5. In △ABC, CE⊥AB is in E, CAD = ∠ Cab is outside △ABC, CF⊥AD is on C, and the extension line of intersection AD is in F, and ∠ FDC = ∠ B. Verification: BE = DF.
8.(6 points) Known: As shown in the figure, △ABC is an equilateral triangle. Take point D on BC side and point E on AC side extension line, so that DE = AD. Verification: BD = CE.
9. Decomposition factor ①
②
10. Calculation
1 1. As shown in the figure, AD‖BC, AD = CB and AE = CF are known. Proof: df = be.
14. As shown in the figure, in △ABC, ∠ BAC = 900, AB = AC, BD is the center line, AF⊥BD, F is the vertical foot, and the extension line of the parallel line intersecting point AB and point C intersects point E. ..
Verification: (1) ≈1= ∠ 2; (2)AB=2CE
16. In the right angle △ABC, the bisector of acute angle C intersects with the opposite side of E, and the high AD on the hypotenuse intersects with O, and ‖ AB ‖ F is drawn from ‖ CB through O, which proves that AE = BF.
17. Decomposition factor ①
②
③
18, calculation
19. Known: as shown in figure AB = AC, ∠ B = ∠ C.
Verification: am = an
22. Known as the figure: AB = AD, AC = AE, ∠ 1 = ∠ 2, guess the relationship between ∠ 1 and ∠3, and prove your own guess.
25. Calculation
26. Calculation
, in which
27. As shown in the figure, in △ABC, the vertical line DE of AB side passes through BC to E, and the vertical foot is D. If BC = 7 cm, AC = 4.
Cm, find the circumference of △ACE.
28. As we all know, as shown in figure bd⊥cd ac⊥cd, M is the midpoint of AB, connecting CM and extending BD to E ... Proof: AC = be.
29.(7 points) As shown in Figure (5), point C is on line AB and △ACM and △CBN are equilateral triangles. Try to judge the shape of △PQC and prove your conclusion.
33. It is known that in △ABC, ∠ ACB = 900 and ∠ B = 600,
Verification:
BC=
A
B
35. (8 points in this question) As shown in the figure, AD is the center line of BC side in △ABC, and ∠ADC is the acute angle. Fold △ADC along the straight line AD, and point C falls on the position of point E. Try to guess the positional relationship between the straight line be and the straight line DA, and prove your guess.
36.(6 points) It is known that A2+B2- 10a-6B+34 = 0, so.
The value.
37.(7 points) It is known that AC⊥BC, AD⊥BD, AD=BC, CE⊥AB, DF⊥AB, and the vertical scales are E and F respectively.
Proof: ce = df.
38.(6 points) Students play billiards and want to hit the ball to A,
Make the ball bounce back and hit the MN on the edge of the table and then hit the colored ball B. Please point out the location where the cue ball hits MN on the map to achieve the goal. (Any writing method will not leave traces of drawing.)
42. It is known that when △AB=AC, AB=AC, D is on AC, BD = BC = AD, the degree of ∠ C can be found.