sat-dog

The correct answer is $123$. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is $180$ degrees. It's given that the measure of $\angle SQX$ is $48°$ and the measure of $\angle SXQ$ is $86°$. Since points $S$, $Q$, and $X$ form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle QSX$ is $180-48-86$, or $46$. It's also given that the measure of $\angle SWU$ is $85°$. Since $\angle SWU$ and $\angle SWR$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $\angle SWR$ is $180-85$, or $95$. Since points $R$, $S$, and $W$ form a triangle, and $\angle RSW$ is the same angle as $\angle QSX$, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle WRS$ is $180-46-95$, or $39$. It's given that the measure of $\angle VTU$ is $162°$. Since $\angle VTU$ and $\angle STU$ are supplementary angles, the sum of their measures is $180$ degrees. It follows that the measure, in degrees, of $\angle STU$ is $180-162$, or $18$. Since points $R$, $T$, and $U$ form a triangle, and $\angle URT$ is the same angle as $\angle WRS$, it follows from the triangle angle sum theorem that the measure, in degrees, of $\angle TUR$ is $180-39-18$, or $123$.