以数学老师的身份，解答这几个题目---Here is the extraction of content from the image: **Problem 4** **Question Stem:** 4. 如图, 将□ABCD的对角线BD向两个方向延长, 分别至点E和点F, 且使BE=DF. 求证: 四边形AECF是平行四边形. (As shown in the figure, extend the diagonal BD of parallelogram ABCD in both directions to points E and F respectively, such that BE=DF. Prove: Quadrilateral AECF is a parallelogram.) **Chart/Diagram Description:** * Type: Geometric figure. * Main Elements: * Parallelogram ABCD. Vertices A, B, C, D are labeled. * Diagonal BD is drawn. * Line BD is extended beyond D to point E and beyond B to point F. * Points E and F are on the extension of line BD. * Line segments AE, EC, CF, FA are drawn, forming quadrilateral AECF. * The diagram visually represents the conditions given in the problem statement. **Problem 5** **Question Stem:** 5. 矩形对角线组成的对顶角中, 有一组是两个50°的角. 对角线与各边组成的角是多少度? (In the vertical angles formed by the diagonals of a rectangle, one pair consists of two 50° angles. What is the angle formed by the diagonal and each side?) **Problem 6** **Question Stem:** 6. 如图, 你能用一根绳子检查一个书架的侧边是否和上、下底都垂直吗? 为什么? (As shown in the figure, can you use a piece of string to check if the side edge and the top/bottom of a bookshelf are perpendicular? Why?) **Chart/Diagram Description:** * Type: Illustration of a bookshelf. * Main Elements: * A drawing depicting a bookshelf with shelves and books. * A piece of string stretched diagonally across one of the shelves, connecting two opposite corners. * The drawing illustrates a method potentially used to check for square corners (90 degree angles) on the shelf. **Problem 7** **Question Stem:** 7. 如图, 矩形ABCD的对角线AC, BD相交于点O, 且DE∥AC, CE∥BD. 求证: 四边形OCED是菱形. (As shown in the figure, diagonals AC, BD of rectangle ABCD intersect at point O, and DE∥AC, CE∥BD. Prove: Quadrilateral OCED is a rhombus.) **Other Relevant Text:** The symbol "求证:" means "Prove:". **Chart/Diagram Description:** * Type: Geometric figure. * Main Elements: * Rectangle ABCD. Vertices A, B, C, D are labeled. * Diagonals AC and BD are drawn, intersecting at point O. * Point E is outside the rectangle. * Line segment DE is drawn, with DE parallel to AC. * Line segment CE is drawn, with CE parallel to BD. * Quadrilateral OCED is formed by lines OC, CE, ED, DO. **Problem 8** **Question Stem:** 8. 如图, E, F, G, H分别是正方形ABCD各边的中点. 四边形EFGH是什么四边形? 为什么? (As shown in the figure, E, F, G, H are respectively the midpoints of each side of square ABCD. What type of quadrilateral is quadrilateral EFGH? Why?) **Chart/Diagram Description:** * Type: Geometric figure. * Main Elements: * Square ABCD. Vertices A, B, C, D are labeled. * Points E, F, G, H are marked on sides AB, BC, CD, DA respectively. * Labels indicate E is on AB, F on BC, G on CD, H on DA. * E, F, G, H are midpoints of the sides. * Line segments EF, FG, GH, HE are drawn, forming quadrilateral EFGH inside the square. **Problem 9** **Question Stem:** 9. 如图, 四边形ABCD是平行四边形, BE∥DF, 且分别交对角线AC于点E, F. 连接ED, BF. 求证∠1=∠2. (As shown in the figure, quadrilateral ABCD is a parallelogram, BE∥DF, and they intersect diagonal AC at points E, F respectively. Connect ED, BF. Prove ∠1 = ∠2.) **Other Relevant Text:** The symbol "求证" means "Prove". Angles are labeled ∠1 and ∠2. **Chart/Diagram Description:** * Type: Geometric figure. * Main Elements: * Parallelogram ABCD. Vertices A, B, C, D are labeled. * Diagonal AC is drawn. * Line segments BE and DF are drawn, intersecting AC at points E and F respectively. * Condition BE || DF is given. * Line segments ED and BF are drawn. * Angle ∠1 is labeled inside triangle ADE, near vertex D. It appears to be ∠ADE. * Angle ∠2 is labeled inside triangle CBF, near vertex B. It appears to be ∠CBF.