Question

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 ∠B≅∠D by the Reflexive Property of Congruence. angle upper B congruent to angle upper D by the Reflexive Property of Congruence. 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. 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. AC¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.

Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent using the ASA (Angle-Side-Angle) Congruence Theorem, you already have the following:

1. \( \angle BCA \cong \angle DAC \) (Given)
2. \( \angle BAC \cong \angle DCA \) (Given)

To complete the proof, you need to establish the congruence of one pair of corresponding sides. The additional information required is:

**\( AC \cong AC \) by the Reflexive Property of Congruence.**

This is necessary because \( AC \) is a common side to both triangles \( \triangle ABC \) and \( \triangle CDA \). Therefore, your final statement becomes:

- \( AC \cong AC \) by the Reflexive Property of Congruence.

Now you have:
- Two pairs of congruent angles: \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \).
- One pair of congruent sides: \( AC \cong AC \).

Thus, you can conclude by the ASA Congruence Theorem that \( \triangle ABC \cong \triangle CDA \).