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The problem involves a triangle \( \triangle RSU \) where \( RS = RT = TU \). Given that \( \angle STU = 50^\circ \), we are asked to find the measure of \( \angle TRU \).

### Step-by-Step Solution:

1. **Identify Triangle Properties:**
   - Since \( RS = RT \) and \( RT = TU \), \( \triangle RSU \) is an **isosceles triangle**.
   - \( \triangle RTU \) is also an isosceles triangle since \( RT = TU \).

2. **Base Angles of \( \triangle RTU \):**
   - In \( \triangle RTU \), the base angles \( \angle RUT \) and \( \angle RTU \) are equal. 
   - Let \( \angle RUT = \angle RTU = x \).

3. **Sum of Angles in \( \triangle RTU \):**
   \[
   x + x + 50^\circ = 180^\circ
   \]
   \[
   2x + 50^\circ = 180^\circ
   \]
   \[
   2x = 130^\circ
   \]
   \[
   x = 65^\circ
   \]

   So, \( \angle RUT = \angle RTU = 65^\circ \).

4. **Find \( \angle TRU \):**
   - \( \angle TRU \) is the exterior angle for \( \triangle RTU \) and equals the sum of the remote interior angles \( \angle RUT \) and \( \angle STU \).
   \[
   \angle TRU = \angle RUT + \angle STU
   \]
   \[
   \angle TRU = 65^\circ + 50^\circ = 115^\circ
   \]

Thus, the measure of \( \angle TRU \) is \( 115^\circ \).

Would you like more details on any part of the solution?

### Related Questions:
1. What is the measure of \( \angle RSU \) in the same triangle?
2. How do you determine the exterior angle of a triangle?
3. If \( RS = RT = RU \), how does this affect the angles in \( \triangle RSU \)?
4. What are the properties of an isosceles triangle?
5. How would the problem change if \( \angle STU \) were different?

### Tip:
In isosceles triangles, the angles opposite the equal sides are always equal. This is a fundamental property often used in solving geometric problems like this one.

In figure, RS = RT = TU. What is the measure of ∠TRU?

Discover how to calculate the measure of ∠TRU in an isosceles triangle where RS = RT = TU, and ∠STU is given as 50°. Learn step-by-step how to apply the Angle Sum and Exterior Angle Theorems. Suitable for students in Grades 9-10.

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