How do you match the rotation of an object based on the edges of another object? How do you match the pentagon's edges with the edges circled in blue?
You could use snapping, but I would use vertex snapping, instead of edges: enable snapping, choose vertex, then in edit mode start the rotation (on the correct axis). 

While rotating, move your cursor over a vertex of the reference shape, to get the alignment. Sometimes, depending on your exact setup, and the various object transforms, it could work also in object mode.

Here is a simplified example: