1. **Problem statement:** We are given a circle with center P, points W and X on the circle, and a tangent line SU touching the circle at point T. We need to find the measure of angle $\angle TWX$. 2. **Given angles:** $\angle WUV = 74^\circ$ and $\angle VTX = 67^\circ$. 3. **Key fact:** The tangent line at point T is perpendicular to the radius PT, so $\angle STP = 90^\circ$. 4. **Step 1:** Since $\angle VTX = 67^\circ$ and T lies on the circle, $\angle PTX$ (angle between radius PT and chord TX) is related to $\angle VTX$. 5. **Step 2:** The angle between the tangent SU and chord TX at point T equals the angle in the alternate segment, which is $\angle TWX$. 6. **Step 3:** By the Alternate Segment Theorem, $m \angle TWX = m \angle STX$. 7. **Step 4:** Since SU is tangent at T, $m \angle STX = m \angle VTX = 67^\circ$. 8. **Final answer:** $$m \angle TWX = 67^\circ$$