In this figure, we are asked to find the measure of angle $\angle AGE$.

Here's the reasoning:

• $\angle CGF$ is given as $51^\circ$.
• $\angle AGE$ and $\angle CGF$ are vertical angles (since they are formed by intersecting lines), and vertical angles are congruent.
• Therefore, the measure of $\angle AGE = \angle CGF = 51^\circ$.

Thus, the solution is:

$\boxed{51}$

Would you like more details on the reasoning? Let me know if you have any questions!

Related Questions:

1. What are vertical angles and how do we know they are always congruent?
2. Can two intersecting lines form more than one pair of vertical angles?
3. How would the solution change if the given angle was supplementary to $\angle AGE$?
4. How can you find the measure of an adjacent angle in this figure?
5. What is the sum of the angles around point $G$ in this configuration?

Tip:

Always remember that vertical angles are congruent, which simplifies many angle-finding problems when lines intersect.