In triangle abc, measure angle a=35 degrees, measure angle b=65 degrees, and c =15. find b. round your answer to the nearest tenth

Given only one side and two angles, the law of sines is the appropriate choice. To use that, you need to know the angles opposite the side of interest and the given side.

∠C = 180° - ∠A - ∠B
∠C = 180° - 35° - 65° = 80°

Now, the law of sines tells you
  b/sin(B) = c/sin(C)
  b = c·sin(B)/sin(C) . . . . . multiply by sin(B)
  b = 15·sin(65°)/sin(80°)
  b ≈ 13.8

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Or you can use a triangle solver.