Based on the markings in the diagram, what additional information must be stated in order to prove ABC =DBC by the congruence theorem?

Question

AB=BD
<a=<d
bc=bc
ac=dc

Answer 1

To prove that triangles \( ABC \) and \( DBC \) are congruent using a congruence theorem, we need to match the criteria of a specific triangle congruence theorem (like Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), etc.).

From the information you've provided:

1. \( AB = BD \) (Side)
2. \( \angle A = \angle D \) (Angle)
3. \( BC = BC \) (Common Side)
4. \( AC = DC \) (Side)

To prove the triangles \( ABC \) and \( DBC \) are congruent, we need to verify a congruence theorem. Based on the information you have, we can use the Side-Angle-Side (SAS) congruence theorem.

SAS states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Additional Information Required:

1. We need to confirm that the angle between sides AB and BC in triangle ABC is equal to the angle between sides BD and BC in triangle DBC. Thus:

   • We need the statement that \(\angle ABC = \angle DBC\) (This angle should be the \(\angle B\) in both triangles).

Conclusion:

With that additional piece of information stated:

1. \( AB = BD \) (Side)
2. \( BC = BC \) (Common Side)
3. \( \angle ABC = \angle DBC \) (Angle)

We can conclude that \( \triangle ABC \cong \triangle DBC \) by the Side-Angle-Side (SAS) theorem.