Let's begin by understanding the problem. We have a complex geometric figure where points Q, R, S, and T all lie on line segment PV. Line segment RU intersects line segment SX at point W. We are given several angle measurements and need to find the measure of angle TUR.Let's list the given information. We know that angle SQX measures 48 degrees, angle SXQ measures 86 degrees, angle SWU measures 85 degrees, and angle VTU measures 162 degrees. These angles will help us find the unknown angle TUR.First, let's find angle QSX using triangle QSX. The sum of angles in any triangle equals 180 degrees. So angle SQX plus angle SXQ plus angle QSX equals 180 degrees. Substituting the known values: 48 degrees plus 86 degrees plus angle QSX equals 180 degrees. Therefore, angle QSX equals 180 minus 48 minus 86, which gives us 46 degrees.Since points Q, R, S, and T all lie on line segment PV, angles QSX and RSX form a linear pair on a straight line. Angles on a straight line sum to 180 degrees. So angle QSX plus angle RSX equals 180 degrees. We found that angle QSX is 46 degrees, so angle RSX equals 180 minus 46, which is 134 degrees.At point W, where line RU intersects line SX, we have vertical angles. Vertical angles are always equal. Since angle SWU measures 85 degrees, its vertical angle RWS also measures 85 degrees.Let's try to find angle SRW using triangle RSW. The sum should be 180 degrees. But when we calculate 134 plus 85 plus angle SRW equals 180, we get angle SRW equals negative 39 degrees. This is impossible! A negative angle means we need to reconsider our approach.Let me correct this. The angle RSW on the other side of the line is 180 minus 134, which equals 46 degrees. Now in triangle RSW, we have 46 degrees plus 85 degrees plus angle SRW equals 180 degrees. Therefore, angle SRW equals 180 minus 46 minus 85, which is 49 degrees.Now, angles SRW and TRU are vertical angles at point R. Since vertical angles are equal, angle TRU also measures 49 degrees.Now let's use the given angle VTU which measures 162 degrees. Angles VTU and RTU form a linear pair on line PV, so they are supplementary and sum to 180 degrees. Therefore, angle RTU equals 180 minus 162, which is 18 degrees.Finally, we can find angle TUR using triangle TRU. The sum of angles in the triangle equals 180 degrees. We have angle RTU equals 18 degrees and angle TRU equals 49 degrees. So 18 plus 49 plus angle TUR equals 180. Therefore, angle TUR equals 180 minus 18 minus 49, which gives us 113 degrees.Let's summarize our solution. We found angle QSX equals 46 degrees, angle RSW equals 46 degrees, angle RWS equals 85 degrees, angle SRW equals 49 degrees, angle TRU equals 49 degrees, angle RTU equals 18 degrees, and finally angle TUR equals 113 degrees. The answer is 113 degrees.