1. **State the problem:** We are given two supplementary angles $\angle C$ and $\angle D$ with measures $m\angle C = 12x - 7$ and $m\angle D = 16x + 19$. We need to find the measure of $\angle C$. 2. **Recall the rule for supplementary angles:** Supplementary angles add up to 180 degrees. So, $$m\angle C + m\angle D = 180$$ 3. **Set up the equation:** Substitute the given expressions: $$ (12x - 7) + (16x + 19) = 180 $$ 4. **Simplify the equation:** $$ 12x - 7 + 16x + 19 = 180 $$ $$ (12x + 16x) + (-7 + 19) = 180 $$ $$ 28x + 12 = 180 $$ 5. **Solve for $x$:** $$ 28x = 180 - 12 $$ $$ 28x = 168 $$ $$ x = \frac{168}{28} = 6 $$ 6. **Find $m\angle C$:** Substitute $x = 6$ into $12x - 7$: $$ m\angle C = 12(6) - 7 = 72 - 7 = 65 $$ **Final answer:** $$ m\angle C = 65 \text{ degrees} $$