Final answer:
The condenser lens is the part of the microscope responsible for gathering diffuse rays from the light source and illuminating the specimen with a small cone of bright light.
Step-by-step explanation:
The part of a microscope that is responsible for gathering diffuse rays from the microscope light source and illuminating the specimen with a small cone of bright light is the condenser lens. The condenser lens focuses light through the specimen to match the aperture of the objective lens above. It maximizes the illumination of the specimen to make it easily visible, especially at higher magnifications.