In the given figure, it is provided that the measure of angle $\angle 7 = 100^\circ$. To find the measure of $\angle 9$, we can use the following reasoning based on the properties of parallel lines and transversals:

1. Lines $m$, $t$, and $u$ appear to be parallel, and line $s$ serves as a transversal.
2. $\angle 7$ and $\angle 9$ are corresponding angles because they are on the same side of the transversal $s$ and on the same relative positions with respect to lines $t$ and $u$.
3. For corresponding angles formed by a transversal crossing parallel lines, the angles are congruent, meaning their measures are equal.

Therefore, $\angle 9 = \angle 7 = 100^\circ$.

Answer:

$\boxed{100^\circ}$

Let me know if you'd like more details! Here are five related questions to explore:

1. What is the measure of $\angle 5$ in this figure?
2. How do you determine the relationship between alternate interior angles?
3. What is the measure of $\angle 11$ based on this diagram?
4. How do vertical angles relate in this configuration?
5. What is the total sum of angles formed at point $t$ by the intersecting lines?

Tip: Remember that corresponding angles are always equal when a transversal cuts across parallel lines.