1. **State the problem:** We are given a figure with vertex $S$ and rays to points $P$, $Q$, and $R$. The angles at $S$ are labeled as follows: $\angle QSP = 4x + 3$, $\angle QSR = 3x + 7$, and $\angle PSR = 87^\circ$. We need to find the measure of $\angle QSP$. 2. **Understand the relationships:** The angles $\angle QSP$, $\angle QSR$, and $\angle PSR$ are adjacent angles around point $S$. Since $\angle QSR$ is between $\angle QSP$ and $\angle PSR$, the sum of $\angle QSP$ and $\angle PSR$ equals $\angle QSR$ plus the other angles around $S$. However, from the figure description, it appears that $\angle QSR$ is the sum of $\angle QSP$ and $\angle PSR$ because $QSR$ is the angle formed by rays $Q$ and $R$, which includes $\angle QSP$ and $\angle PSR$. 3. **Set up the equation:** Since $\angle QSR = \angle QSP + \angle PSR$, we write: $$ 3x + 7 = (4x + 3) + 87 $$ 4. **Simplify the equation:** $$ 3x + 7 = 4x + 90 $$ 5. **Isolate $x$:** $$ 3x + 7 - 4x = 90 $$ $$ \cancel{3x} + 7 - \cancel{4x} = 90 - x $$ $$ -x + 7 = 90 $$ $$ -x = 90 - 7 $$ $$ -x = 83 $$ $$ x = -83 $$ 6. **Calculate $\angle QSP$:** Substitute $x = -83$ into $4x + 3$: $$ 4(-83) + 3 = -332 + 3 = -329 $$ 7. **Interpretation:** The angle measure cannot be negative, so we must reconsider the angle relationships. Since $\angle PSR = 87^\circ$ and the other two angles are adjacent, the sum of $\angle QSP$ and $\angle QSR$ should be $180^\circ - 87^\circ = 93^\circ$ if they form a straight line with $\angle PSR$. 8. **Revised equation:** $$ (4x + 3) + (3x + 7) = 93 $$ $$ 7x + 10 = 93 $$ $$ 7x = 83 $$ $$ x = \frac{83}{7} $$ 9. **Calculate $\angle QSP$ with correct $x$:** $$ 4 \times \frac{83}{7} + 3 = \frac{332}{7} + 3 = \frac{332}{7} + \frac{21}{7} = \frac{353}{7} \approx 50.43^\circ $$ **Final answer:** The measure of $\angle QSP$ is approximately $50.43^\circ$.