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We investigate the effects of a massive ($\gtrsim4\times10^{10}M_\odot$) Sagittarius dwarf spheroidal galaxy (Sgr) on stellar streams using test particle simulations in a realistic Milky Way potential. We find that Sgr can easily disrupt streams formed more than $\sim3$ Gyr ago, while stars stripped more recently are generally unaffected. In certain realizations, Sgr is able to produce asymmetry between the leading and trailing tails of Pal 5, qualitatively similar to observations. Using data from the Gaia space telescope and elsewhere, we fit models to the GD-1 stream in the presence of a Sgr with various initial masses. While the best-fitting models do show perturbations resulting from interactions with Sgr, we find that the level of disruption is not significantly greater than in the observed stream. To investigate the general effects of Sgr on a population of streams, we generate 1000 mock streams on GD-1-like orbits with randomized orientations. Some streams show clear evidence of disruption, becoming folded on the sky or developing asymmetry betweeen their two tails. However, many survive unaffected and the peak surface brightness of stars is decreased by no more than $\sim0.3$ mag/arcsec$^2$ on average. We conclude that Sgr having an initial mass of $\gtrsim4\times10^{10}M_\odot$ is compatible with the survival and detection of streams formed more than 3 Gyr ago.