What is the sum of the measures of the exterior angles of this triangle?

Your answer is 360°, but there are two ways to answer this question.

The first is knowing that the exterior angles of any shape always add up to 360°, so that is your answer.

The second is by figuring out all of the exterior angles and adding them up, so you would first need to subtract 112 from 180 to get 68° as angle C. Then you can add this to 51° to get 119, and subtract that from 180 to get 61° as angle A as angles in a triangle add to 180°. You can then subtract this from 180 again as angles on a straight line add to 180, and get angle B as 119°. Finally you just need to subtract 51 from 180 to get 129°, and then you add 129, 112, and 119 to get 360°.
I hope this helps!