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To solve for \( m\angle RTU \) in triangle \( STU \) formed by connecting the midpoints of the sides of triangle \( PQR \), let's break down the steps:

1. **Identify the interior angles of \( \triangle PQR \):**
   - Angle \( \angle P = 53^\circ \)
   - Angle \( \angle R = 37^\circ \)
   - Since the sum of angles in any triangle is \( 180^\circ \), we can calculate \( \angle Q \):
     \[
     \angle Q = 180^\circ - 53^\circ - 37^\circ = 90^\circ
     \]

2. **Properties of the Midpoint Triangle \( \triangle STU \):**
   - Since \( \triangle STU \) is formed by the midpoints of the sides of \( \triangle PQR \), it is a **midsegment triangle**.
   - The angles of \( \triangle STU \) will be congruent to the angles of \( \triangle PQR \) due to similarity. This is a property of midsegment triangles, which are always similar to the original triangle.

3. **Angle \( \angle RTU \) in Triangle \( STU \):**
   - Since \( \triangle STU \) is similar to \( \triangle PQR \), each corresponding angle in \( \triangle STU \) will be equal to the corresponding angle in \( \triangle PQR \).
   - Therefore, \( \angle RTU \) (which corresponds to \( \angle R \) in \( \triangle PQR \)) will have the same measure as \( \angle R \).
   
4. **Conclusion:**
   \[
   m\angle RTU = 37^\circ
   \]

Thus, the measure of \( m\angle RTU \) is \( 37^\circ \).

Would you like more details or have any questions?

### Related Questions
1. Why are the triangles \( PQR \) and \( STU \) similar?
2. How do the midsegment properties apply to angles in \( \triangle STU \)?
3. Can we determine the lengths of sides in \( \triangle STU \) relative to \( \triangle PQR \)?
4. How would the solution change if the midpoints formed a different type of triangle?
5. How can we use this concept in other types of geometric problems?

**Tip:** Remember, midsegment triangles always form a triangle similar to the original triangle and scale down all sides by half.

Triangle STU is formed by connecting the midpoints of the side of triangle PQR. The measures of the interior angles of triangle PQR are shown. Find the measure of ∠RTU.

Solve for the measure of angle RTU in triangle STU, formed by connecting the midpoints of triangle PQR. This problem explores triangle similarity and midsegment properties, providing a step-by-step solution. Suitable for high school geometry students.

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