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.
≈EAF = 45 ∴∠2+∠3=∠bad-∠eaf=90-45 = 45。
∵∠ 1=∠2,∴∠ 1+∠3=45 .
That is ∠GAF =∞.
FAE
.
And AG=AE, AF=AF.
∴△GAF≌
△ EAF
.
∴
girlfriend
=EF, so de+BF = ef.
(2) Use the accumulated experience and knowledge in (1) solution to solve the following problems:
As shown in Figure 2, in the right-angled trapezoidal ABCD, AD ∥ BC (AD > BC), ∠D = 90°, AD=CD= 10, E is a point on the CD, and ∠BAE = 45°, DE=4. Find the length of BE.
(3) The proof idea of analogy (1) completes the following problems: put two isosceles right triangles ABC and AFG together in the same plane, where A is the common vertex and ∠ BAC = ∠ AGF = 90. If △ABC is fixed, △AFG rotates around point A, AF and AFG.
Decomposition solution: (1) According to equivalent substitution, ∠GAF=∠FAE.
The △ GAF △ electric arc furnace is obtained by SAS.
∴GF=EF，
So the answer is: FAE；; ; △ electric arc furnace; GF；
(2) A is AG⊥BC, and the extension line of CB is at G 。
In the right-angled trapezoidal ABCD,
∵AD∥BC,∴∠C=∠D=90，
∠ CGA = 90，AD=CD，
∴ Quadrilateral AGCD is a square.
∴CG=AD= 10.
Given BAE = 45,
According to (1), be = GB+de.
Let BE=x, then BG=x-4,
∴BC= 14-x.
In Rt△BCE, ∫BE2 = BC2+CE2, that is, x2 = (14-x) 2+62.
To understand this equation, we get: x=
58
seven
.
∴BE=
58
seven
(3) Proof: As shown in the figure, rotate △ACE 90 degrees clockwise around point A to the position of △ABH.
Then CE=HB, AE=AH, ∠ abh = ∠ c = 45, and the rotation angle ∠ eah = 90.
Connect HD, in △EAD and △HAD,
AE = AH，∠HAD=∠EAH-∠FAG=45 =∠EAD，AD=AD。
∴△EAD≌△HAD，
∴DH=DE，
And < hbd = < abh+< Abd = 90,
∴BD2+HB2=DH2，
That is BD2+Ce2 = DE2 ... (11)
(1) Get ∠GAF=∠FAE by equivalent substitution between angles, then get △ GAF △ EAF by SAS, and get the answer;
(2) Let A BE AG⊥BC, pass BC and continue to G, then CG=AD= 10 can be obtained from the property of square, and then the length of BE can be obtained by Pythagorean theorem and equation;
(3) The equation is proved by using the rotation property and Pythagorean theorem.