For statement 3, which asserts that $\triangle MEG \cong \triangle EMO$, we need to identify the correct congruence postulate that justifies this.

Let's analyze the information provided:

1. Statement 1: $\angle GEM \cong \angle OME$ and $\angle GME \cong \angle OEM$, suggesting pairs of congruent angles.
2. Statement 2: $EM = EM$, which is the reflexive property stating that the segment $EM$ is equal to itself.

Now, we have two pairs of congruent angles ($\angle GEM \cong \angle OME$ and $\angle GME \cong \angle OEM$) and a congruent side between these two pairs of angles ($EM = EM$).

This aligns with the ASA (Angle-Side-Angle) congruence criterion, where two triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle.

Therefore, the reason for statement 3 is ASA.

Would you like further explanation on this problem?

Here are five related questions to deepen understanding:

1. What other congruence criteria could be used if we had information about all three sides of the triangles?
2. How does the reflexive property of equality help in proving congruence?
3. Why is it important to identify the "included side" in the ASA criterion?
4. How would the solution change if we only had one pair of congruent angles and a pair of congruent sides?
5. Could we use any other congruence criterion in this specific problem?

Tip: Always look for pairs of angles and sides that are common between two triangles when trying to establish congruence. This simplifies the identification of the correct postulate or theorem to use.