1. **State the problem:** We need to find the measure of angle $x$ given that it and a $59^\circ$ angle are adjacent and together form a right angle of $90^\circ$. 2. **Use the Angle Addition Postulate:** The postulate states that if two angles are adjacent, their measures add up to the measure of the larger angle they form together. 3. **Set up the equation:** Since the two angles form a right angle, $$59^\circ + x = 90^\circ$$ 4. **Solve for $x$:** $$x = 90^\circ - 59^\circ$$ 5. **Calculate:** $$x = 31^\circ$$ 6. **Answer:** The measure of angle $x$ is $31^\circ$.