1. **State the problem:** We have a triangle with points T, S, Q, and R, S, Q, where angles at T and R are right angles. We want to find the value of $c$ given the angles $5c$ and $c + 64^\circ$ and the lengths $TQ = RQ = 73$. 2. **Analyze the figure:** Since $TQ$ and $RQ$ are equal, and both angles at T and R are right angles, triangle $TSQ$ and $RSQ$ share segment $SQ$ and have right angles at T and R respectively. 3. **Use the fact that the sum of angles around point S on the straight line $SQ$ is $180^\circ$:** $$5c + (c + 64^\circ) = 180^\circ$$ 4. **Simplify the equation:** $$5c + c + 64 = 180$$ $$6c + 64 = 180$$ 5. **Isolate $c$:** $$6c = 180 - 64$$ $$6c = 116$$ 6. **Divide both sides by 6:** $$c = \frac{116}{6}$$ $$c = \cancel{\frac{116}{6}}\frac{58}{3}$$ 7. **Final answer:** $$c = \frac{58}{3} \approx 19.33^\circ$$