Question

​ Quadrilateral ABCD ​ is inscribed in this circle.

Answer 1

When the quadrilateral is inscribed in a circle (a "cyclic" quadrilateral), opposite angles are supplementary. This fact gives a relationship between angles B and D that lets us find the value of x.
  ∠B + ∠D = 180
  (2x-4) + (3x+9) = 180
  5x +5 = 180
  x = (180 -5)/5 = 35

Now, we can find angle C as
  ∠C = 180 - ∠A
  ∠C = 180 - (2×35+3)
  ∠C = 107°