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?

Answer 1

To prove triangles \( \triangle ABC \) and \( \triangle CDA \) are congruent using the Angle-Side-Angle (ASA) Congruence Theorem, we need two angles and the included side of those triangles to be congruent.

We already have:

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

To apply the ASA criterion, we need to demonstrate that the side between these two angles (which is side \( AC \)) is congruent in both triangles. That is, we need to show that:

\[ AC \cong AC \]

This is true because:

• \( AC \) is a common side of both triangles \( \triangle ABC \) and \( \triangle CDA \).

Thus, the additional information needed is to state and justify that \( AC \) is congruent to itself (CPCTC—Corresponding Parts of Congruent Triangles are Congruent).

With this information, we can conclude that:

• Since \( \angle BCA \cong \angle DAC \),
• \( \angle BAC \cong \angle DCA \), and
• \( AC \cong AC \),

it follows by the ASA Theorem that:

\[ \triangle ABC \cong \triangle CDA \]