1. **State the problem:** We are given a triangle $\triangle HIJ$ with angles $m\angle H = (4x - 9)^\circ$, $m\angle I = (3x - 17)^\circ$, and $m\angle J = (3x + 16)^\circ$. We need to find the measure of angle $J$. 2. **Recall the triangle angle sum property:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$m\angle H + m\angle I + m\angle J = 180^\circ$$ 3. **Set up the equation:** Substitute the given expressions: $$ (4x - 9) + (3x - 17) + (3x + 16) = 180 $$ 4. **Combine like terms:** $$ 4x - 9 + 3x - 17 + 3x + 16 = 180 $$ $$ (4x + 3x + 3x) + (-9 - 17 + 16) = 180 $$ $$ 10x - 10 = 180 $$ 5. **Solve for $x$:** $$ 10x - 10 = 180 $$ Add 10 to both sides: $$ 10x - \cancel{10} + \cancel{10} = 180 + 10 $$ $$ 10x = 190 $$ Divide both sides by 10: $$ \frac{10x}{\cancel{10}} = \frac{190}{\cancel{10}} $$ $$ x = 19 $$ 6. **Find $m\angle J$:** Substitute $x=19$ into $m\angle J = 3x + 16$: $$ m\angle J = 3(19) + 16 = 57 + 16 = 73^\circ $$ **Final answer:** $$ m\angle J = 73^\circ $$