1. **Problem statement:** Given rectangle STUV with angles at S and U expressed as $(x-4)^\circ$ and $(3x-22)^\circ$ respectively, find the measure of angle $\angle SUT$. 2. **Recall properties of rectangles:** All angles in a rectangle are right angles, so each angle measures $90^\circ$. 3. **Set up equations:** Since $\angle S = (x-4)^\circ$ and $\angle U = (3x-22)^\circ$ are angles inside the rectangle, they must each equal $90^\circ$. 4. **Solve for $x$ using $\angle S$:** $$x - 4 = 90$$ $$x = 90 + 4 = 94$$ 5. **Check $\angle U$ with $x=94$:** $$3(94) - 22 = 282 - 22 = 260$$ This contradicts the rectangle property, so the given angle expressions likely represent angles formed by diagonals or other segments, not the rectangle's interior angles. 6. **Analyze $\angle SUT$:** This angle is formed at vertex U by points S and T. 7. **Use diagonal properties:** In rectangle STUV, diagonals SU and TV are equal and bisect each other. 8. **Calculate $\angle SUT$ using vectors or triangle properties:** Since STUV is rectangle, triangle SUT is right-angled at T or U. 9. **Use given angle expressions to find $x$ by noting that angles $(x-4)^\circ$ and $(3x-22)^\circ$ are adjacent angles on a straight line formed by diagonal SU, so their sum is $180^\circ$: $$ (x-4) + (3x-22) = 180 $$ $$ 4x - 26 = 180 $$ $$ 4x = 206 $$ $$ x = \frac{206}{4} = 51.5 $$ 10. **Calculate $\angle SUT$:** Since $\angle SUT = (3x - 22)^\circ$, $$ 3(51.5) - 22 = 154.5 - 22 = 132.5^\circ $$ **Final answer:** $$\boxed{132.5^\circ}$$