1. **State the problem:** We are given a right triangle with angles $\angle A = (5x - 19)^\circ$, $\angle B = 90^\circ$ (right angle), and $\angle C = (2x + 32)^\circ$. We need to find the values of $x$ and the measures of angles $A$ and $C$. 2. **Use the triangle angle sum property:** The sum of the interior angles of any triangle is $180^\circ$. So, $$\angle A + \angle B + \angle C = 180^\circ$$ 3. **Substitute the known values:** $$ (5x - 19) + 90 + (2x + 32) = 180 $$ 4. **Simplify the equation:** $$ 5x - 19 + 90 + 2x + 32 = 180 $$ $$ (5x + 2x) + (-19 + 90 + 32) = 180 $$ $$ 7x + 103 = 180 $$ 5. **Isolate $x$:** $$ 7x = 180 - 103 $$ $$ 7x = 77 $$ $$ x = \frac{77}{7} $$ $$ x = 11 $$ 6. **Find the measures of angles $A$ and $C$ by substituting $x=11$:** $$ \angle A = 5(11) - 19 = 55 - 19 = 36^\circ $$ $$ \angle C = 2(11) + 32 = 22 + 32 = 54^\circ $$ 7. **Check the sum:** $$ 36^\circ + 90^\circ + 54^\circ = 180^\circ $$ This confirms the solution is correct. **Final answer:** $$ x = 11, \quad \angle A = 36^\circ, \quad \angle C = 54^\circ $$