1. **State the problem:** We have a triangle UST with an exterior angle at vertex S measuring 105°. 2. **Given:** Side US = 12x, side ST = 9x, and the exterior angle at S = 105°. 3. **Recall the exterior angle theorem:** The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. 4. **Apply the theorem:** The exterior angle at S (105°) equals the sum of the interior angles at U and T. 5. **Express the interior angles at U and T:** Since sides US and ST are opposite these angles and labeled 12x and 9x respectively, we assume the angles are proportional to these sides. 6. **Set up the equation:** $$105 = 12x + 9x$$ 7. **Simplify:** $$105 = 21x$$ 8. **Solve for x:** $$x = \frac{105}{21} = 5$$ **Final answer:** The value of $x$ is 5.