1. As shown in figure 1, AD is the center line, E and F are points on the extension lines of AD and AD respectively, and BF and CE are connected. The following statements: ① CE = BF; ② The areas of △ Abd and △ACD are equal; ③BF∑CE； ④△BDF?△CDE .. The correct one is ().
1。
2. As shown in Figure 2, the following conclusion is wrong ().
A.△ABE?△ACD b .△ABD?△ACE C .∠DAE = 40d .∠C = 30
3. As shown in Figure 3, if △AB=AC, AB=AC, D is the midpoint of BC, DE⊥AB is in E, and DF⊥AC is in F, then there is congruent triangles () in the figure.
A.5 to B.4 to C.3 to d.2.
4. Fold a rectangular piece of paper as shown in Figure 4.
Is a crease, then the degree is ()
60-75 AD
5. According to the following known conditions, the only thing that can draw △ABC is ()
A.AB=3，BC=4，CA=8 B.AB=4，BC=3，∠A=30
C.∠A = 60°，∠B = 45°，AB = 4d .∠C = 90°，AB = 6°
6. The following proposition is true ()
A. congruent triangles is of equal height. The median lines of congruent triangles are equal.
C the angular bisectors of congruent triangles are equal. D the bisectors of the corresponding angles in congruent triangles are equal.
7. As shown in Figure 5, in △ABC, ∠A:∠B:∠C=3:5: 10, while △ MNC △ ABC, ∠ BCM: ∠ BCN is equal to ().
1:2 b . 1:3 c . 2:3d . 1:4
8. As shown in Figure 6, the lengths of the three sides AB, BC and CA of △ABC are 20, 30 and 40 respectively. If the three bisectors divide △ABC into three triangles, then S △ ABO s △ BCO s △ Cao is equal to () A. 1 1 65438+.
9. As shown in Figure 7, from the following four conditions: ①BC = b'c, ②AC = a'c, ③∞ ∑∑∑∑∑∑∑.
1。
10. As shown in Figure 8, △ABE and △ADC are formed by △ABC folded along AB and AC edges 180 respectively. If ∠ 1: ∠ 2: ∠ 3 = 28: 5: 3, then ∠
Second, fill in the blanks
1 1. As shown in Figure 9, AB and CD intersect at point O, and AD = CB. Please add a condition to make △ aod △ cob. Your supplementary condition is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
12. As shown in figure 10, AC and BD intersect at point O, AC = BD, AB = CD, and write two pairs of equal angles _ _ _ _.
13. As shown in figure 1 1, when △ABC, ∠ C = 90, AD bisects ∠BAC, AB = 5, and CD = 2, the area of △ABD is _ _ _ _ _ _.
14. As shown in figure 12, the straight line AE∨BD and point C are on BD. If AE = 4, BD = 8 and the area of △ABD is 16, the area of is _ _ _ _ _.
15. In △ABC, ∠ c = 90, BC=4CM, in D, ∠ the bisector of ∠BAC intersects BC, and BD \u DC = 5 \u 3, then the distance from D to AB is _ _ _ _ _ _ _ _ _.
16. As shown in figure 13, △ABC is an equilateral triangle with DE=BC, and triangles with D and E as two vertices are made in different positions.
Form, so that triangles are congruent with △ABC, so that at most _ _ _ triangles can be drawn.
17. As shown in figure 14, they are the heights of acute triangle and the middle edge of acute triangle respectively. If yes, please supplement the condition _ _ _ _ _ _. (Fill in only one condition you think is appropriate)
18. As shown in figure 14, if the height correspondence between two sides of two triangles and one of them is equal, then the relationship between the angles opposite to the third side of these two triangles is _ _ _ _ _ _ _ _ _.
19. As shown in figure 15, it is known that in the middle, it is equally divided, and now, if, then the circumference is. Figure 16
20. In the math activity class, Xiaoming asked such a question: ∠B=∠C=90, E is the midpoint of BC, and DE bisects ∠ADC, ∠CED=35, as shown in figure 16. What is the degree of ∠EAB? Everyone had a heated discussion and exchange, and Xiaoying was the first to get the correct answer, which was _ _ _ _ _.
Third, think hard.
2 1. Please draw ∠ POQ = 60 with a triangle, compasses or protractor, cut OA = 50 mm on its side OP, cut OB = 70 mm on OQ, connect AB, draw the bisector of ∠AOB and AB at point C, and measure the length of AC and OC.
22. as shown in figure 17, where ∠ b = ∠B=∠C, d, e and f are on, and respectively.
Verification:
Proof: ∫∠dec =∠b+∠bde (),
Also ≈def =∠b (known),
∴∠ _ _ _ = ∞ _ _ _ _ _ (equal property).
At △EBD and △FCE,
∞ _ _ _ _ _ =∞ _ _ _ _ _ (authentication),
_ _ _ _ _ _ = _ _ _ _ _ (known),
∠ b =∠ c (known),
∴ ( ).
∴ED=EF()。
23. As shown in Figure 18, O is the wharf, two lighthouses A and B are equidistant from the wharf, and OA and OB are the coastlines. A ship left the dock and planned to sail along the bisector of ∠AOB. During the voyage, the distance between the ship and the lighthouses A and B was measured to be equal. Did the ship deviate from the course at this time? Draw a picture and explain your reasons.
24. As shown in Figure 19, fold △ABC paper along DE. When the point a falls within the quadrilateral BCDE,
(1) Write a pair of congruent triangles in the diagram and write all their corresponding angles;
(2) Let the number of times be x, and the number of times be ∠ 1 and ∠2, respectively. (represented by an algebraic expression containing x or y)
(3) There is a quantitative relationship between ∠ a and ∠ 1+∠2. Please find out this rule.
25. As shown in Figure 20, there is a winding road in the park, and there is a small stone bench at every place, which is the midpoint of the park. Are three small stone stools in a straight line? State the reasons for your inference.
26. As shown in Figure 2 1, five equivalence relations are given: ① ② ③ ④.
Please take two of them as the conditions and one of the other three as the conclusion to derive a correct one.
Conclusion (write only one case) and prove it.
Known:
Verification:
Prove:
27. as shown in figure 22, OM=ON, OD=OE, DN and EM intersect at point C.
Prove that point C is on the bisector of ∠AOB.
28.( 1) As shown in Figure 23( 1), the sides of a square are made into squares and squares respectively.
, contact, try to judge the relationship with the region and explain the reasons.
(2) The garden path is winding and secluded, as shown in Figure 23(2), which is paved with white square marble and black triangular marble. It is known that the sum of all squares in the middle is square meters and the sum of all triangles in the inner circle.
It's square meters. How many square meters is this path?