In parallelogram ABCD, diagonal AC¯¯¯¯¯¯¯¯ is drawn, and ∠BCA≅∠DAC and ∠BAC≅∠DCA . In order to prove △ABC≅△CDA by the ASA Congruence Theorem, what additional information must be stated and justified?

Question

Possible Answers:
DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence. Modifying above upper D upper A with bar congruent to Modifying above upper B upper C with bar by the Reflexive Property of Congruence. AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence. Modifying above upper A upper C with bar congruent to Modifying above upper A upper C with bar by the Reflexive Property of Congruence. AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯ by the Triangle Angle Sum Theorem. Modifying above upper A upper B with bar congruent to Modifying above upper C upper D with bar by the Triangle Angle Sum Theorem. ∠B≅∠D by the Reflexive Property of Congruence.

Answer 1

To prove that triangles ABC and CDA are congruent using the ASA (Angle-Side-Angle) Congruence Theorem in parallelogram ABCD, we need one additional piece of information that confirms the congruence of another angle or side.

Given the conditions, we know:

1. ∠BCA ≅ ∠DAC (Angle)
2. ∠BAC ≅ ∠DCA (Angle)

To apply the ASA Congruence Theorem, we need to establish that the included side between these two angles is congruent.

The correct additional piece of information you need to state and justify is:

DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence. Modifying above upper D upper A with bar congruent to Modifying above upper B upper C with bar by the Reflexive Property of Congruence.

In parallelogram ABCD, opposite sides are equal in length. Therefore, by the properties of parallelograms, we know that DA ≅ BC, which provides the necessary side for the ASA theorem to conclude that triangles ABC and CDA are congruent.

Thus, stating the congruence of DA and BC will complete the conditions needed for the ASA theorem, establishing triangle congruence.