1. **Problem statement:** A figure containing angle $\angle FGH$ is dilated by a scale factor of 5 to form a new figure containing angle $\angle F'G'H'$. Given that $\angle FGH = 129^\circ$, find the measure of $\angle F'G'H'$. 2. **Formula and rule:** Dilation changes the size of a figure but does not change the measure of its angles. This means that angles remain congruent after dilation. 3. **Explanation:** Since dilation only affects lengths and not angles, the measure of $\angle F'G'H'$ is the same as the measure of $\angle FGH$. 4. **Calculation:** $$\angle F'G'H' = \angle FGH = 129^\circ$$ 5. **Answer:** The measure of $\angle F'G'H'$ is $129^\circ$.