请用初中知识求解第13题---**Extracted Content:** **Question Number:** 13 **Question Stem:** 如图, 在四边形 ABCD 中, AG ⊥ BC, ∠CBD = 30°, AB || CD, BG = 2√5, CG = √3, AG = 4, E 为四边形对角线 BD 上一点, F 为 CD 边上一点, 且 BE = CF, 连接 AE、AF, 则 AE + AF 的最小值为_____. **Translation of Question Stem:** As shown in the figure, in quadrilateral ABCD, AG ⊥ BC, ∠CBD = 30°, AB || CD, BG = 2√5, CG = √3, AG = 4, E is a point on the diagonal BD of the quadrilateral, F is a point on side CD, and BE = CF. Connect AE, AF, then the minimum value of AE + AF is_____. **Given Conditions from Text:** * Geometric figure: Quadrilateral ABCD * AG ⊥ BC * ∠CBD = 30° * AB || CD * BG = 2√5 * CG = √3 * AG = 4 * E is a point on diagonal BD * F is a point on side CD * BE = CF **Expression to Minimize:** AE + AF **Diagram Description:** * **Type:** Geometric figure. * **Points:** Labeled points are A, B, C, D, E, G. * **Lines/Segments:** Segments AB, BC, CD, DA, BD (diagonal), AC (diagonal), AE, AF, AG. Line segment AG is drawn from A to point G on BC. * **Perpendicularity:** Right angle symbol at G on BC, indicating AG ⊥ BC. * **Angle Labels (handwritten):** * ∠ABG ≈ 70° * ∠CBD = 30° * ∠BCG ≈ 20° * ∠BCD ≈ 55° * Angle near A ≈ 70° * Angle near D ≈ 20° * Angles within the figure involving E have handwritten labels ≈ 70° and ≈ 20°. * **Equal Markings (handwritten):** * Single tick marks on segments AE and CE (indicating AE = CE). * Double tick marks on segments AD and CD (indicating AD = CD). * **Note:** The position of point E in the diagram appears inconsistent with the question text, which states E is on the diagonal BD. The diagram shows E connected to A and C with equal length markings and situated such that triangle ACE is visible. F is mentioned in the text as being on CD but is not explicitly shown in the diagram.