Final answer:
The presence of a sudden, severe headache is a piece of crucial information that can distinguish a hemorrhagic stroke from a thrombotic (ischemic) stroke.
Step-by-step explanation:
Information provided by the patient that would help differentiate a hemorrhagic stroke from a thrombotic stroke includes the sudden onset of a severe headache. Hemorrhagic strokes, which occur when an artery in the brain ruptures and causes bleeding, can lead to symptoms such as a headache due to the accumulation of blood, which in turn causes pressure on the brain tissue. This is in contrast to ischemic strokes, where the loss of blood flow to an area is due to blocked or narrowed vessels, often without the sudden, severe headache characteristic of hemorrhagic strokes.