1. **Problem Statement:** We are given a triangle with angles labeled 5, 6, and 7 at the top vertex, and angles 1, 2 at the bottom left vertex, and angles 3, 1 at the bottom right vertex. We know $m\angle 5 = 38^\circ$ and $m\angle 6 = 62^\circ$. We need to find $m\angle 3$. 2. **Recall the Triangle Angle Sum Theorem:** The sum of the interior angles of any triangle is always $180^\circ$. That is, $$m\angle 5 + m\angle 6 + m\angle 7 = 180^\circ.$$ 3. **Calculate $m\angle 7$:** Using the given values, $$m\angle 7 = 180^\circ - m\angle 5 - m\angle 6 = 180^\circ - 38^\circ - 62^\circ = 80^\circ.$$ 4. **Identify relationship between $\angle 7$ and $\angle 3$:** From the diagram description, angles 7 and 3 are vertically opposite angles or corresponding angles formed by intersecting lines, so they are equal. 5. **Conclusion:** Therefore, $$m\angle 3 = m\angle 7 = 80^\circ.$$ **Final answer:** $m\angle 3 = 80$