1. **State the problem:** We need to find the measure of angle QA in the triangle with angles expressed as $x+4$ and $2x-7$ at vertex O. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. 3. **Set up the equation:** Let the three angles of the triangle be $x+4$, $2x-7$, and $QA$. Then, $$ (x+4) + (2x-7) + QA = 180 $$ 4. **Simplify the equation:** $$ 3x - 3 + QA = 180 $$ 5. **Express $QA$ in terms of $x$:** $$ QA = 180 - 3x + 3 = 183 - 3x $$ 6. **Use the fact that angles $x$ and $y$ are related:** Since the problem does not provide $y$ or $z$ explicitly, assume $QA$ corresponds to one of the given multiple-choice values and solve for $x$ to check consistency. 7. **Test each multiple-choice value for $QA$:** - For $QA=20$: $$ 20 = 183 - 3x \implies 3x = 163 \implies x = \frac{163}{3} \approx 54.33 $$ - For $QA=28$: $$ 28 = 183 - 3x \implies 3x = 155 \implies x = \frac{155}{3} \approx 51.67 $$ - For $QA=39$: $$ 39 = 183 - 3x \implies 3x = 144 \implies x = 48 $$ - For $QA=14$: $$ 14 = 183 - 3x \implies 3x = 169 \implies x = \frac{169}{3} \approx 56.33 $$ 8. **Check if $x$ values make sense for angles $x+4$ and $2x-7$:** Angles must be positive and less than $180^\circ$. - For $x=48$ (when $QA=39$): $$ x+4 = 48 + 4 = 52 $$ $$ 2x - 7 = 2(48) - 7 = 96 - 7 = 89 $$ All angles: $52$, $89$, and $39$ sum to $180$ and are valid. 9. **Conclusion:** The measure of angle QA is $39$ degrees. **Final answer:** $QA = 39$