Based on the diagram, what additional information must be stated in order to prove △ABD ≅ △DBC by the ASA Congruence Theorem? (1 point)

Question

∠A≅∠D
AC¯ ≅ DC¯ ¯
AB¯ ≅ BD¯
BC¯¯¯¯¯¯¯¯ ≅ BC¯¯¯¯¯¯¯¯

Answer 1

To prove triangles \( \triangle ABD \) and \( \triangle DBC \) congruent by the ASA (Angle-Side-Angle) Congruence Theorem, we need to establish two angles and the included side.

Based on the options provided:

1. \( \angle A \cong \angle D \) - This provides one pair of angles.
2. \( AC \cong DC \) - This is not helpful for ASA since it doesn't involve the angles directly at \( A \) or \( D \).
3. \( AB \cong BD \) - This provides a side, but we need the included side between two angles.
4. \( BC \cong BC \) - This is a common side and does not contribute to proving angle congruence.

From the information, you can prove \( \triangle ABD \) and \( \triangle DBC \) are congruent by the ASA theorem if you know two angles and the included side.

Thus, in addition to \( \angle A \cong \angle D \), you would also need to state one of the angles at \( B \) or \( C \) to satisfy the ASA criteria. Specifically, you would need:

To prove \( \triangle ABD \cong \triangle DBC \) by ASA, you need to state \( \angle B \cong \angle C \) (or the necessary angle congruence between \( A \) and \( D \), and the included side \( BD \) which is the base side).

This means you must establish that the angles at \( B \) and \( D \) are congruent (or the angle at \( C \) is congruent to \( B \)—whichever fits your diagram) to apply the ASA theorem correctly.