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

Question

GPT-4o mini · Answer

To use the ASA (Angle-Side-Angle) Congruence Theorem to prove that triangles ABC and CDA are congruent, you have established the angles \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \). However, you need an additional piece of information relating to one of the sides.

The correct additional information to state and justify is:

DA¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ by the Reflexive Property of Congruence.

This can be understood as follows:

In parallelogram ABCD, by definition, opposite sides are congruent. Thus, \( AB \cong CD \) and \( DA \cong BC \). You can specifically use the reflexive property on side AC: each triangle shares the side AC, which enables you to conclude that the triangles are congruent.

Once you establish that \( DA \cong BC \), along with the given angle congruences, you can apply the ASA theorem as follows:

You have two angles from both triangles:
- \( \angle BAC \cong \angle DCA \)
- \( \angle BCA \cong \angle DAC \)
And you have the side \( DA \cong BC \) (which means the side between the angles is congruent).

With this information, you can conclude that triangles \( ABC \) and \( CDA \) are congruent by ASA.