1. **Problem Statement:** Given $m\angle 2 = 55^\circ$ and $m\angle 3 = 80^\circ$, find $m\angle 4$.\n\n2. **Understanding the figure and relationships:** The angles are part of a triangle and adjacent angles on a straight line. We know that angles on a straight line sum to $180^\circ$. Also, the sum of angles in a triangle is $180^\circ$.\n\n3. **Step 1: Find $m\angle 5$ using the triangle sum:** Angles $3$, $4$, and $5$ form a triangle. So,\n$$m\angle 3 + m\angle 4 + m\angle 5 = 180^\circ.$$\nWe want $m\angle 4$, so we need $m\angle 5$.\n\n4. **Step 2: Use the linear pair relationship:** Angles $2$ and $5$ are adjacent on a straight line, so\n$$m\angle 2 + m\angle 5 = 180^\circ.$$\nGiven $m\angle 2 = 55^\circ$, then\n$$m\angle 5 = 180^\circ - 55^\circ = 125^\circ.$$\n\n5. **Step 3: Substitute $m\angle 3$ and $m\angle 5$ into the triangle sum:**\n$$80^\circ + m\angle 4 + 125^\circ = 180^\circ.$$\nSimplify:\n$$m\angle 4 + 205^\circ = 180^\circ.$$\n\n6. **Step 4: Solve for $m\angle 4$:**\n$$m\angle 4 = 180^\circ - 205^\circ = -25^\circ.$$\nThis is impossible for an angle measure, so we must reconsider the relationships.\n\n7. **Re-examining the figure:** Angles $1$ and $4$ are adjacent on a straight line, so\n$$m\angle 1 + m\angle 4 = 180^\circ.$$\nAlso, angles $1$, $2$, and $3$ form a triangle, so\n$$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ.$$\n\n8. **Step 5: Find $m\angle 1$:**\n$$m\angle 1 = 180^\circ - m\angle 2 - m\angle 3 = 180^\circ - 55^\circ - 80^\circ = 45^\circ.$$\n\n9. **Step 6: Find $m\angle 4$ using the linear pair:**\n$$m\angle 4 = 180^\circ - m\angle 1 = 180^\circ - 45^\circ = 135^\circ.$$\n\n**Final answer:** $m\angle 4 = 135^\circ$.