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.