Chapter V Triangular Relations (1)
First, multiple-choice questions (3 points for each small question, 30 points for * * *)
1. As shown in the figure, the number of * * triangles is ().
a3 b . 4 c . 5d . 6
There are three small sticks, the length of which is as follows (cm). If connected end to end in sequence, what can be nailed into a triangle is ().
A. 10、 14、24 B. 12、 16、32 C. 16、6、4 D.8、 10、 12
3. The triangle with the condition ∠A =∠B = ∠C must be ().
A acute triangle b obtuse triangle c right triangle d arbitrary triangle
4. As shown in figure AB‖CD, AD and BC intersect at point O, ∠A=420, ∠C=580, then ∠AOB= ().
420 B.580 C.800 D. 1000
5. The following statement is wrong ()
A. The midline, angle bisector and height line of a triangle are all line segments; B The sum of the internal angles of any triangle is180;
C the degree of each internal angle in a triangle cannot be less than 500;
D triangle can be divided into acute triangle and obtuse triangle according to angle.
6. Draw the height on the side of △ABC. The following drawing method is correct ()
7. If two triangles have the following elements, they are not necessarily congruent ().
A. two angles on one side B. angles between two sides C. angles between two sides and one side D. three sides
8. As shown in the figure, in △ABC, △ ∠ACB is an obtuse angle. Let point C move to the right on ray BD, then ()
A.△ABC will first become a right triangle, then an acute triangle, not an obtuse triangle.
B.△ABC will become an acute triangle instead of an obtuse triangle.
C.△ABC will first become a right triangle and then an acute triangle.
Then change from an acute triangle to an obtuse triangle.
D.△ABC changes from an obtuse triangle to a right triangle and then to an acute angle.
Triangle, then right triangle, then obtuse triangle.
9. As shown in the figure, AB//ED, CD=BF, if △ ABC △ def, the additional condition can be ().
A.ac = ef b.ab = de c. ∠ b = ∠ e d. No need to add.
10. The following statement is incorrect ()
A. Two right-angled triangles with a hypotenuse and a right-angled side are congruent.
Two right-angled triangles with hypotenuse and an acute angle are congruent.
C. Two right-angled sides correspond to congruences of two right-angled triangles. D. Two right-angled triangles with hypotenuse correspond to congruences.
Fill in the blanks: (3 points for each small question, ***30 points)
1 1. As shown in the figure, at △ABC, ∠ ABC = 90, BD⊥AC, then the complementary angle in the figure is right.
12. If two sides of a triangle are 2 and 4 respectively, and the third side is odd, then the third side is, and if the third side is even, then the circumference of the triangle is.
On 13. △ABC, AD⊥BC in D, AD divides ∠BAC into two angles of 400 and 600, then ∠ B = _ _ _ _ _ _
The point 14.D is the midpoint of BC in △ABC. If AB=3 and AC=4, the difference between the perimeters of △ABD and △ACD is:
15. When carpenters make wooden rectangular door frames, they often need to nail wooden strips on the adjacent sides of the door. The purpose of this is that the mathematical truth involved is.
16. As shown in the figure, it is made of the same small patterns, without gaps and overlaps. Draw the small patterns that make it in the box on the right.
On 17. Rt△ABC, if the acute angle ∠ABC and the bisector of ∠CAB intersect at point O, ∠ boa = _ _ _ _.
18. As shown in the figure, it is known that ∠B=∠DEF and AB=DE. Please add a condition to make △ ABC△ def, and the condition to be added is.
19. Xiao Ming made a kite as shown in the figure, where ∠EDH=∠FDH, ED=FD=a, EH=b, then the perimeter of the quadrilateral kite is.
20. As the picture shows, there are two slides with the same length. The height AC of the left slideway is equal to the horizontal length DF of the right slideway. If ∠CBA=320, Fed =, EFD =
Iii. Answering questions (***60 points)
2 1. (8 points in this question) As shown in the figure, divide a square square with a size of 4×4 into two congruent graphs, such as graph 1. Please draw four different equal parts along the dotted line in the figure below to divide a square square with a size of 4×4 into two congruent figures.
22.( 10) As shown in the figure, DB is the height of △ABC, AE is the angular bisector, ∠BAE=260, and the degree of ∠BFE is found.
