1. **State the problem:** We need to find the measure of angle $m\angle TQW$ given that $m\angle ZQY = 67^\circ$ and $m\angle RQX = 67^\circ$.\n\n2. **Analyze the figure:** Point $Q$ is the center with six rays: $R$, $X$, $Z$, $Y$, $T$, and $W$. The angles $\angle ZQY$ and $\angle RQX$ are each $67^\circ$. These two angles are adjacent and together form part of the full $360^\circ$ around point $Q$.\n\n3. **Use the full circle rule:** The sum of all angles around point $Q$ is $360^\circ$. The six rays divide the circle into 6 angles. We know two of these angles are $67^\circ$ each.\n\n4. **Calculate the sum of the known angles:**\n$$67^\circ + 67^\circ = 134^\circ$$\n\n5. **Calculate the sum of the remaining four angles:**\n$$360^\circ - 134^\circ = 226^\circ$$\n\n6. **Assuming symmetry:** Since the rays are evenly spaced, the remaining four angles are equal. So each of these four angles, including $m\angle TQW$, is:\n$$\frac{226^\circ}{4} = 56.5^\circ$$\n\n7. **Final answer:**\n$$m\angle TQW = 56.5^\circ$$