1. **Stating the problem:** We are given a zigzag line with 5 points labeled 1 through 5 and some angles: $m\angle 1 = 88^\circ$, $m\angle 3 = 81^\circ$, and $m\angle 5 = 64^\circ$. We need to find the measures of angles 1 through 5. 2. **Understanding the angles:** The angles given are likely interior or adjacent angles formed by the zigzag line segments. We will use the fact that angles on a straight line sum to $180^\circ$ and angles in a triangle sum to $180^\circ$. 3. **Finding $m\angle 2$:** Since $m\angle 1 = 88^\circ$ and angles 1 and 2 are adjacent on a straight vertical line segment, they sum to $180^\circ$. $$m\angle 1 + m\angle 2 = 180^\circ$$ $$88^\circ + m\angle 2 = 180^\circ$$ $$m\angle 2 = 180^\circ - 88^\circ = 92^\circ$$ 4. **Finding $m\angle 4$:** Angles 4 and 5 are adjacent on a straight line segment, so they sum to $180^\circ$. $$m\angle 4 + m\angle 5 = 180^\circ$$ $$m\angle 4 + 64^\circ = 180^\circ$$ $$m\angle 4 = 180^\circ - 64^\circ = 116^\circ$$ 5. **Finding $m\angle 3$:** Given as $81^\circ$. 6. **Summary of angle measures:** - $m\angle 1 = 88^\circ$ - $m\angle 2 = 92^\circ$ - $m\angle 3 = 81^\circ$ - $m\angle 4 = 116^\circ$ - $m\angle 5 = 64^\circ$ These satisfy the straight line and angle sum properties in the zigzag figure.