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

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.

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

Answer 1

To prove that triangles \( \triangle ABC \cong \triangle CDA \) by the ASA (Angle-Side-Angle) Congruence Theorem, we need the following information:

1. Information about the angles: We have already stated that \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \). These two pairs of angles provide us with two angles in each triangle.

2. Additional information about the side: The side you must establish as congruent for ASA is \( AC \) since both triangles share this side. \[ AC \cong AC \quad \text{(by the Reflexive Property of Congruence)} \]

3. Conclusion: With the congruence of two angles and the included side \( AC \cong AC \) established, we can conclude that: \[ \triangle ABC \cong \triangle CDA \quad \text{(by the ASA Congruence Theorem)} \]

It appears that you already have the congruences set up, but didn't explicitly state the necessary additional information about congruence of side \( AC \). Here's the summary of information needed for the proof:

• Given angles: \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \).
• Shared side: \( AC \cong AC \) (Reflexive Property of Congruence).

By clarifying this information, you ensure that all necessary elements for the ASA theorem are covered, allowing you to conclude triangle congruence effectively.