SAT Question bd7f6e30: Lines, Angles, and Triangles answer and explanation with Desmos solution

Choice C is correct. Since $RT=TU$, it follows that triangle $RTU$ is an isosceles triangle with base $RU$. Therefore, $\angle TRU$ and $\angle TUR$ are the base angles of an isosceles triangle and are congruent. Let the measures of both $\angle TRU$ and $\angle TUR$ be $t^{\circ }$. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is $180^{\circ }$. Therefore, $114^{\circ }+2t^{\circ }=180^{\circ }$, so $t=33$. Note that $\angle TUR$ is the same angle as $\angle SUV$. Thus, the measure of $\angle SUV$ is $33^{\circ }$. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, $x^{\circ }$ is equal to the sum of the measures of $\angle VSU$ and $\angle SUV$; that is, $31^{\circ }+33^{\circ }=64^{\circ }$. Thus, the value of $x$ is $64$.