1. **Problem Statement:** We have a graph representing land masses (vertices) A, B, C, D, and E connected by bridges (edges). We want to determine if there is a circuit (Eulerian circuit) that uses each bridge exactly once. 2. **Key Concept:** An Eulerian circuit exists in a graph if and only if the graph is connected and every vertex has an even degree (an even number of edges). 3. **Calculate the degree of each vertex:** - Vertex A: connected to C by 3 bridges and to D by 3 bridges, so degree $= 3 + 3 = 6$ (even) - Vertex B: connected to C by 1 bridge and to E by 1 bridge, so degree $= 1 + 1 = 2$ (even) - Vertex C: connected to A by 3 bridges and to B by 1 bridge, so degree $= 3 + 1 = 4$ (even) - Vertex D: connected to A by 3 bridges and to E by 2 bridges, so degree $= 3 + 2 = 5$ (odd) - Vertex E: connected to D by 2 bridges and to B by 1 bridge, so degree $= 2 + 1 = 3$ (odd) 4. **Check degrees:** Vertices D and E have odd degrees (5 and 3 respectively). 5. **Conclusion:** Since there are vertices with odd degree, an Eulerian circuit does not exist. 6. **Answer:** The answer is **DNE** (Does Not Exist).