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

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.

Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent by the ASA (Angle-Side-Angle) Congruence Theorem, you already have the congruent angles \( \angle BCA \cong \angle DAC \) and \( \angle BAC \cong \angle DCA \). The only remaining piece required for ASA is to show that the sides opposite these angles are congruent.

In a parallelogram, opposite sides are congruent. Therefore, you can state:

**Additional Information:**
\( AB \cong CD \) (by the property of parallelograms that states opposite sides are congruent).

Thus, the complete justification is:
- \( \angle BCA \cong \angle DAC \) (given)
- \( \angle BAC \cong \angle DCA \) (given)
- \( AB \cong CD \) (by the property of parallelograms)

This establishes \( \triangle ABC \cong \triangle CDA \) by ASA congruence.