1. **State the problem:** We need to find the measure of angle $x$ at vertex $Q$ formed by rays $QR$ and $QS$. 2. **Analyze the given information:** Angle $PQR$ is given as $76^\circ$. Rays $QP$ and $QR$ form this angle. 3. **Understand the geometry:** Since rays $QP$, $QR$, and $QS$ all emanate from point $Q$, and $QR$ is between $QP$ and $QS$, the angles around point $Q$ on a straight line sum to $180^\circ$. 4. **Use the straight angle rule:** The sum of angles $PQR$ and $RQS$ (which is $x$) is $180^\circ$ because they form a straight line. 5. **Write the equation:** $$ 76^\circ + x = 180^\circ $$ 6. **Solve for $x$:** $$ x = 180^\circ - 76^\circ $$ $$ x = 104^\circ $$ 7. **Final answer:** The measure of angle $x$ is $104^\circ$.