给出这三道题的完整解题过程---Here is the extracted content from the image:
**Title:**
高中数学《立体几何》大题及答案解析(理)
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**Problem 1:**
**Question Stem:**
1. (2009全国卷Ⅰ) 如图, 四棱锥 S-ABCD 中, 底面 ABCD 为 矩形, SD ⊥ 底面 ABCD, AD = √2, DC = SD = 2, 点 M 在 侧棱 SC 上, ∠ABM = 60°.
**Sub-questions:**
(I) 证明: M 是侧棱 SC 的中点;
(II) 求二面角 S-AM-B 的大小。
**Mathematical Formulas/Equations:**
* AD = √2
* DC = SD = 2
* ∠ABM = 60°
**Chart/Diagram Description:**
* **Type:** 3D Geometric Figure (Pyramid S-ABCD).
* **Main Elements:**
* **Vertices:** S, A, B, C, D, M.
* **Base:** Rectangle ABCD.
* **Apex:** S.
* **Edges:** Visible edges are SA, SB, BC, AB. Hidden edges (represented by dashed lines) are SD, AD, CD.
* **Point M:** Located on the side edge SC.
* **Lines:** Lines connecting points A-B, B-C, C-D, D-A form the base. Lines connecting S to A, B, C, D form the pyramid. A line segment is drawn from B to M. A line segment is drawn from A to M (partially visible, suggesting a triangle AMB).
* **Annotations/Labels:** Vertices are labeled A, B, C, D, S, M.
* **Relative Position and Direction:**
* S is the apex, ABCD is the base.
* SD is depicted as a vertical line segment from S to the base, indicating SD ⊥ base ABCD.
* M is on SC.
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**Problem 2:**
**Question Stem:**
2. (2009全国卷Ⅱ) 如图, 直三棱柱 ABC-A₁B₁C₁ 中, AB ⊥ AC, D, E 分别为 AA₁, B₁C 的中点, DE ⊥ 平面 BCC₁.
**Sub-questions:**
(I) 证明 AB = AC.
(II) 设二面角 A-BD-C 为 60°, 求 B₁C 与平面 BCD 所成的角的大小。
**Mathematical Formulas/Equations:**
* AB ⊥ AC
* ∠A-BD-C = 60°
**Chart/Diagram Description:**
* **Type:** 3D Geometric Figure (Right Triangular Prism ABC-A₁B₁C₁).
* **Main Elements:**
* **Vertices:** A, B, C, A₁, B₁, C₁, D, E.
* **Bases:** Triangle ABC and its parallel congruent top base A₁B₁C₁.
* **Vertical Edges:** AA₁, BB₁, CC₁.
* **Points:** D is the midpoint of AA₁. E is the midpoint of B₁C.
* **Lines:** Visible edges are AB, BC, AC₁, B₁C₁, A₁B₁, AA₁, BB₁, CC₁. Dashed lines represent hidden edges/segments: AC, A₁C₁, DE, DB, DC.
* **Annotations/Labels:** Vertices are labeled A, B, C, A₁, B₁, C₁, D, E.
* **Relative Position and Direction:**
* AA₁, BB₁, CC₁ are lateral edges, perpendicular to the bases ABC and A₁B₁C₁.
* D is on AA₁.
* E is on B₁C.
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**Problem 3:**
**Question Stem:**
3. (2009卷) 如图, DC ⊥ 平面 ABC, EB // DC, AC = BC = EB = 2DC = 2, ∠ACB = 120°, P, Q 分别为 AE, AB 的中点.
**Sub-questions:**
(I) 证明: PQ // 平面 ACD;
(II) 求 AD 与 面 ABE 所成角的正弦值。
**Mathematical Formulas/Equations:**
* AC = BC = EB = 2DC = 2
* ∠ACB = 120°
**Chart/Diagram Description:**
* **Type:** 3D Geometric Figure (Composite shape, resembles a prism with a triangular base and an extended edge).
* **Main Elements:**
* **Vertices:** A, B, C, D, E, P, Q.
* **Base:** Triangle ABC.
* **Lines/Segments:**
* Visible edges: AB, BC, AC, EB, AE.
* Hidden edges (dashed lines): DC, AD, ED, PQ, AQ.
* **Points:** P is the midpoint of AE. Q is the midpoint of AB.
* **Annotations/Labels:** Vertices are labeled A, B, C, D, E, P, Q.
* **Relative Position and Direction:**
* DC is depicted as perpendicular to the plane ABC.
* EB is parallel to DC.
* P is on AE.
* Q is on AB.