1. As shown in the figure, in the isosceles triangle ABC, the top angle ∠A=36°, BD biseminates ∠ABC, and point D is the golden segmentation point of AC . If AC=4cm, then BD=______cm (the result is retained with three significant numbers)
2. In △ABC, A1, A2, ...A5 are different points on the edge of AC. Connect BA1, and there are 3 different triangles in the figure; then connect BA2, and there are 6 different triangles in the figure; continue like this, when connecting BA5, there are ______ in the figure.
3. As shown in the figure, ∠ACD is the outer angle of △ABC, the bisector of ∠ABC and the bisector of ∠ACD intersect at point A1, the bisector of ∠A1BC and the bisector of ∠A1CD intersect at point A2,…, the bisector of ∠An-1BC and the bisector of ∠An﹣1CD intersect at point An, assuming ∠A=θ, then:
(1)∠A1=( );
(2)∠An=( ).
4. It is known that in △ABC, ∠A:∠B:∠C=1:3:5, then △ABC is a ______ triangle.
5. The ratio of the three sides of a triangle is 2:3:4 and the perimeter is 36cm. Find the length of the three sides of this triangle.
6. Complete the drawing as required and answer the question: As shown in the figure, in the triangle ABC,
(1) draw the midpoint D of the line segment BC and connect AD;
(2) draw the vertical line of BC through point A, and the foot is E;
(3) According to the above drawing, if ∠ABC=60°, then ∠BAE= _______, if BD=2cm, then BC= ______cm;
(4) Use "", "=", "" to fill in the blanks: AB+AC_______BC, according to __________.
7. In the following proposition:
(1) Rotation does not change the shape and size of the figure
(2) Axial reflection does not change the shape and size of the figure
(3) Among all the lines connecting the two points, the shortest line segment
(4) The sum of the inner corners of the triangle equals 180°
belongs to the axial ones ( )
A. 1
B. 2
C. 3
D. 4
8. It is known that the sum of two inner angles in a triangle is n°, and the maximum angle is 24° larger than the minimum angle, so the value range of n is ______.
9. The following four statements:
① If the degree ratio of three inner corners of a triangle is 2:3:4, then this triangle is acute triangle .
② "The sum of the points of two uniformly textured cube dice must be greater than 6" is an inevitable event.
③Purchase , a lottery may win.
④ It is known that an inner angle of an isosceles triangle is 40°, so the top angle of this isosceles triangle is 100°.
The correct sequence number is _______
10. Among the following judgment statements, the wrong one is ( )
A. The sum of the three inner corners of a triangle is equal to 180 degrees
B. There is only one straight line perpendicular to the known straight line
C after passing through a point outside the line. Ray AB and ray BA represent the same ray
D. There are two triangles with an inner angle equal to 60°, which must be an equilateral triangle
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1.
【Analysis】
∵AB=AC, ∠A=36° ,
∴∠ABC=∠C=72°,
BD and BD divide ∠ABC,
∴∠ABD=36°,
∴BD=AD,
∵D point is the golden segmentation point of AC,
∴BD=AD≈4×0.618≈2.47cm
【Answer】2.47
2.
【Analysis】 When
connects to BA1, there are 3 △AA1B, △ACB, and △A1CB. When
connects to BA2, there are 6 △A1B, △AA2B, △ACB, △A1A2B, △A1CB, and △A2CB.
…,
and so on. When BAn is connected to BAn, there are 6 triangles:
(n+1)(n+2)
2
2
and so on. l0 So, when connecting BA5, the different triangles have
(5+1)(5+2)
2
=21
[Answer] 21
3.
[Answer]
(1)
; (2)
4.
【Analysis】
Set ∠A=x, ∠B=3x, ∠C=5x,
∵∠A+∠B+∠C=180°,
∴x+3x+5x=180°,
Solution: x=20°,
∴∠A=20°, ∠B=60°, ∠C=100°,
∴△ABC is an obtuse triangle
【Answer】Obdominant angle
5.
【Analysis】
Set the three sides lengths to be 2x, 3x, 4x,
from the question, 2x+3x+4x=36,
3
Solution: x=4
[Answer]
Three sides are: 8cm, 12cm, 16cm
6.
[Answer]
(1) Draw with B and C respectively, and
is greater than
BC is the radius Circle, two circles intersect at F,
pass F to FD⊥BC to D, then point D is the midpoint of BC;
(2) draw the perpendicular line of BC through point A, and the drooping foot is E;
(3) ∵In △ABE, ∠ABC=60°,
∠AEB=90°,
∴∠BAE=180°﹣60°﹣30°=30°,
BD=2cm, then BC=2BD=4cm;
(4) AB+ACBC, the line segment between two points is the shortest.
7.
[Analysis]
(1) Rotation does not change the shape and size of the figure, which is correct and is axiom;
(2) Axial reflection does not change the shape and size of the figure, which is correct and is axiom;
(3) Among all the lines connecting the two points, the line segment is the shortest, correct and is axiom;
(4) The sum of the inner corners of the triangle is equal to 180°, which is correct but not an axiom.
To sum up, we can get (1) (2) (3) Correct
[Answer] C
8.
[Analysis]
Suppose the inner angle of △ABC is ∠A, ∠B, ∠A+24°, and ∠A≤∠B≤∠A+24°.
When ∠A=∠B, n=∠A+∠B, you can get n has a minimum value of 104°, that is, n≥104°.
When ∠B=∠A+24°, n=∠B+ (∠A+24°), you can get n has a maximum value of 136°, that is, n≤136°
【Answer】104°≤n≤136°
9.
【Analysis】
①If the degree ratio of three inner angles of a triangle is 2:3:4, you can get 2x+3x+4x=180°, and the solution is: x=20°,
∴Three triangles within three inner angles The degrees of angles are: 40°, 60°, 80°,
∴ This triangle is an acute triangle; therefore this option is correct;
② "The sum of the points of two uniform cube dices must be greater than 6" is an inevitable event; according to the rolling of two uniform cube dices, the sum of the two points may be less than 6, so it is a random event, so this option is wrong;
③ may win a lottery ticket; it is a random event, so this option is correct;
④ It is known that an internal angle of an isosces triangle is 40°, so the top angle of this isosces triangle is 100°, and the top angle of this triangle may also be 40°, so this option is wrong.
【Answer】①③
10.
[Analysis]
A, conforms to the inner angle and theorem of a triangle, so this option is correct and does not meet the question;
B, is the nature of a perpendicular line, so this option is correct and does not meet the question;
C, ray AB and ray BA have different end points and extension directions, so this option is wrong and meets the question;
D, conforms to the judgment of an equilateral triangle, so this option is correct and does not meet the question.
【Answer】C