23.( 10) As shown in the figure, AB = AD, ∠ B = ∠ D, ∠ BAC = ∠ DAE, are AC and AE equal? (8 points)
Xiao Ming's thinking process is as follows:
AB=AD
∠B =∠D△ABC?△ADE AC = AE
∠BAC=∠DAE
Explain the reasons for each step.
24. (This title is 10) Do you still remember the jigsaw puzzle we learned last semester? This is an outstanding creation of our ancestors. Although there are only seven pieces, they can spell out various shapes. As shown in the picture, it is a puzzle. These seven pieces just form a square with four corners at right angles. There are three pairs of congruent triangles, such as ⊿ABN≌⊿ADN, and several pairs of congruent quadrangles.
(1) Please calculate the degree of ∠BAN according to the characteristics of congruence graph;
(2) Please write a pair of congruent quadrangles and two other pairs of congruent triangles (please write letters representing the corresponding vertices in the corresponding positions).
25. (Question 12) Drawing and discussion: As shown in the figure, it is required to draw a triangle so that it has a common vertex C with △ABC and is congruent with △ABC.
A classmate's painting method is: (1) extending BC and AC; (2) Take point D on BC extension line, make CD = BC(3) Take point E on AC extension line, make CE = AC(4) add DE to get △ dec 。
The drawing method of student B is: (1) extending AC and BC; (2) Take a point M on the extension line of BC, make cm = AC(3) take the point N on the extension line of AC, and make CN = BC(4) add MN to get △MNC.
What kind of painting is right has the following possibilities:
(1) A is drawn correctly and B is drawn incorrectly; (2) A is not drawn correctly and B is drawn correctly; (3) A and B are all drawn correctly; (4) A and B are not drawn correctly; The correct conclusion is.
This problem can also be completed as follows: (1) Measure the degree of ∠ACB with a protractor; (2) Draw a ray CP outside ∠ACB so that ∠ ACP = ∠ ACB; (3) Take point D on ray CP, so that CD = CB⑷ (4) is connected with AD, and △ADC is the triangle to be drawn. The result of this drawing can be written as △ ABC.
How many triangles can be drawn to meet the requirements of the topic? The answer is.
Please design another drawing method and draw a picture.
26.( 10) has a conical hill, as shown in the figure. To measure the distance between A and B at both ends of a conical hill, first take a point C on the flat ground that can directly reach A and B, connect it with AC, and extend it to D, so that CD=CA, BC and E, CE=CB and d E can be measured, which is the length of A and B.
The seventh grade mathematics second volume (Beijing Normal University Edition) reaches the standard to detect the problem six.
The fifth chapter triangle (b)
First, multiple-choice questions (3 points for each small question, 30 points for * * *)
There are five kinds of 1. battens, namely 12cm, 10cm, 8cm, 6cm and 4cm. The probability that any three pieces of wood can form a triangle is ().
A.B. C. D。
2. Make the following judgments: ① There is at most one obtuse angle among the three internal angles of the triangle; ② There are at least two acute angles among the three internal angles of the triangle; ③ Two triangles with internal angles of 500 and 200 must be obtuse triangles; (4) the sum of two acute angles in a right triangle is 900, and the correct judgment is ().
1。
3. Under the following conditions: ①∠A+∠B=∠C, ② ∠ A: ∠ B: ∠ C = 1: 2: 3, ③∠A = 900-∞.
(4) ∠ A = ∠ B = 12 ∠ C, and the condition for judging △ABC as a right triangle is ().
1。
4. As shown in the figure, BE⊥ac cd⊥ab, the vertical feet are D and E respectively, and be and CD intersect at point O,
∠ 1 =∠ 2. In the figure, congruent triangles * * * has ().
A. 1 pair B.2, pair C.3 and pair d.4.
As the picture shows, a classmate broke a triangular glass into three pieces, and now he is going to break the glass.
The most convenient way for a shop to match an identical piece of glass is ().
A. Go ① to B. Go ② to C. Go ③ to D. Go ① and ②.
6. The number of triangles in the picture on the right is ()
a6 b . 7 c . 8d . 9
7. If two triangles are congruent, then the following conclusion is incorrect ().
A. The corresponding sides of these two triangles are equal. The perimeters of these two triangles are equal.
C. the areas of these two triangles are equal. Both triangles are acute triangles.
8. In the following four groups of conditions, we can judge △ ABC△ A/B/C/IS ().
A.AB=A/B/，BC= B/C/，∠A=∠A/ B.∠A=∠A/，∠C=∠C/，AC= B/C/
C.∠A=∠B/, ∠B=∠C/, AB= B/C/ D.AB=A/B/, BC= B/C/, and the circumference of △ABC is equal to that of △A/B/C/.
