To determine which triangle congruence theorem proves that triangle $ADE$ is congruent to triangle $CBE$, let’s analyze the given diagram:

1. Shared angle:

   • The triangles $ADE$ and $CBE$ share angle $E$, so this angle is congruent between the two triangles.

2. Side proportionality:

   • The segments $AD$ and $CB$ are marked as congruent (indicated by the matching single tick marks), which tells us that $AD = CB$.

3. Additional side:

   • The segments $AE$ and $BE$ are also marked as congruent (matching double tick marks), so $AE = BE$.

Given the congruence of two sides and the included angle (side, angle, side), we can apply the Side-Angle-Side (SAS) Triangle Congruence Theorem to prove the congruence of triangles $ADE$ and $CBE$.

The correct answer is: Side-Angle-Side Triangle Congruence Theorem.

Would you like any further explanation on the solution?

Here are 5 related questions to consider:

1. What conditions are necessary for the Angle-Side-Angle Congruence Theorem?
2. Can two triangles be congruent if only two angles are congruent without a corresponding side?
3. How does the Side-Side-Side (SSS) Congruence Theorem differ from SAS?
4. How would the proof change if $AD \neq CB$ but the angles at $A$ and $B$ were congruent?
5. Can the SSA (Side-Side-Angle) theorem prove triangle congruence?

Tip: When proving triangle congruence, always check for matching sides and angles in the correct order based on the theorem you're using.