To prove that triangles $\Delta ABC$ and $\Delta DBC$ are congruent using the Side-Angle-Side (SAS) criterion, we need to demonstrate the following:

1. One pair of corresponding sides is congruent (AB ≅ DB or AC ≅ DC).
2. The included angle between those sides is congruent ($\angle ABC = \angle DBC$).
3. The other pair of corresponding sides is congruent (BC ≅ BC, since it is a common side).

Let's evaluate the given options:

1. $m\angle ABC = 125^\circ$ and $AB \cong DB$:

   • $AB \cong DB$ gives us a congruent side pair, and if $\angle ABC$ and $\angle DBC$ are the same (since $\overline{CD}$ bisects $\angle ACD$, the two angles $\angle ABC$ and $\angle DBC$ could be congruent), this satisfies the SAS condition.

2. $\Delta ACD$ is isosceles with base $AD$:

   • Since $\Delta ACD$ is isosceles, we know $AC = DC$. This provides congruent sides, and since $\overline{AB} \cong \overline{DB}$, and the included angle $\angle ABC = \angle DBC$, this can be used to prove SAS congruence.

3. $\Delta ABD$ is isosceles with base $AD$:

   • This doesn't directly help with proving $\Delta ABC \cong \Delta DBC$ under the SAS criterion, as it doesn't specify the congruency of sides that would link to the other triangle $\Delta DBC$.

4. $CD = 52$ cm:

   • This is just a length of side $CD$, but without context, it doesn't directly aid in proving the congruency of the two triangles under SAS.

5. $AB = 29$ cm:

   • This specifies the length of $AB$ but doesn't directly compare with any other side or help with the angles, so it isn't enough by itself to prove congruency using SAS.

The three options that contribute to proving $\Delta ABC \cong \Delta DBC$ by SAS are:

1. $m\angle ABC = 125^\circ$ and $AB \cong DB$.
2. $\Delta ACD$ is isosceles with base $AD$.
3. $AB = 29$ cm (as it supports congruent side pairing with $DB$).

Would you like further explanation or any additional clarifications?

Related questions:

1. How do you prove triangle congruence using other criteria like ASA or AAS?
2. What does the "Angle Bisector Theorem" state and how does it relate to this problem?
3. What happens if two angles and a non-included side are known? How can that be used in triangle congruence?
4. Can the Pythagorean Theorem be used in triangle congruence, and if so, when?
5. How does symmetry affect triangle congruence?

Tip: In geometric proofs, drawing clear diagrams often helps you see how certain sides and angles are related. This can make recognizing congruency conditions easier!