给我答案---**17. (15 分)**
**Question Stem:**
As shown in the figure, in quadrilateral ABCD, AB // CD, ∠DAB = 90°. F is the midpoint of CD, point E is on AB, EF // AD. Given AB = 3AD, CD = 2AD. Quadrilateral EFDA is folded along EF to quadrilateral EFD′A′, such that the dihedral angle formed by plane EFD′A′ and plane EFCB is 60°.
**Sub-questions:**
(1) Prove: A′B // plane CD′F;
(2) Find the sine value of the dihedral angle formed by plane BCD′ and plane EFD′A′.
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**Chart/Diagram Description:**
* **Type:** 3D geometric figure (solid geometry diagram).
* **Main Elements:**
* **Points:** Labeled points are A, B, C, D, E, F, A', D'.
* Points A, E, B appear collinear, representing segment AB.
* Points D, F, C appear collinear, representing segment CD.
* **Initial Configuration (implied):** A plane quadrilateral ABCD is the base.
* AB is parallel to CD (AB // CD).
* Angle DAB is a right angle (∠DAB = 90°).
* Point E is on AB.
* Point F is the midpoint of CD.
* Segment EF is parallel to AD (EF // AD). This implies that ADEF is a rectangle.
* The lengths are related as AB = 3AD and CD = 2AD.
* **Transformation:** The quadrilateral EFDA is folded along the line segment EF.
* Point A is moved to A'.
* Point D is moved to D'.
* This creates a new plane EFD'A' which is lifted above the original plane.
* **Resulting 3D Figure:**
* **Base Plane (Unfolded part):** The quadrilateral EBCF remains in the original plane.
* **Visible Edges (Solid):** EB, BC, CF, EF.
* **Implied Connections:** EC, FB.
* **Folded Plane:** Quadrilateral EFD'A'.
* **Visible Edges (Solid):** A'E, A'D', D'F, EF.
* **Implied Connection (Dashed):** A'F.
* **Connecting Edges (Solid):** D'C (connects D' to C), A'B (connects A' to B).
* **Hidden/Construction Edges (Dashed):**
* AD (original side of ABCD, implied).
* AE (part of AB, implied).
* DF (part of CD, implied).
* BD' (connects B to D').
* ED' (connects E to D').
* CD' (connects C to D').
* **Relative Position and Direction:** The points A' and D' are elevated relative to the plane containing E, B, C, F. The line segment EF serves as the hinge for the fold.
* **Given Angle (Textual):** The dihedral angle between plane EFD′A′ and plane EFCB is 60°.