9. In the picture below, the pattern exactly the same as the one on the left is ().
10. To measure the distance between two opposite points A and B on the river bank, first take two points C and D on the vertical line BF of AB, so that CD=BC, and then determine the vertical line DE of BF, so that A, C and E are in a straight line. As shown in the figure, it can be explained that △ EDC △ ABC gets ED=AB, so the measured length of ED is the distance of AB.
A. Nordic Airlines
Fill in the blanks: (3 points for each small question, ***30 points)
1 1. As shown in the figure, the bisector of ∠ABC and ∠ACB in △ABC passes through point O,
If ∠ BOC = 120, ∠ A = _ _ _ _ _
12. Divide an equilateral triangle into three congruent figures in three ways.
13. The lengths of the two sides of the triangle are 2 cm and 4 cm respectively. If it is known that the length of the third side is twice as long as that of one side, the circumference of the triangle is.
14.△ABC, where AD is the angular bisector and AE is high. If ∠B=500 and ∠C=700, ∠DAE=.
15. The shape of the part is shown in the figure. If ∠A=600, ∠B=200 and ∠D=300, ∠BCD=.
16. As shown in the figure, extend the midline AD of △ABC to E, make DE=AD, and connect BE, then △ ADC △ EDB, in which the judgment method used is that the positional relationship between BE and AC is
17. As shown in figure △ ABC△ def, write a set of equal angles, two sets of parallel lines and four sets of equal line segments.
18. As shown in figure 17, in △ABC and △DEF, AB=DE. When △ ABC △ def, the reason is that.
19. As shown in the figure, if two triangles are known to be congruent and the lengths of some sides and the degrees of some angles are known, then x=
20. As shown in the figure, in Rt△ABC and Rt△DEF, ∠B=∠E=900, AC=DF, AB=DE, ∠A=500, then ∠DFE=
Iii. Answering questions (***60 points)
2 1. (9 points in this question) As shown in the 8×8 square grid diagram,
There are twelve small trees. Please divide this square into four small pieces.
Each block has the same shape and size, and each block has exactly three small trees.
Do you think you can do that?
22. (This is entitled 10) As shown in the figure, in △ABC, ∠ABC=520, ∠ACB=680, CD and BE are the heights on the sides of AB and AC, respectively, and BE and CD intersect at point O, so find the degree of ∠BOC.
23. As shown in the figure, straight lines AC‖DF, C and E are on AB and DF respectively. Xiaohua wants to know whether ∠ACE and ∠DEC are complementary, but he has a protractor and only a set of triangles, so he thinks of such a method: connect CF first, and then find CF's. The following is his idea, please fill in the basis.
Xiaohua thinks this way:
Because CF and BE intersect at point o,
According to ∠ cob = ∠ eof;
And o is the midpoint of CF, then co = fo and EO = bo are known.
According to △ cob △ foe,
According to BC = ef,
According to bco = f,
Since ∠ BCO = ∠ F, according to AB ∠ DF,
Due to AB‖DF, according to ∠ACE and ∠DEC are complementary.
24. As shown in the figure, there is a right triangle △ABC, ∠C=900, AC = 10 cm, BC=5cm, a line segment △APQ=AB, P, Q moves on AC respectively, and the ray AM passes through point A and is perpendicular to AC. Point p is required to move.
25. (This title is 10) The school pole vault competition depends on whether the heights AC and BD at both ends of the crossbar AB are the same as the ground. Xiao Ming found that the lengths CE and FD of the shadows of AC and DB on the ground were the same at this time, so he concluded that the height of both ends of the wooden pole was the same as the ground. Is he right? Why?
26. (9 points in this question) We know that two triangles corresponding to only two sides and one angle are not necessarily congruent. How to deal with and arrange these three conditions to make these two triangles congruent? Would you please write out the plans (2), (3) and (4) according to the plan (1)? Can you do it?
Scheme (1): If the opposite side of this angle happens to be the major side of these two sides, then these two triangles are congruent.
Scheme (2):
Option 3:
Option 4: