The ASA Congruence Theorem Quick Check

Question

4 of 54 of 5 Items

Question
Use the image to answer the 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

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.

∠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.

Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent by the ASA Congruence Theorem, you need to demonstrate that the two angles and the included side are congruent. You have already established that \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \).

The additional information you need to prove is the congruence of the sides \( AC \):

AC ≅ AC by the Reflexive Property of Congruence.

This allows you to establish the side-angle-side configuration needed for the ASA criteria, completing the proof that \( \triangle ABC \cong \triangle CDA \).