Let's analyze the problem using the information provided in the image.

We have a triangle $STU$ with angles at $S$, $T$, and $U$. The diagram also shows a smaller triangle $SRV$ inside triangle $STU$, where we need to find the value of $x$ at angle $\angle RVU$.

Given:

• $\angle STU = 114^\circ$
• $\angle TSR = 31^\circ$
• $RT = TU$, meaning triangle $RTU$ is isosceles.

Step 1: Calculate $\angle RTU$

Since triangle $RTU$ is isosceles, angles $\angle RTU$ and $\angle TUR$ must be equal. To find the measure of these two angles, we first calculate the third angle in triangle $STU$:

$\angle STU = 114^\circ$

The sum of the angles in a triangle is always $180^\circ$. Therefore, the remaining angle $\angle SRT$ is:

$\angle SRT = 180^\circ - 114^\circ - 31^\circ = 35^\circ$

Since $RT = TU$, the angles at $R$ and $U$ are equal. Let $\angle RTU = \angle TUR = x$.

Now, using the fact that the sum of angles in triangle $RTU$ is $180^\circ$, we write:

$x + x + 35^\circ = 180^\circ$

$2x + 35^\circ = 180^\circ$

$2x = 145^\circ$

$x = 72.5^\circ$

Thus, the value of $x$ is $72.5^\circ$.

Would you like more details on this solution or have any questions?

Related Questions:

1. How do you calculate angles in an isosceles triangle?
2. Why is the sum of the interior angles in a triangle always $180^\circ$?
3. Can you explain how angle properties of an isosceles triangle help in solving this problem?
4. What other angle relationships exist in an isosceles triangle?
5. How do external angles relate to the internal angles in triangles?

Tip: Remember that in isosceles triangles, the base angles are always equal, which often simplifies problems involving angles.