1. **State the problem:** We need to find a series of transformations that map polygon ABCDE onto polygon A'B'C'D'E'. 2. **Analyze the positions:** Polygon ABCDE is in the fourth quadrant (mostly negative y, positive x) and polygon A'B'C'D'E' is in the second quadrant (negative x, positive y). 3. **Identify possible transformations:** Common transformations include translations, reflections, rotations, and dilations. Since the polygons are congruent and only repositioned, we focus on reflections and rotations. 4. **Step 1: Reflect polygon ABCDE across the x-axis.** - Reflection across the x-axis changes each point $(x,y)$ to $(x,-y)$. - This moves the polygon from the fourth quadrant to the first quadrant. 5. **Step 2: Reflect the resulting polygon across the y-axis.** - Reflection across the y-axis changes each point $(x,y)$ to $(-x,y)$. - This moves the polygon from the first quadrant to the second quadrant, matching the location of polygon A'B'C'D'E'. 6. **Summary of transformations:** - First, reflect across the x-axis: $(x,y) \to (x,-y)$. - Then, reflect across the y-axis: $(x,y) \to (-x,y)$. 7. **Verification:** Applying these two reflections in sequence is equivalent to a rotation of 180 degrees about the origin: $$ (x,y) \xrightarrow{180^\circ \text{ rotation}} (-x,-y) $$ 8. **Final conclusion:** The series of transformations that map polygon ABCDE onto polygon A'B'C'D'E' is a rotation of 180 degrees about the origin. **Answer:** Rotate polygon ABCDE by 180 degrees about the origin to obtain polygon A'B'C'D'E'.