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In the provided figure, we are given the following:

- \( \overline{RS} = \overline{RT} = \overline{TU} \), which indicates that triangle \( \triangle RST \) and triangle \( \triangle RTU \) are isosceles.
- \( \angle STR = 50^\circ \), which is shown in the figure.
- We need to determine the measure of \( \angle TRU \).

### Step-by-Step Solution:

1. **Isosceles Triangle \( \triangle RST \):**
   Since \( \overline{RS} = \overline{RT} \), triangle \( \triangle RST \) is isosceles. Therefore, the base angles \( \angle RST \) and \( \angle RTS \) are equal.
   
   Let \( \angle RST = \angle RTS = x \).

2. **Angle Sum in \( \triangle RST \):**
   The sum of the angles in a triangle is always \( 180^\circ \). Thus, for triangle \( \triangle RST \):
   \[
   x + x + 50^\circ = 180^\circ
   \]
   Simplifying:
   \[
   2x + 50^\circ = 180^\circ
   \]
   \[
   2x = 130^\circ
   \]
   \[
   x = 65^\circ
   \]
   So, \( \angle RST = \angle RTS = 65^\circ \).

3. **Isosceles Triangle \( \triangle RTU \):**
   Now, since \( \overline{RT} = \overline{TU} \), triangle \( \triangle RTU \) is also isosceles. Therefore, the base angles \( \angle RTU \) and \( \angle RUT \) are equal.
   
   Let \( \angle RTU = \angle RUT = y \).

4. **Finding \( \angle TRU \):**
   The angle \( \angle TRU \) is the exterior angle of triangle \( \triangle RST \) at point \( R \). The exterior angle is equal to the sum of the opposite interior angles. Therefore:
   \[
   \angle TRU = \angle RST + \angle RTS
   \]
   Since both \( \angle RST \) and \( \angle RTS \) are \( 65^\circ \), we have:
   \[
   \angle TRU = 65^\circ + 65^\circ = 130^\circ
   \]

Thus, the measure of \( \angle TRU \) is **130°**.

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Would you like more details on any part of the explanation? Here are some related questions you might find useful:

1. How do you calculate exterior angles in any polygon?
2. What are the properties of an isosceles triangle?
3. Can the angle sum property of triangles be extended to quadrilaterals?
4. How does the exterior angle theorem apply to polygons in general?
5. Why are the base angles of an isosceles triangle always equal?

**Tip:** The exterior angle theorem is a powerful tool that relates an exterior angle to the sum of the two opposite interior angles, helping to simplify many geometry problems.

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

Solve for the measure of angle ∠TRU in a triangle with RS = RT = TU, utilizing properties of isosceles triangles and the exterior angle theorem. This problem is perfect for students learning advanced geometry concepts.

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