As shown in figure 1, in the square ABCD, points E and F are points on the sides of DC and BC respectively, and satisfy ∠ EAF = 45, connect EF, and verify DE+BF = EF.

Feel how to solve the problem and complete the following blanks:

Rotate △ADE clockwise by 90 degrees around point A to get △ABG. At this time, AB and AD overlap, which can be obtained by rotation:

AB=AD，BG=DE，∠ 1=∠2，∠ABG=∠D=90，

∴∠ABG+∠ABF=90 +90 = 180，

Therefore, point G, point B and point F are on the same straight line.

Electric arc furnace = 45 ∴∠ 2+∠ 3 = ∠ bad ∠∠ EAF = 90 ∠∠ 45 = 45.

∵∠ 1=∠2,∴∠ 1+∠3=45 .

That is ∠ GAF = ∠ FAE.

And AG=AE, AF=AF.

∴△gaf≌△ EAF.

∴ GF =EF, so de+BF = ef.

(2) Method migration:

As shown in Figure 2, rotate Rt△ABC along the hypotenuse to get△ △ADC, where point E and point F are points on the sides of DC and BC respectively, and ∠ EAF = ∠ DAB. Try to guess the quantitative relationship between DE, BF and EF, and prove your guess.

(3) problem expansion:

As shown in Figure ③, in quadrilateral ABCD, AB=AD, E and F are points on DC and BC, respectively, and satisfy ∠EAF= ∠DAB. Try to guess what relation ∠B and ∠D satisfy, so that DE+BF = EF. Please write your guess directly (no need to explain the reason).

Test center: the nature of the square; Congruent triangles's judgment and nature; Folding transformation (folding problem); The essence of rotation.

Analysis: (1) Get ∠GAF=∠FAE by equivalent substitution between angles, then get △ GAF △ EAF by SAS, and get the answer;

(2) Let ∠4=∠ 1, use the known ∠GAF=∠FAE, and then prove △ AGF △ AEF to get the answer;

(3) According to the relationship between the angles, as long as ∠ b+∠ d = 180, we can get the congruence of the triangle and get the answer.

Solution: Solution: (1) Replace by equivalence, ∠GAF=∠FAE.

The △ GAF △ electric arc furnace is obtained by SAS.

∴GF=EF，

So the answer is: FAE；; ; △ electric arc furnace; GF；

(2) Proof: Generalize CF, as ∠4=∠ 1,

∵ Turn Rt△ABC along the hypotenuse to get △ADC, and point E and point F are points on the sides of DC and BC respectively ∠EAF= ∠DAB.

∴∠ 1+∠2=∠3+∠5,

∠2+∠3=∠ 1+∠5,

∵∠4=∠ 1,

∴∠2+∠3=∠4+∠5,

∴∠GAF=∠FAE，

∠∠4 =∠ 1，∠ABG=∠ADE，AB=AD，

∴△AGB≌△AED，

∴AG=AE,BG=DE，

AF = AF，∠GAF=∠FAE，

∴△AGF≌△AEF，

∴GF=EF，

∴de+bf=ef；

(3) When ∠B and ∠D satisfy ∠ b+∠ d = 180, DE+BF = ef.

Comments: This topic mainly examines the nature of congruent triangles's judgment, folding and rotation transformation. According to the meaning of the question, it is the key to solve the problem to make an angle equal to a known angle and find it by triangle congruence.