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(1) As shown in Figure 1, fold the △ ABC paper along de so that point a falls at the position of point a 'inside the quadrilateral bced, and try to explain that 2 ∠ a = ∠ 1 + ∠ 2; (2) as shown in Figure 2, if the △ ABC paper is folded along de so that point a falls at the position of point a' outside the quadrilateral bced, then the equivalent relationship between ∠ A and ∠ 1, ∠ 2 is______ (no need to explain the reason); (3) as shown in Figure 3, if the quadrilateral ABCD is folded along EF, so that points a and d fall on the positions of internal points a ′, D ′ of quadrilateral BCFE, please explore the quantitative relationship between ∠ a, D, 1 and 2 at this time, write down the conclusion you found and explain the reason
(1) As shown in the figure, according to the nature of the fold, ∠ 3 = 12 (180 - 1), ∠ 4 = 12 (180 - 2), ∵ a + 3 + 4 = 180 ° and ∵ a + 12 (180 - 1) + 12 (180 - 2) = 180 °, we can get 2 ∠ a = 1 + 2; (2) according to the nature of the fold, ∠ 3 = 12 (180 - 1), ∠ 4 = 12 (180
The triangle ABC is folded along De, and the point a falls inside the quadrilateral bced, ∠ 1 + ∠ 2 = 100 ° to find ∠ a
∠1=180-2∠def
∠2=180-2∠edf
∠def+∠edf=180-∠f
∴∠1+∠2=360-2(∠def+∠edf)=360-2(180-∠f)=2∠f=2∠a
As shown in the figure, fold a triangular piece of paper ABC along De, and point a falls on the inside of the quadrilateral bced. If ∠ a = 30 degrees, find ∠ 1 ＋ 2
2∠ADE+∠1=180°…… ① 2∠AED+∠2=180°…… ② ∠ADE+∠AED+∠A=180°…… ③ Please click the "adopt the answer" button below to send us a little red flower for encouragement!
RELATED INFORMATIONS
- 1. As shown in the figure, D and E are the points on △ ABC sides AB and AC respectively, de ‖ BC. If ad = BD, find the ratio of the area of triangle ade to the area of trapezoid bced
- 2. It is known that a (0, - 4), B (- 2,0) C (3,0) are the three vertices of △ ABC. A straight line L passing through the coordinate origin o intersects the line segment AB at point D, and the bisector of ∠ ADO and ∠ ABO intersects at point M?
- 3. As shown in the figure, in △ ABC, ab = AC, and points D, e and F are the midpoint of △ ABC
- 4. Given that the triangle ABC is equal to the triangle def, ab = 3, AC = 6, if the perimeter of the triangle DEF is even, find the length of EF
- 5. AE is the bisector of the angle BAC in the triangle ABC, and the angle c is greater than the angle B. if the point F D on AE is perpendicular to B C and D, then the quantitative relationship between the angle EFD and the angle c · B is? A reward of 100 yuan is offered
- 6. As shown in the figure, in △ ABC, AE bisects ∠ BAC, ∠ C ＞ B, f is the point above AE, and FD ⊥ BC is at point D. try to deduce the relationship between ∠ EFD, ∠ B and ∠ C When the point F is on the extension line of AE, as shown in Figure 2, the other conditions remain unchanged. Is the conclusion you deduced in question 1 still valid? Please give reasons.
- 7. As shown in the figure, in △ ABC, de and FG are the vertical bisectors of AB and AC, respectively. If BC = 10, what is the perimeter of △ ADF?
- 8. In triangle ABC, we know EF / / BC, | BC | = radical 2 | BF |. It is proved that the area of triangle AEF is equal to the area of trapezoidal ebcf
- 9. Given that the exterior angles CBE and BCF of DB and DC bisecting triangle ABC intersect at point D, the relationship between angle D and angle a is proved?
- 10. In △ ABC, ∠ a = 90 ° and △ ABC rotates 45 ° counterclockwise around ∠ A. if AB = 3 and BC = 5, connect CC & # 185;, and calculate the area of △ ACC & # 185
- 11. As shown in Figure 1, fold the △ ABC paper along De, so that point a falls at the position of point a 'in the quadrilateral bced. Through calculation, we know that: 2 ∠ a = ∠ 1 + ∠ 2 (1) If the △ ABC paper is folded along the de so that the point a falls on the position of the outer point a 'of the quadrilateral bced, as shown in Fig. 2, what is the relationship between ∠ A and ∠ 1, 2? Why? Please explain the reason. (2) if the quadrilateral ABCD is folded along EF so that the points a and d fall on the positions of a 'and d' inside the quadrilateral BCFE, as shown in Figure 3, can you find out the relationship between ∠ a, ∠ D, ∠ 1 and ∠ 2? (write the relation directly)
- 12. As shown in the figure △ ABC, ∠ a = 90, de bisects AB vertically, intersects BC with EAB = 20, AC = 12, finds the length of be and the area of quadrilateral Adec
- 13. The side length of equilateral triangle ABC is a, and the square defg is inscribed with △ ABC, D, E on AB, AC, G, f on BC respectively. Find the side length of square defg
- 14. If AB is 1:4 to de, the sum of the circumference of the equilateral triangle ABC and the square defg is 76 cm, what is the circumference of the triangle? AB is one side of an equilateral triangle, De is one side of a square,
- 15. It is known that: as shown in the figure, in △ ABC, ∠ BCA = 90 °, D and E are the midpoint of AC and ab respectively, and point F is on the extension line of BC, and ∠ CDF = ∠ A. (1) prove that the quadrilateral decf is a parallelogram; (2) bcab = 35, the perimeter of quadrilateral ebfd is 22, and calculate the area of quadrilateral decf. (Note: the middle line on the hypotenuse of a right triangle is equal to half of the hypotenuse.)
- 16. In △ ABC, ab = AC, ad ⊥ BC is at point D, and points E and F are the midpoint of AB and AC sides respectively. When △ ABC satisfies what conditions, the quadrilateral AEDF is a square
- 17. RT △ ABC, ∠ a = 90 °, the bisector of ∠ B intersects AC at D, the vertical line from a as BC intersects BD at e, and the bisector from D as DF ⊥ BC
- 18. In the triangle ABC, ad is the middle line on the edge of BC, ad bisects ∠ a, and makes de ⊥ AB, DF ⊥ AC through point D. prove: be = CF
- 19. It is known that: as shown in the figure, one side De of rectangular defg is on the side BC of △ ABC, the vertices g and F are on the sides AB and AC respectively, ah is the height of the side BC, ah and GF intersect at the point K, BC = 12, ah = 6, EF: GF = 1:2 are known, and the perimeter of rectangular defg is calculated
- 20. As shown in the figure, in the right angle △ ABC, CD is the height on the hypotenuse AB, ∠ BCD = 35 °, find: (1) the degree of ∠ EBC; (2) the degree of ∠ a