Final answer:
Detection of dark matter annihilation products and observation of gravitational lensing would strongly support the existence of dark matter, the latter being visual confirmation of mass affecting space-time.
Step-by-step explanation:
Among the hypothetical observations listed, the detection of dark matter annihilation products in space and the observation of gravitational lensing caused by dark matter would provide substantial evidence for the existence of dark matter. Finding a new particle that only interacts with dark matter would be significant, but might not directly confirm dark matter's existence. The measurement of the rotation curves of galaxies showing a consistent pattern has already been observed and suggests dark matter's gravitational influence but does not directly detect dark matter itself.
Discovering dark matter annihilation products would be akin to observing the 'footprints' of dark matter, providing direct evidence for its interactions. Gravitational lensing has been a powerful tool because it relies on the effect of mass on space-time, a well-understood principle in physics. It offers visual confirmation via distortions in the path of light from distant galaxies, indicating the presence of unseen mass.