A and D are two points lying on the same side of line segment BC, such that AB =5 cm, AC=8 cm, BC =4 cm, BD =8 cm and CD =5 cm. Then,

The correct option is ∆ABC ≅ ∆DCB, by SSS.

To determine the correct congruence relationship between the two triangles, let us analyze the given side lengths of triangle ABC and triangle DCB.

We are given the following side lengths:
In ∆ABC:
AB = 5 cm
AC = 8 cm
BC = 4 cm

In ∆DCB:
DC = 5 cm
DB = 8 cm
CB = 4 cm (common side to both triangles)

Now, let us match the corresponding sides of ∆ABC and ∆DCB with equal lengths:

1. $\mathrm{AB}=\mathrm{DC}=5\text{cm}$

2. $\mathrm{AC}=\mathrm{DB}=8\text{cm}$

3. $\mathrm{BC}=\mathrm{CB}=4\text{cm}$

Since all three corresponding sides of ∆ABC and ∆DCB are equal in length, the two triangles are congruent by the Side-Side-Side (SSS) congruence criterion.

Writing the vertices in exact correspondence:
Vertex A corresponds to vertex D (since AB = DC and AC = DB).
Vertex B corresponds to vertex C.
Vertex C corresponds to vertex B.

Therefore, we can state that $∆\mathrm{ABC}\cong ∆\mathrm{DCB}$ by SSS criterion.