
9. As shown in the figure, ∠ 1=∠2, Abd, ∠ A = 105, ∠ Abd = 35, then ∠ BDE = _ _ _ _ _
10, as shown in the figure, AB∥CD, and ∠ 1 = 42, AE⊥EC in E, then ∠ 2 = _ _ _ _ _ _.
Third, answer carefully (each small question 10, ***60)
1. On the rectangular billiard table as shown in the figure, if ∠ 1 = ∠ 2 = 30, how many degrees is ∠3? What is the relationship between ∠ 1 and ∠3?
2. Write down the reasons for the following certification process.
Known: as shown in the figure, AB⊥BC in B, CD⊥BC in C, ∠ 1=∠2, verification: Be ∨ CF.
Proof: AB⊥BC in ∫b and CO⊥BC () in C.
∴∠ 1+∠3=90 ,∠2+∠4=90 ( )
∴∠ 1 and ∠3 are complementary, and ∠2 and ∠4 are complementary ()
∵∠ 1=∠2 (),
∴__________=___________( )
∴BE∥CF()。
3. As shown in the figure, it is known that AF bisects ∠BAC, DE bisects ∠BDF, and ∠ 1=∠2.
Can (1)DF∑AC be judged? Why?
(2) Can DE ∑ AF be judged? Why?
4. As shown in the figure, AB∨CD and AD∨BC are known, and the verification: ∠A=∠C, ∠ B = ∠ D. 。
5. As shown in the figure, AB∨CD, ∠ 1=∠2 is known, and verification: ∠BEF=∠EFC.
6. Given ∠ α and ∠β, make an angle with a ruler to make it equal to 2 ∠α-∠β.
Answer:
Third, 1. ∠3 = 60, ∠ 1 and ∠3 are complementary.
2. It is known that the complementary angle of vertical definition is equal to that of equal angle.
∠3 ∠4 Internal dislocation angles are equal and two straight lines are parallel.
3.( 1) DF∨AC can be determined, and it can be proved that ∠BDF=∠BAC are determined from the same angle and two straight lines are parallel.
(2) If DE∨AF can be judged and it can be proved that ∠ 1=∠BAF, it is equal to the complementary angle and the two straight lines are parallel.
4.AB∥CD,∴ ∠B+ ∠C= 180，∠A+∠D= 180
It was ∨ BC again.
∴ ∠A+∠B= 180，∠C+∠D= 180
∴ ∠B=∠D，∠A=∠C