1. As shown in figure (1), in △ABC, AD is the angular bisector and AE⊥BC is at point E. 。
(1) If ∠ C = 80 and ∠ B = 50, the number of times to find ∠DAE. (2) If ∠ c > ∠B, verification: ∠DAE= 0.5 (∠C-∠B).
(3) As shown in Figure (2), if point A moves to A? Department, a? E⊥BC is at point E, and at this time ∠DAE becomes ∠DA? E. Is the conclusion in [2] correct? Why? (4) Change AE in the figure to a straight line perpendicular to AD. Is ∠DAE= 0.5(∠C-∠B) in this diagram?
2. As shown in the figure, AM and CM are divided into ∠BAD and ∠BCD respectively. (1) Verification: ∠ M = 0.5 (∠B+∠D). (2) E is on the extension line of BA, and the bisector of ∠DAE and ∠ BCD intersects at point n (.
3. As shown in the figure, BH and CH share ∠ABC and ∠ACB respectively, BP and CP share ∠DBC and ∠ECB respectively, and BH and PC intersect at G point, which proves that: (1) ∠ HBP = ∠ HCP = 90; ⑵∠G = 0.5∠A； ⑶ Explore the quantitative relationship between ∠BHC and ∠A, ∠P and ∠A respectively.
4.( 1) As shown in figure 1, ∠ xOy = 90, point A and point B move on rays Ox and oy respectively, BE is the bisector of ∠ABy, and the reverse extension line of BE intersects with the bisector of ∠OAB at point C. Could you tell me the bisector of ∠ACB? If it does not change, please provide evidence; If it changes with the movement of point A and point B, request the change range. (2) As shown in Figure 2, if BC and AC are bisectors of ∠ABx and ∠BAy respectively, then ask: Did the degree of ∠C change during the movement of B and A on Ox and Oy? If it is unchanged, find its value; If there are any changes, explain the reasons. (3) As shown in Figure 3, if BC and AC are bisectors of ∠BAO respectively, Q: Does the degree of ∠C change when B and A move on Ox and Oy? (4) If ∠ AOB = 70 in the figure, what is the degree of ∠C in the above figure?
5. As shown in the figure, in the plane rectangular coordinate system, point A (-3,2), point B (2,0) and point C are on the negative semi-axis of the X axis. (1) Fold △∠BCQ along the X axis, so that point A falls on point D, write the coordinates of point D, and find the extension line of AD. (2) DC makes AB intersect with E, EF divides equally ∠AED, if∞. If it does not change, find its degree, if it changes, find its range of change.
6. As shown in Figure (1), in a pair of triangular plates, the right-angled side AC with an angle of 45 is on the Y axis, the hypotenuse AB intersects with the X axis at point G, the vertex of the triangular plate with an angle of 30 coincides with point A, and the right-angled side AE and the hypotenuse AD intersect with the X axis at points F and H respectively.
(1) If AB∑ED, find the degree of ∠AHO;
(2) As shown in Figure 2, the triangle ADE is rotated around point A, during which the bisector GM of ∠AGH and HM of ∠AHF intersect at point M, and the bisector on of ∠COF and the bisector FN of ∠OFE intersect at point N. 。
(1) when ∠ AHO = 60, the number of times to find ∠M;
② Has the degree of ∠ n+∠ m changed? If so, find out the scope of change; If not, please explain why.
7. As shown in the figure: In the rectangular coordinate system, B(b, 0), C(0, c) are known, and |b+3|+(2c-8)2=0.
(1) Find the coordinates of b and c; (2) Points A and D are points in the second quadrant, points M and N are points on the negative semi-axis of X axis and Y axis respectively, and the straight lines where ∠ABM=∠CBO, CD∨AB, MC and NB are located intersect AB and CD at E and F respectively, if ∠ MEA = 70, ∠ CF.
(3) As shown in the figure: AB∨CD, Q is the moving point on CD, CP bisects ∠DCB, BQ and CP intersect at point P, and the verification value remains unchanged; Find this value.
As shown in the figure, both points A and B start from the origin O at the same time. Point A moves in the negative direction of X axis with unit length per second, and point B moves in the positive direction of Y axis with unit length per second. (1) as shown in figure 1, if |a+2b-5|+(2a-b)2=0, try to find out 65433 respectively.
(2) As shown in Figure 2, extend BA to E, and make a ray BF in ∠ABO, and the X axis intersects with point C. If the bisector of the intersection point G of ∠EAC, ∠FCA and ∠BGC, the intersection point G is the vertical line of BE, and the vertical foot is H, then ∠AGH, ∞? Please write your conclusion and prove it;
(3) As shown in Figure 3, the straight line passing through point A and point O intersects point N, and the extension line of AB intersects point M. If ∠MAN=∠NOB, ∠ bao-∠ n = m, try to find the degree of ∠AMO.
9. As shown in the figure: Fold the △ABC paper into Figure ① along the DE, and at this time point A falls within the quadrilateral BCDE, then there is a quantitative relationship between ∠A and ∠ 1 and ∠2. Please find out this quantitative relationship and explain the reasons.
(1) If it is folded into Figure ② or Figure ③, that is, when point A falls on BE or CD, write ∠A and ∠ 2 respectively; The relationship between ∠A and ∠ 1; (No proof required)
(2) If folded into Figure ④, write the relationship between ∠A and ∠ 1 and ∠2; (No proof required)
(3) Write the relationship between ∠A and ∠ 1 and ∠2 if it is folded into Figure ⑤. (No proof required)
10. As shown in figure 1, in the plane rectangular coordinate system, △AOB is a right triangle, ∠ AOB = 90, and the hypotenuse AB intersects the Y axis at point C. 。
(1) If ∠A=∠AOC, verify: ∠ B = ∠ BOC;
(2) As shown in Figure 2, extend the intersection of AB and X axis at point E, and the intersection with O is OD⊥AB. If ∠DOB=∠EOB, ∠A=∠E, find the degree of ∠A;
(3) As shown in Figure 3, the bisector bisecting ∠AOM and ∠BCO and the extension line of FO intersect at point P, ∠ A = 40. When △ABO rotates around point O (hypotenuse AB and positive half axis of Y axis always intersect at point C), does the degree of △ P change? If it does not change, find its degree; If so, please explain why. (12)
1 1. It is known that A(a, 0) and B(b, 0), point C is on the positive semi-axis of Y axis, and │a+4│+(b-2)2=0.
(1) if S△ABC =6, find the coordinates of point C.
(2) Move point C to the right, so that OC divides ∠ACB equally, point P is the point where point B moves to the right on the X axis, and point PQ⊥OC is at point Q. When ∠ ABC-∠ BAC = 60, find the number of times of ∠APQ.
(3) Under the condition of (2), translate the line segment AC to make it pass through the line segment EF of point P, so that the angular bisector of ∠APE and the extension line of OC intersect at point M. When point P moves on the X axis, find the value of ∠M-0.5∠ABC.