1. **Problem statement:** Prove that quadrilateral BCED is a parallelogram given that ADE, AOC, and BOD are straight lines, OA = OC, OB = OD, and AD = DE. 2. **Recall the definition of a parallelogram:** A quadrilateral is a parallelogram if both pairs of opposite sides are equal and parallel. 3. **Given:** AD = DE (given), OA = OC, OB = OD, and lines ADE, AOC, BOD are straight. 4. **Step 1: Show BC = ED.** Since OA = OC and OB = OD, triangles OAB and OCD are congruent by SAS (side-angle-side) because they share angle AOC and BOD respectively. 5. **Step 2: Since AD = DE and points A, D, E are collinear, segment DE is equal and parallel to AD. This implies DE is parallel to AB (since AD is part of line ADE and AB is connected to B). Similarly, BC is parallel to AD because of the congruence and equal lengths. 6. **Step 3: Show BC = ED and BC is parallel to ED.** Since AD = DE and BC is parallel and equal to AD, BC = ED and BC is parallel to ED. 7. **Step 4: Show BE = CD.** Using the congruent triangles and given equalities, BE and CD are equal and parallel. 8. **Conclusion:** Since both pairs of opposite sides BC and ED, and BE and CD are equal and parallel, quadrilateral BCED is a parallelogram. **Final answer:** BCED is a parallelogram.