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. 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.

Answer 1

To prove triangles \( \triangle ABC \cong \triangle CDA \) by the ASA (Angle-Side-Angle) Congruence Theorem, you already have the following information:

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

To complete the proof using the ASA criterion, you need to establish the congruence of one pair of sides that is included between the angles. In a parallelogram, opposite sides are congruent, so the additional information you need would be:

DA\(\cong\)BC by the properties of a parallelogram.

Justifying this, you can state:

• Opposite sides of a parallelogram are congruent. Therefore, \( DA \cong BC \).

Now you have:

• Two pairs of angles \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \), and
• One pair of included sides \( DA \cong BC \).

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