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In this problem, you are asked to find \( m \angle POR \), given that \( RS \) and \( PT \) are altitudes, and that \( m \angle Q = 62^\circ \).

### Step-by-Step Solution:
1. **Recognize the Altitudes**: 
   The altitudes \( RS \) and \( PT \) indicate that \( \triangle PQR \) is a right triangle, with the altitudes dividing the triangle into smaller triangles that also have special properties.

2. **Use Triangle Angle Sum Theorem**:
   In \( \triangle PQR \), the sum of the angles in a triangle must equal \( 180^\circ \). Therefore:
   \[
   m \angle P + m \angle Q + m \angle R = 180^\circ
   \]
   You're given that \( m \angle Q = 62^\circ \).

3. **Relationship Between \( m \angle POR \)**:
   \( O \) is the intersection of the altitudes, and thus \( O \) is the orthocenter. The angle \( \angle POR \) is formed by the rays extending from \( P \) and \( R \) and passing through \( O \).

4. **Conclusion**:
   Since the diagram suggests the triangle \( PQR \) with given conditions, the angle \( m \angle POR \) (the angle at the orthocenter involving altitudes) must be complementary to the known angles in the triangle.

The final answer is \( m \angle POR = 28^\circ \).

Would you like further clarification or details on any step?

### Follow-up Questions:
1. What is the significance of the orthocenter in a triangle?
2. How do altitudes help in determining angles within a triangle?
3. Can you calculate the other angles in \( \triangle PQR \)?
4. What happens to the angles when a triangle has an orthocenter?
5. How do special points (like orthocenters or centroids) help in geometric constructions?

### Tip:
Always remember that the sum of interior angles in any triangle is \( 180^\circ \), which is a key fact for many triangle problems.

In the diagram, RS and PT are altitudes, and m∠Q = 62°. Find m∠POR.

Solve this geometry problem where RS and PT are altitudes in triangle PQR, and m∠Q = 62°. Learn how to apply the triangle angle sum theorem and understand the role of altitudes in determining the orthocenter. Suitable for high school students in Grades 9-12.

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