1. As shown in the figure, it is known that in △ABC, AQ=PQ, PR=PS, PR⊥AB in R, PS⊥AC in S, and there are three conclusions: ① As = AR; ②QP AR； ③△BRP?△CSP, where ().

(a) All correct (b) Only ① correct (c) Only ① and ② correct (d) Only ① and ③ correct.

2. Given a line segment, if you want to do line segment AB, make AB=, and the correct way is (line m ‖ n in the figure) ().

3. Exchange the conditions and conclusions of the following propositions, and the obtained proposition is still true ().

(a) the vertex angles are equal; (b) The corresponding angles of congruent triangles are equal.

(c) The two acute angles of a right triangle are complementary (d) If >:, >; And then >

4. As shown in the figure, fold the right-angle △ABC paper with AC = 6 cm and BC = 8 cm in half, so that point B coincides with point A, and the crease is DE, then CD is equal to ().

(A) (B) (C) (D)

5. As shown in the figure, make the following judgment or reasoning with the graph:

① As shown in Figure A, CD⊥AB and D are vertical feet, so the distance from point C to AB is equal to the distance between points C and D;

② As shown in Figure B, if AB‖CD, then ∠ B = ∠ D;

③ As shown in Figure C, if ∠ACD=∠CAB, then AD ∠ BC;

④ As shown in Figure D, if ∠ 1=∠2, ∠ D = 120, then ∠ BCD = 60. The correct number is ().

1 (B)2 (C)3 (D)4

6. As shown in the figure, BE and CF are the heights of ABC, and m is the midpoint of BC, so the triangle in the figure must be an isosceles triangle ().

2 (B)3 (C)4 (D)5。

7. As shown in the figure, AD and BE are the heights of △ABC and intersect at point F, so there is a similar triangle () in the figure.

(A)6 pairs (B)5 pairs (C)4 pairs (D)3 pairs

8. As shown in the figure, ABG, D, E, C and F are the bisectors of AG and BG respectively. Four conclusions are given below:

( 1) (2)

(3)SδEGF:SδGAB = 2:3(4)

The number of correct conclusions is ()

1 (B)2 (C)3 (D)4

Second, fill in the blanks

1. As shown in the figure, if a pair of triangular plates are stacked together so that the vertex of the right angle coincides with the O point, the degree of ∠AOC+∠DOB is degrees.

2. As shown in the figure, in △ABC, ∠ c = 90, AD is the bisector of ∠CAB in △ABC, and DE⊥AB is in E. In order to make △ ADC △ BDE, a condition needs to be added. This condition is.

3. Steel triangular frame with 2m, 5m and 6m sides respectively. Now it is required to make a similar steel triangular frame. Now there are only two steel bars 3m and 5m long. It is required to take one of them as an edge and cut two sections from the other as the other two edges. Then the length of the other two sides is.

4. As shown in the figure, the coordinates of five points A, B, C, D and E are known as (1, 2), (3,2), (4,3), (2,6) and (3,5) respectively. If point f is in the first quadrant, d, e and f are taken as.

5. On a square piece of paper with the unit of 1cm, according to the rules shown on the right, set points A 1, A2, A3, A4…, An, connection points A 1, A2, A3 to form a triangle, marked as …, and connection points An, A3 to form a triangle.

Third, answer questions.

1. Draw △DEF and △ deg on the square paper as shown in the figure (f and g cannot coincide), so △ ABC △ def ≌ deg. Can you explain why they are all equal?

2. As shown in the figure, there is a lake, and the distance between shore A and shore B cannot be directly measured. In order to get the distance between A and B, please use a goniometer and a tape measure to design two kinds of measurement schemes on the shore (draw graphs to explain the schemes respectively, and the schemes should be based on the relevant knowledge of this unit), and write the line segments to be measured for the schemes. (The measured line segment is long-term (or)

3. Measure the diameter of the small glass tube on the measuring tool CDE, CD=l0mm, DE=80mm ... If the diameter AB of the small tube faces the scale of 50mm on the measuring tool, what is the length of the diameter AB of the small tube?

4. As shown in the figure, the area of small squares in the square grid is 1, and there are △ABC and △ DFE in the grid.

(1) Are these two triangles similar? Tell your reasons;

(2) Please draw a triangle with an area of 4 and similar to △ABC with the grid point as the vertex.

5. As shown in the figure, it is known that △ABC, △DCE and △FEG are three congruent isosceles triangles, with the bases BC, CE and EG on the same straight line, AB=, BC = 1. The connection BF, AC, DC and DE intersect at points P, Q and R respectively.

(1) Are △ BFG and △FEG similar? Why?

(2) Write all triangles similar to △ABP in the diagram (without proof).

6. As shown in Figure (), it is represented by acute angle △ABC, BC >; AB & gtAC, D and E are the moving points on BC and AB respectively, connecting AD and D E.

(1) When D and E move, draw the positions of D and E in the other three pictures respectively; Draw only a set of similar triangles in Figure (); Only two groups of similar triangles are drawn in Figure (); Draw three groups of similar triangles in Figure ().

(2)BC=9，AB=8，AC=6。 Find the length of DE from Figure ().

7. In rectangular coordinate system, three points, A (-4,0), B (1 0) and C (0 0,2), are known. Please design two schemes according to the following requirements: make a straight line that does not coincide with the axis and intersects with both sides of △ABC, so that the cut triangle is similar to △ABC and the area is △AOC.

8.( 1) As shown in Figure ①, BD and CE are bisectors of the outer corner of △ABC, the passing point A is AG⊥ce af⊥bd, and the vertical feet are F and G respectively, which connect FG, extend AF and AG and intersect with the straight line BC. Verification: FG = (AB+BC

(2) If BD and CE are bisectors of the internal angle of △ABC respectively, and other conditions remain unchanged (Figure ②), what is the quantitative relationship between the three sides of the line segment FG and △ABC? Write your guess and prove it.

Reference answer:

First, multiple-choice questions:1-4: accd; 5-8: BDBC

2. Fill in the blanks:1.180; 2.∠ b = 30 degrees; 3. 1,2.5; 4.(2,8); 5. 10

Third, answer questions:

1. Omit; 2. Omission; 3.； 4. Similarity; 5. Similarity is proved by the ratio of numerical values; 6. Omit; 7. Omission; 8. Tip: Extend the extension lines where AG and AF intersect BC and the reverse extension lines at points M and N, and prove it with the center line.