1. **Problem Statement:** Find the value of angle $z$ given that it is adjacent to an angle measuring $34.1^\circ$ and the two angles are formed by perpendicular lines. 2. **Understanding the problem:** Since the lines are perpendicular, they form a right angle, which measures $90^\circ$. 3. **Formula used:** Adjacent angles on a straight line or forming a right angle add up to $90^\circ$ (complementary angles). 4. **Calculation:** $$z + 34.1^\circ = 90^\circ$$ 5. **Solving for $z$:** $$z = 90^\circ - 34.1^\circ$$ $$z = 55.9^\circ$$ 6. **Answer:** The value of angle $z$ is $55.9^\circ$. This means angle $z$ complements the $34.1^\circ$ angle to make a right angle.