Parts of Mt Pleasant are very close to Rock Creek Park. You might be near the bottom of the zoo where you can enter the park easily. There is quite a bit of path there that is heavily used by runners and bikers. It is paved path with quite a bit of dirt path next to it that was worn-down out of grass by the runners. If you head downtown on the paths - you'll go by the Potomac River and some very nice views.

It's quite possible you're not near Rock Creek Park though. You could be within a mile of the Mall which is popular with runners as well (as long as you go early - before it is swarming with tourists).

I know there are some small park areas in Mount Pleasant but I don't know them and definitely can't vouch for their safety.

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