Question

note: figure not drawn to scale. in the figure shown, points q, r, s, and t lie on line segment pv, and line segment ru intersects line segment sx at point w. the measure of ∠sqx is 48°, the measure of ∠sxq is 86°, the measure of ∠swu is 85°, and the measure of ∠vtu is 102°. what is the measure, in degrees, of ∠tur?

Explanation:

Step1: Find $\angle QSW$ in $\triangle SQX$

In $\triangle SQX$, by the angle - sum property of a triangle ($\angle QSX+\angle SQX+\angle SXQ = 180^{\circ}$), we have $\angle QSW=180^{\circ}-\angle SQX - \angle SXQ$. Given $\angle SQX = 48^{\circ}$ and $\angle SXQ=86^{\circ}$, so $\angle QSW=180-(48 + 86)=46^{\circ}$.

Step2: Find $\angle SWU$'s vertical - angle

$\angle SWU = 85^{\circ}$, and its vertical - angle $\angle RWX$ also has a measure of $85^{\circ}$.

Step3: Find $\angle SRW$ in $\triangle SWU$

In $\triangle SWU$, we know one angle $\angle SWU = 85^{\circ}$ and $\angle QSW = 46^{\circ}$. In $\triangle SRW$, $\angle SRW=180^{\circ}-\angle SWU-\angle QSW$. So $\angle SRW=180-(85 + 46)=49^{\circ}$.

Step4: Find $\angle VTR$'s supplementary angle

$\angle VTR = 162^{\circ}$, and its supplementary angle $\angle STR=180 - 162=18^{\circ}$.

Step5: Find $\angle TUR$ in $\triangle TUR$

In $\triangle TUR$, using the exterior - angle property (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles) or the angle - sum property. Let $\angle TUR=x$. We know that in $\triangle TUR$, considering the angles related to the lines and triangles formed.
We know that $\angle TUR=180^{\circ}-(18^{\circ}+49^{\circ})=113^{\circ}$.

Answer:

$113$