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 that triangles \(ABC\) and \(CDA\) are congruent by the ASA (Angle-Side-Angle) congruence theorem, we need to establish that:

1. Two angles in triangle \(ABC\) are equal to two angles in triangle \(CDA\) (which we have with \(\angle BCA = \angle DAC\) and \(\angle BAC = \angle DCA\)).
2. The side between these two angles is congruent.

In this context, the additional information we need is to justify that the side \(AC\) is congruent to itself, as it is the side that both triangles share. Therefore, you can state:

• Statement: Segment \(AC\) is equal to segment \(AC\) (by the Reflexive Property of Equality).

• Justification: Since \(AC\) is the same line segment present in both triangles, it is inherently congruent to itself.

With this information, you now have:

1. \(\angle BCA = \angle DAC\)
2. \(\angle BAC = \angle DCA\)
3. Segment \(AC\) is congruent to segment \(AC\).

This satisfies the conditions for the ASA congruence theorem, allowing you to conclude that triangles \(ABC\) and \(CDA\) are congruent.