Question: In the figure below, m\angle 1 = (x - 9)^\circ and m\angle 2 = 8x^\circ.\nFind the angle measures.\n\nm\angle 1 = (x - 9)^\circ \nm\angle 2 = 8x^\circ
1. **State the problem:** We are given two angles, m\angle 1 = (x - 9)^\circ and m\angle 2 = 8x^\circ, that together form a right angle. We need to find the measures of these angles.\n\n2. **Understand the relationship:** Since the two angles split a right angle, their measures add up to 90^\circ.\n\n3. **Write the equation:**\n$$ (x - 9) + 8x = 90 $$\n\n4. **Simplify the equation:**\n$$ x - 9 + 8x = 90 $$\n$$ 9x - 9 = 90 $$\n\n5. **Isolate the variable:**\n$$ 9x - 9 + 9 = 90 + 9 $$\n$$ 9x = 99 $$\n\n6. **Solve for x:**\n$$ x = \frac{99}{9} $$\n$$ x = 11 $$\n\n7. **Find the angle measures:**\n$$ m\angle 1 = x - 9 = 11 - 9 = 2^\circ $$\n$$ m\angle 2 = 8x = 8 \times 11 = 88^\circ $$\n\n8. **Check the sum:**\n$$ 2^\circ + 88^\circ = 90^\circ $$ which confirms the solution is correct.\n\n**Final answer:**\n$$ m\angle 1 = 2^\circ, \quad m\angle 2 = 88^\circ $$