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Question

Parallelogram upper A upper D upper C upper B is graphed on a plane, with diagonal upper A upper C.

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?

(1 point)
Responses

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.

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.

∠B≅∠D by the Reflexive Property of Congruence.

Answer 1

In order to prove \( \triangle ABC \cong \triangle CDA \) by the ASA (Angle-Side-Angle) Congruence Theorem, you need to establish two angles and the side between them as congruent in both triangles.

You mentioned that \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \). For the ASA theorem, you now need to confirm that the side \( AC \) is congruent to itself in both triangles.

Thus, the additional information you must state and justify is:

AC ≅ AC by the Reflexive Property of Congruence.

This justification indicates that the diagonal \( AC \) is the same segment in both triangles \( ABC \) and \( CDA \), fulfilling the requirements of the ASA theorem for congruence.